TRANSIENT POPULATION DYNAMICS AND POPULATION MOMENTUM
IN VERTEBRATES
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
______________________________
David Nelson Koons
Certificate of Approval:
_________________________ _________________________
Gary R. Hepp James B. Grand, Chair
Professor Associate Professor
Forestry and Wildlife Sciences Forestry and Wildlife Sciences
_________________________ _________________________
Robert F. Rockwell F. Stephen Dobson
Professor Professor
American Museum of Natural Biological Sciences
History, New York, NY
_________________________ _________________________
Bertram Zinner Stephen L. McFarland
Associate Professor Dean
Mathematics and Statistics Graduate School
TRANSIENT POPULATION DYNAMICS AND POPULATION MOMENTUM
IN VERTEBRATES
David Nelson Koons
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 13, 2005
DISSERTATION ABSTRACT
TRANSIENT POPULATION DYNAMICS
AND POPULATION MOMENTUM
IN VERTEBRATES
David Nelson Koons
Doctor of Philosophy, May 13, 2005
(M.S., Montana State University, 2001)
(B.S., Montana State University, 1998)
163 Typed Pages
Directed by James B. Grand
Maintenance of sustainable wildlife populations is one of the primary purposes of
wildlife management. Thus, it is important to monitor and manage population growth
and size over time. Population structure (i.e., age, stage, or size distribution) can affect
both population size and growth over time; however, the effects of population structure
on vertebrate population dynamics are poorly studied. Here, I examine and compare the
effects of population structure on short (i.e., transient) and longterm (i.e., asymptotic)
population dynamics across selected vertebrate taxa.
A general formula for relating sensitivity of transient population growth rate to
changes in life history parameters was developed. Using this tool and others, I found that
variation in transient growth rates and their sensitivities to changes in life history
iii
parameters were largely dependent upon a population?s initial net reproductive value.
Furthermore, transient population dynamics of longlived, slow reproducing species were
more variable and different than asymptotic dynamics when compared to shortlived, fast
reproducing species.
Management actions have strong potential to change population structure, which
can produce strong residual effects on population size, commonly known as population
momentum. I examined management actions targeted at halting the growth or decline of
hypothetical populations and measured the effects of instantaneous changes in fertility or
survival on shortterm transient dynamics and population momentum. Population
momentum following changes in fertility actually reduced population size in growing
populations and increased population size in declining populations. Changes in survival
rarely reversed the direction of shortterm population growth, and resulting population
momentum was generally in the direction of historical population growth.
Because population momentum has not been well studied across vertebrates,
novel formulas and hypotheses relating the dynamics of population momentum to life
history parameters were developed. Using algebra, calculus and computer simulation, I
compared the dynamical relationship between life history and population momentum
across a variety of bird, mammal, turtle, lizard, snake, and bony fish life histories.
Population momentum was related to age at maturity within each taxon, and was largest
in late maturing bony fishes. Lastly, I provide examples that illustrate how studies of
population momentum can be used to better understand life histories, sourcesink
metapopulation dynamics, and management actions.
iv
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to Dr. James B. Grand for his
supervision and guidance. His support made this doctoral research possible and he has
been most helpful in helping me prepare for my future endeavors. I would also like to
thank members of my Advisory Committee, Drs. Gary R. Hepp, Robert F. Rockwell, F.
Stephen Dobson, and Bertram Zinner for making me think rigorously about science and
for their constructive criticism of my ideas and assistance with manuscripts. Discussions
with Drs. James B. Grand, Robert F. Rockwell, and James D. Nichols initially set me on
a path away from ?Asymptopia?. I would also like to thank Dr. Randall R. Holmes for
invigorating discussions about mathematics and for his assistance with developing
manuscripts, as well as Dr. Michael C. Wooten for serving as my outside reader.
I thank my lab mates Travis Folk and Dave Miller, and colleagues in Zoology and
Wildlife Science for interesting discussion about population ecology, evolution,
conservation, natural resource management in the 21
st
century, as well as other things. I
thank Joe and my other cycling teammates for keeping me outdoors and enjoying life.
Finally, I would like to thank Val for her love and support. Her friendship has
helped me focus, and with her, Auburn was a little more enjoyable.
I dedicate this work to my parents, Robert and Lorrie Koons for their continual
love and support. To my mom, whose bout with cancer has put life in perspective. Her
displays of strength and courage help one realize that work is work, life is about living it
and sharing it with others.
v
Style manual or journals used: Ecological Modelling, Journal of Wildlife Management,
Ecology, Theoretical Population Biology.
Computer software used: Microsoft Word XP, MathType 5.2, SigmaPlot 8.0.2, SAS 8.2,
Matlab 7.0.1, Maple 9.5.
vi
TABLE OF CONTENTS
LIST OF TABLES............................................................................................................. ix
LIST OF FIGURES .............................................................................................................x
I. GENERAL INTRODUCTION................................................................................1
II. TRANSIENT POPULATION DYNAMICS:
RELATIONS TO LIFE HISTORY AND
INITIAL POPULATION STATE .........................................................................11
Abstract......................................................................................................11
Introduction................................................................................................12
Methods......................................................................................................14
Results........................................................................................................19
Discussion..................................................................................................21
References..................................................................................................27
Appendix A................................................................................................36
Appendix B ................................................................................................39
III. POPULATION MOMENTUM: IMPLICATIONS
FOR WILDLIFE MANAGEMENT......................................................................52
Abstract......................................................................................................52
Introduction................................................................................................53
Methods......................................................................................................55
Results........................................................................................................59
Discussion..................................................................................................62
Research and management implications....................................................65
Literature cited...........................................................................................67
IV. POPULATION MOMENTUM ACROSS VERTEBRATE
LIFE HISTORY STRATEGIES............................................................................86
Abstract......................................................................................................86
Introduction................................................................................................87
Methods......................................................................................................88
Results........................................................................................................94
vii
Discussion..................................................................................................96
Literature cited.........................................................................................101
Appendix 1...............................................................................................106
V. GENERAL FORMULAS FOR THE SENSITIVITY
OF POPULATION MOMENTUM TO CHANGES
IN POPULATION VITAL RATES AND
INITIAL CONDITIONS .....................................................................................116
Abstract....................................................................................................116
Introduction..............................................................................................117
Derivation ................................................................................................118
Applications and examples ......................................................................127
Discussion................................................................................................135
References................................................................................................138
VI. GENERAL CONCLUSIONS..............................................................................156
viii
LIST OF TABLES
III. 1. Vital rate elasticities for initially growing
and declining populations across
three life histories.......................................................................................73
V. 1. The sensitivity of population momentum
to unit changes in vital rates across
different initial conditions........................................................................143
ix
LIST OF FIGURES
II. 1. The relationship between the Keyfitz distance and
initial net reproductive value among species.............................................42
2. The relationship between the initial net reproductive
value and early transient population growth rates
among species ............................................................................................44
3. The relationship between the initial net reproductive
value and transient sensitivities at time step 1 to 2
among species ............................................................................................46
4. The relationship between the initial net reproductive
value and transient sensitivities at time step 4 to 5
among species ............................................................................................48
5. Interspecific relationships of the transient population
growth rates and sensitivities.....................................................................50
III. 1. A graphical example of transient dynamics leading
to population momentum ...........................................................................74
2. Population momentum caused by changes in fertility
or adult survival in the standard experiments for
the ? = 3 life history...................................................................................76
3. Population momentum across 3 life histories for
the standard virtual experiments ................................................................78
4. The magnitude of population momentum as it relates
to the proportional change made to a vital rate..........................................80
5. Population momentum values across all population
experiments for the ? = 3 life history.........................................................82
6. Biased elasticity predictions relative to the proportional
change made to each vital rate...................................................................84
x
IV. 1. Plots of population momentum following change
in fertility across vertebrate life histories.................................................108
2. Plots of population momentum following change
in adult survival across vertebrate life histories.......................................110
3. Comparison of asymptotic stable age distributions
and reproductive values between increasing and
stationary populations for selected life histories......................................112
4. Comparison of asymptotic stable age distributions
and reproductive values between declining and
stationary populations for selected life histories......................................114
V. 1. Plots of population momentum for bird and
mammal life histories from the experiments
in Chapter 4..............................................................................................144
2. Population momentum sensitivities for bird and
mammal life histories following change
in fertility .................................................................................................146
3. Population momentum sensitivities for bird and
mammal life histories following change
in adult survival........................................................................................148
4. The relationship between perturbation size of
adult survival and population momentum
for life histories with late maturity...........................................................150
5. The relationship between perturbation size of
adult survival and asymptotic
population structure .................................................................................152
6. The relationship between perturbation size of
adult survival and reproductive value......................................................154
xi
I. GENERAL INTRODUCTION
In wildlife management, agency and stakeholder goals are often centered on the
population, its size, and changes in size over time. To meet these goals, biologists often
direct management at the population growth rate because it can be manipulated to
increase population size in the case of conservation (e.g., Fujiwara and Caswell 2001) or
decrease size in the case of control (e.g., Rockwell et al. 1997, Merrill et al. 2003).
Because population structure (i.e., the age, stage, size, or state distribution) is
rarely known, it is usually assumed that the population is in a stable population structure.
This assumption allows use of the stablestate metrics of population growth (e.g., ?
1
, r, ?
s
,
and a), which are known as longterm, or asymptotic dynamics. All else being equal,
theory suggests that convergence to asymptotic conditions will occur in most populations
(Lopez 1961, Cull and Vogt 1973, Cohen 1976, 1977a, b, 1979, Tuljapurkar 1982, 1990).
Yet, environmental catastrophes, natural disturbances, selective harvest regimes,
and animal release and relocation programs can disrupt the stability of population
structure. If given enough time between perturbations, an unstable population structure
will undergo damped fluctuations until the stable population structure is reached.
Meanwhile, the population dynamics are ?transient? because they change according to the
fluctuating population structure.
1
Empirical evidence has shown that stable population structures rarely exist in
nature (Bierzychudek 1999, CluttonBrock and Coulson 2002), probably because
environments are so variable. Thus, the assumption of asymptotic population dynamics
in the wild may be unwarranted in most cases, and more attention should be given to the
transient dynamics (Hastings and Higgins 1994, Fox and Gurevitch 2000, Hastings 2001,
2004).
Much is known about the duration and oscillatory behavior of transient dynamics
(Coale 1972, Keyfitz 1972, Trussell 1977, Tuljapurkar 1982); however, few have focused
on the demographic causes of transient change in population size or growth rate. Even in
the transient case, population size and growth rate describe the population?s status. Thus,
it would be useful to understand the demographic mechanisms controlling change in
transient dynamics.
Sensitivity analysis can be used to determine the functional relationship between
population size or growth rate and the constituent vital rates (e.g., fecundity, survival,
growth, maturation, recruitment, movement), and to project changes in population growth
rate and size as vital rates change. Such analyses usually assume a stable population
structure and focus on longterm population dynamics. However, new tools allow one to
examine the sensitivity of transient population size and structure (Fox and Gurevitch
2000) or growth rate (Yearsley 2004) to changes in initial population structure or vital
rates. These new tools are important because transient sensitivities may be very different
from asymptotic sensitivities, and could be very useful in developing management plans
or examining natural selection in highly perturbed environments.
2
In Chapter 2, my aim was to elucidate the biological correlates of intraspecific
variation in transient dynamics across all possible population structures. My secondary
objective was to explain variation in transient dynamics across life histories. To
accomplish these objectives, I calculated the transient population growth rate, and its
sensitivity to changes in vital rates, across the complete range of possible population
structures for 6 bird and mammal species. Longlived birds and mammals tend to have
longer generation lengths and larger disparity in reproductive value across age classes.
Thus, I hypothesized that these properties would cause the transient dynamics in long
lived species to be more variable and different than asymptotic dynamics when compared
to shortlived, fast reproducing species.
Management actions also have strong potential to change vital rates by large
enough amounts to disrupt population structure. Before population structure begins its
approach to a stablestate following a perturbation, the population size can change rapidly
and influence the ultimate abundance (Neubert et al. 2002). For example, when
examining the potential benefit of turtleexcluder devices on shrimp trawls to loggerhead
sea turtle (Caretta caretta) populations, Crowder et al. (1994) found that sudden
improvements in survival rates caused instability in the age structure. The ensuing
transient dynamics resulted in a population size much different than that predicted by
asymptotic projections, which is the phenomenon that is better known as ?population
momentum? (Keyfitz 1971). The classical description of population momentum in
demography is as follows. If per capita fertility rates were high, creating an abundance of
young individuals with high likelihood of surviving to maturity, and then suddenly
dropped to the stationary level (i.e., ?
1
= 1, the level of lifetime individual replacement), a
3
population would keep growing because overabundance of young individuals would
ensure high net fertility rates long after the transition to stationary per capita fertility
(sensu Keyfitz 1971).
In demography, studies of population momentum have influenced international
policy (e.g., Bos et al. 1992, Bos et al. 1994, United Nations 2003); however, population
biologists have paid very little attention to population momentum. Even though
population momentum could occur in fish and wildlife populations when management or
large environmental perturbations (e.g., hurricanes, floods, fires, epidemics) cause any
vital rate to change by a large enough amount to alter the population structure, the
phenomenon has not been explicitly examined.
In Chapter 3, I used computer simulation to measure the transient dynamics and
population momentum resulting from the control or conservation of hypothetical wildlife
populations. To simulate management practices focused on population growth rate, I
halted population growth or decline by changing survival or fertility rates, which often
have very different elasticities (i.e., sensitivities measured on a proportional scale;
Heppell et al. 2000, S?ther and Bakke 2000). It is often suggested that management
should focus on the vital rate with the highest elasticity to get the best return in
population growth per unit of management effort (Caswell 2000). Yet, I show that short
term population growth and eventual size following some virtual management
experiments are so different from the expectations of asymptotic elasticity analysis, that
my findings could modify how we manage populations for control and conservation.
Understanding population momentum and its effect on population dynamics could
increase the efficacy of conservation, natural resource management, and pest control
4
practices. Further, it could help us better understand evolution and the invasion of new
strategies in unstable or nonlinear environments. Chapter 3 is the first study of
population momentum for nonhuman life cycles. Theory describing the behavior of
population momentum across species is nonexistent.
To partially fill this void, in Chapter 4 I used traditional methods for calculating
population momentum, and used computer simulation and theoretical experiments (i.e.,
thought experiments) to examine population momentum across vertebrate life history
strategies. As stated above, changes in vital rates will alter population structure, causing
transient dynamics that can ultimately result in population momentum (Caswell 2001).
Because transient dynamics of longlived, slow reproducing vertebrates with delayed
maturity are more reactive to changes in population structure than they are for short
lived, fast reproducing vertebrates with early maturity (Chapter 2), I predicted that the
magnitude of population momentum would increase with age at maturity.
In Chapter 4, I used thought experiments where every population began on the
same trajectory (i.e., growing or declining at the same rate). I then forced each
population to transition to the longterm stationary population growth rate. Because the
functional relationship between vital rates and population growth rate depends on life
history (Heppell et al. 2000, S?ther and Bakke 2000, Oli and Dobson 2003), I had to
change vital rates by different amounts for each life history in order to achieve stationary
growth. This invoked a new question: how does population momentum respond to equal
unitchanges across organisms? More generally, can we analytically measure the
?change? in population momentum caused by a ?unit change? in a single vital rate or the
initial population structure?
5
I know of no general formula that relates unit changes in vital rates or population
structure to changes in population momentum. Therefore, in Chapter 5 I derived
analytical formulas for the sensitivity of population momentum to unit changes in any
vital rate or the initial population structure. I then compared the sensitivity of population
momentum to an assortment of vital rates across a wide variety of life histories, and
examined population momentum in a simple sourcesink metapopulation. Furthermore, I
examined the effects of direct management of population structure on population
momentum. These examples illustrate an entirely new set of questions for demographers,
population biologists, and mathematicians to explore.
6
LITERATURE CITED
Bierzychudek, P. 1999. Looking backwards: assessing the projections of a transition
matrix model. Ecological Applications 9:12781287.
Bos, E., M.T. Vu, A. Levin, and R.A. Bulatao. 1992. World population projections
19921993. John Hopkins University Press, Baltimore, Maryland, USA.
Bos, E., M.T. Vu, E. Massiah, and R.A. Bulatao. 1994. World population projections
19941995: estimates and projections with related demographic statistics. John
Hopkins University Press, Baltimore, Maryland, USA.
Caswell, H. 2000. Prospective and retrospective perturbation analyses: their roles in
conservation biology. Ecology 81:619627.
Caswell, H. 2001. Matrix population models. Second Edition. Sinauer Associates,
Sunderland, Massachusetts, USA.
CluttonBrock, T.H., and T. Coulson. 2002. Comparative ungulate dynamics: the devil
is in the detail. Philosophical Transactions of the Royal Society of London, B
357:12851298.
Coale, A. J. 1972. The growth and structure of human populations: A Mathematical
Approach. Princeton University Press, Princeton, New Jersey, USA.
Cohen, J.E. 1976. Ergodicity of age structure in populations with Markovian vital rates,
I: countable states. Journal of the American Statistical Association 71:335339.
Cohen, J.E. 1977a. Ergodicity of age structure in populations with Markovian vital
rates, II: general states. Advances in Applied Probability 9:1837.
7
Cohen, J.E. 1977b. Ergodicity of age structure in populations with Markovian vital
rates, III: finitestate moments and growth rate; an illustration. Advances in
Applied Probability 9:462475.
Cohen, J.E. 1979. Ergodic theorems in demography. Bulletin of the American
Mathematical Society 1:275295.
Crowder, L.B., D.T. Crouse, S.S. Heppell, and T.H. Martin. 1994. Predicting the impact
of turtleexcluder devices on loggerhead sea turtle populations. Ecological
Applications 4:437445.
Cull, P., and A. Vogt. 1973. Mathematical analysis of the asymptotic behavior of the
Leslie population matrix model. Bulletin of Mathematical Biology 35:645661.
Fox, G.A., and J. Gurevitch. 2000. Population numbers count: tools for nearterm
demographic analysis. American Naturalist 156:242256.
Fujiwara, M., and H. Caswell. 2001. Demography of the endangered North Atlantic
right whale. Nature 414:537541.
Hastings, A., and K. Higgins. 1994. Persistence of transients in spatially structured
ecological models. Science 263:11331136.
Hastings, A. 2001. Transient dynamics and persistence of ecological systems. Ecology
Letters 4:215220.
Hastings, A. 2004. Transients: the key to longterm ecological understanding. Trends in
Ecology and Evolution 19:3945.
Heppell, S.S., H. Caswell, and L.B. Crowder. 2000. Life histories and elasticity patterns:
perturbation analysis for species with minimal demographic data. Ecology
81:654665.
8
Keyfitz, N. 1971. On the momentum of population growth. Demography 8:7180.
Keyfitz, N. 1972. Population waves. in T.N.E. Greville, editor, Population dynamics.
Academic Press, New York, New York, USA.
Lopez, A. 1961. Problems in stable population theory. Princeton University Press,
Princeton, New Jersey, USA.
Merrill, J.A., E.G. Cooch, and P.D. Curtis. 2003. Time to reduction: factors influencing
management efficacy in sterilizing overabundant whitetailed deer. Journal of
Wildlife Management 67:267279.
Mills, L.S., and M.S. Lindberg. 2002. Sensitivity analysis to evaluate the consequences
of conservation actions. Pages 338366 in S.R. Beissinger, and D.R.
McCullough, editors. Population viability analysis. The University of Chicago
Press, Chicago, Illinois, USA.
Neubert, M.G., H. Caswell, and J.D. Murray. 2002. Transient dynamics and pattern
formation: reactivity is necessary for Turing instabilities. Mathematical
Biosciences 175:111.
Oli, M.K., and F.S. Dobson. 2003. The relative importance of lifehistory variables to
population growth rate in mammals: Cole?s prediction revisited. American
Naturalist 161: 422440.
Rockwell, R. F., E. G. Cooch, and S. Brault. 1997. High goose populations: causes,
impacts and implications. Pages 73100 in B. J. Batt, editor. Arctic ecosystems
in peril: report of the arctic goose habitat working group. Arctic Goose joint
venture special publication. U.S. Fish and Wildlife Service, Washington D.C.,
U.S.A.
9
S?ther, B. ?E. , and O. Bakke. 2000. Avian life history variation and contribution of
demographic traits to the population growth rate. Ecology 81:642653.
Trussell, T. J. 1977. Determinants of roots of Lotka's equation. Mathematical
Biosciences 36:213227.
Tuljapurkar, S. D. 1982. Population dynamics in variable environments. II. Correlated
environments, sensitivity analysis and dynamics. Theoretical Population Biology
21:114140.
Tuljapurkar, S. D. 1990. Population dynamics in variable environments. Springer
Verlag, New York, New York, USA.
United Nations. 2003. World population prospects: the 2002 revision. Volume 1:
Comprehensive Tables. New York, New York, USA
Yearsley, J.M. 2004. Transient population dynamics and shortterm sensitivity analysis
of matrix population models. Ecological Modelling 177:245258.
10
II. TRANSIENT POPULATION DYNAMICS: RELATIONS TO LIFE HISTORY
AND INITIAL POPULATION STATE
Abstract
Most environments are variable with disturbances (e.g., hurricanes, fires) that can
lead to substantial changes in a population?s state (i.e., age, stage, or size distribution). In
these situations, the longterm (i.e., asymptotic) measure of population growth rate (? )
may inaccurately represent population growth in the shortterm. Thus, I calculated the
shortterm (i.e., transient) population growth rate and its sensitivity to changes in the life
cycle parameters for 3 bird and 3 mammal species with widely varying life histories.
Further, I performed these calculations for initial population states that spanned the entire
range of possibilities. Variation in a population?s initial net reproductive value largely
explained the variation in transient growth rates and their sensitivities to changes in life
cycle parameters (all AIC
1
c
? 6.67 units better than the null model, all R
2
? 0.55).
Additionally, the transient fertility and adult survival sensitivities tended to increase with
the initial net reproductive value of the population, whereas the subadult survival
sensitivity decreased. Transient population dynamics in longlived, slow reproducing
species were more variable and different than asymptotic dynamics when compared to
shortlived, fast reproducing species. Because can be a biased estimate of the actual
growth rate in the short term (e.g., 19% difference), conservation and wildlife
1
?
11
biologists should consider transient dynamics when developing management plans that
could affect a population?s state, or whenever population state could be unstable.
1. Introduction
Sensitivity analysis has become popular in ecology (e.g., van Groenendael et al.,
1988; Horvitz et al., 1997; Benton and Grant, 1999; Heppell et al., 2000a) and has been
used to manage and conserve wild populations (e.g., Rockwell et al., 1997; Cooch et al.,
2001; Fujiwara and Caswell, 2001). Such analyses usually assume that the population?s
state (i.e., age, stage, or size distributions) remains stable through time (i.e., the
asymptotic stable state), and that populations grow according to a constant, or stable
distribution of rate(s) (e.g., ?
1
, r, ?
s
, a). All else being equal, theory suggests that the
stable state assumption in population biology is a safe one (Lopez, 1961; Cull and Vogt,
1973; Cohen, 1976, 1977a, b, 1979; Tuljapurkar, 1984, 1990).
Environmental catastrophes, natural disturbances, selective harvest regimes, and
animal release and relocation programs can significantly alter a population, causing
unstable states. When given enough time between environmental perturbations,
population state will approach the stable state. Meanwhile, the population dynamics are
?transient? because they change according to the fluctuating population state until the
asymptotic stable state is achieved. Empirical evidence suggests that stable populations
rarely occur in nature (Bierzychudek, 1999; CluttonBrock and Coulson, 2002). Thus,
the assumption of asymptotic population dynamics in the wild may be unwarranted in
many cases (Hastings and Higgins, 1994; Fox and Gurevitch, 2000; Hastings, 2001,
2004).
12
Although much is known about the mathematics of transient dynamics (Coale,
1972; Keyfitz, 1972; Trussell, 1977; Tuljapurkar, 1982), few have focused on
demographic causes of transient change in population size or growth rate even though
they are the unifying parameters of evolutionary and population biology (Sibly et al.,
2002).
Sensitivity analysis can be used to determine the functional relationship between
population size or growth rate and the constituent vital rates (e.g., fecundity, survival,
growth, maturation, recruitment, movement), and to project changes in population growth
rate and size as vital rates change. New tools allow one to examine the sensitivity of
transient population size and structure (Fox and Gurevitch, 2000) or growth rate
(Yearsley, 2004) to changes in the initial population state or the vital rates. These new
tools are important because transient sensitivities may be very different from asymptotic
sensitivities. For example, in Coryphantha robbinsorum, the asymptotic population
growth rate was most sensitive to adult survival, but transient population growth rate and
size were most sensitive to growth of juvenile stages (Fox and Gurevitch, 2000; Yearsley,
2004). Thus, asymptotic sensitivities might not be informative for guiding shortterm
management decisions.
I examined the sensitivity of ?transient population growth rate? to changes in vital
rates for 6 bird and mammal species across all possible population states. My primary
objective was to elucidate the biological correlates of intraspecific variation in transient
dynamics across the possible population states. My secondary objective was to explain
variation in transient dynamics across life histories. Longlived birds and mammals tend
to have longer generation lengths and larger disparity in reproductive value across age
13
classes. I hypothesized that these properties would cause transient dynamics in long
lived species to be more variable and different than asymptotic dynamics when compared
to shortlived, fast reproducing species.
2. Methods
2.1. Data sets and matrix projection models
To examine the magnitude of difference between transient dynamics and
asymptotic dynamics across species, I chose 3 bird and 3 mammal species that have been
extensively studied and were known to have widely varying life histories. Along the
slowfast continuum of bird and mammal life histories, the ?slowest? species are those
that live a long life, mature late, and have low reproductive rates and long generation
lengths. The ?fastest? species are shortlived, mature early, have high reproductive rates
and short generation lengths (sensu Gaillard et al., 1989; Charnov, 1993).
I attained agespecific vital rate data from published longterm studies of blue tit
Parus caeruleus (Dhondt, 1989 a, b), manatee Trichechus manatus (Eberhardt and
O'Shea, 1995), red deer (Benton et al., 1995; Albon et al., 2000), snow goose Chen
caerulescens (Cooke et al., 1995; Cooch et al., 2001), snowshoe hare Lepus americanus
(Meslow and Keith, 1968), and wandering albatross Diomedea exulans (Weimerskirch,
1992; Weimerskirch et al., 1997) (App. A). Wandering albatross and manatee have slow
life histories, snow goose and red deer have mediumslow life histories, and blue tit and
snowshoe hare have fast life histories (Heppell et al., 2000b; S?ther and Bakke, 2000). I
used agespecific vital rates where the authors reported agespecific differences. Meslow
and Keith (1968) did not detect agespecific differences in vital rates during their long
term study. To examine the effects of agestructured vital rates on transient dynamics in
14
a fast species, I used Meslow and Keith?s (1968) original data, and implemented
hypothetical age structure by increasing fertility by 5% for age 2 and 10% for ages 3 and
older (App. A).
For each species, I assumed birthpulse reproduction and parameterized the vital
rate data into a life cycle projection matrix (A) assuming a prebreeding census
??+n1 ?+n
1
?
?+n1 ?+n
0F FF
P00 0 0
00 00
=
00P 0 0
0
000 P P
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
""
"
%"
"
##%##
"
(1)
where ? was the average age of first breeding and (?+n) was the oldest known age
group with unique vital rates. Because the dynamics of increasing and decreasing
populations can be very different, even within a single population (Mertz 1971), I
multiplied each matrix by a constant K (Appendix A) so that the dominant eigenvalue of
each matrix would equal 1.00.
2.2. Transient sensitivity analysis
For a population at any state, the population growth rate (GR) can be defined as,
t,k
k
t 1,k
k
GR =
?
?
n
n
(2)
where n
t, k
is the kth element of the population state vector at time t. Thus, if the
population is not in the asymptotic stable state, GR is the transient growth rate for a one
timestep interval (see App. B for longer time steps). I sought a solution to the sensitivity
15
of the transient GR to infinitely small changes in a vital rate ( ), which can be defined
as,
ij
TS
t, k t 1, k
kk
ij
ij
TS .
a
??
???
?
=
?
??
nn
?
(3)
The solution of TS is a twopart equation as follows,
ij
()
()
'
ij 0
'
0
t  2
' t 2 ' ' t  1 ' t  1 ' t  1
ij
ij 0 0 0 ij 0 0
0
2
't  1
0
for t 1
TS
+
for t 2, 3, . . .
ll
l
??
=
?
?
=
?
?
?
=
?
?
?
?
=
?
?
?
?
e? n
en
eA? A Ane neAA n eA ? neA n
eA n
and derivation of the solution can be found in appendix B, where I provide further
explanation of notation and the similarities and differences of my derivation to Yearsley?s
(2004).
2.3. Simulations and projection analysis
For each lifecycle matrix, I attained the stable population state vector and
systematically generated 1000 state vectors, each normalized to one (1200 for wandering
albatross because of the larger statevector dimension), by systematically drawing
numbers from a random uniform distribution. To examine transient dynamics under
stable population state and random initial conditions, I projected each lifecycle matrix 5
time steps (years) with each state vector using equation B1 (i.e., 1201 initialcondition
state vector projections for wandering albatross and 1001 for all other species). To
calculate the distance between each initialcondition state vector and the stable
population state vector, I used Keyfitz?s ? (1968),
16
()
1 k 1, k
k
1
x, w x w
2
?= ?
?
(4)
where and were the kth elements of the initial population state and stable state
vectors, respectively. The maximum value of Keyfitz?s ? is 1 and its minimum is 0 when
the population state vectors are identical. A population state vector that has
proportionately more breeding adults than the stablestate vector and one that has
proportionately more subadults could have the same ? value. To rectify this important
biological difference, I assigned (+) values to all ?s when vectors had proportionately
more breeding adults than the stablestate vector and (?) values to all ?s when vectors
had proportionately more subadults than the stablestate vector. Species that mature and
breed at age 1 (i.e., blue tit and snowshoe hare), and are counted with a prebreeding
census, will not have subadults in the population state vector. Thus, the signed Keyfitz
? can only vary between 0 and 1 for these species. I used the signed Keyfitz ? as a
predictor variable in statistical analyses, and linearly mapped ? values from the region
k
x
1, k
w
[ ]
1,1 to the region
[ ]
0, 2 in order to examine models using the exponential distribution,
which ranges from 0 to infinity. In addition, I calculated the initial net reproductive value
( ) of a population for each population state vector as,
1
c
(5)
11
c= ?
?
vn
0
where is the dominant left eigenvector of the A matrix normalized to 1 and represents
statespecific reproductive value (Goodman, 1968).
1
v
Furthermore, I estimated the transient growth rate at time steps 1to2 (GR2), 4
to5 (GR5), and 0to5 (5YRGR). 5YRGR is not the usual measure of growth rate, but
rather a measure of the % change in population size over 5 years. Additionally, I
17
estimated the sensitivity of transient GR to small changes in the vital rates at time steps 1
to2 and 4to5 according to equation B7. I then summed the transient sensitivity
estimates across relevant state classes to obtain transient fertility sensitivity (TFS),
transient subadultsurvival sensitivity (TSASS), and transient adult survival sensitivity
(TASS) for the aforementioned time steps (e.g., Oli and Zinner, 2001: 383). For
comparison, I also estimated the asymptotic growth rate (?
1
= 1 in all cases after adjusting
each life cycle with a constant K, Appendix A) and sensitivities (Caswell, 1978) for each
life history.
2.4. Data Analysis
I used data from the 1001 projections (1201 for wandering albatross) described
above for each species and considered a variety of null, linear, and nonlinear models to
examine the form of the relationship between the initial net reproductive value and the
response variables describing transient dynamics (GR, TFS, TSASS, and TASS at each
of the aforementioned time steps). Because heteroscedasticity was present in the
transient response variables across the initial population states, I used iteratively re
weighted least squares (IRLS) robust regression with the Huber weight function
(Rousseeuw and Leroy, 1987; Carroll and Ruppert, 1988; Neter et al., 1996: 418) to
estimate the intraspecific relationships. Analyses were conducted with Proc NLIN (SAS
Institute, Inc. 2000).
I used Akaike?s Information Criterion adjusted for sample size (AIC
c
) and Akaike
weights (Akaike, 1973; Burnham and Anderson, 1998: 51, 124) to evaluate the amount of
support in my data for each model in my candidate list (see above). I considered the best
18
approximating model to be that with the lowest AIC
c
value and highest Akaike weight
(W
i
) (Burnham and Anderson, 1998).
To examine the magnitudes of differences between transient and asymptotic
population dynamics for each of the 7 life histories, I first measured the difference
between each transient dynamic (e.g., GR2, TFS2, etc.) and the respective asymptotic
dynamic for all simulated projections. I then calculated the absolute value of the
difference, and finally estimated the mean and variance of the absolute values across all
simulated projections (1001, 1201 for wandering albatross) for each life history. I again
used IRLS robust regression with the Huber weight function to estimate the linear
relationship between the generation length of the life history (explanatory variable) and
each of the aforementioned ?difference? estimates (response variable) (Rousseeuw and
Leroy, 1987). I used Ftests to examine the support, or lack thereof, for the a priori
hypothesis that mean and variance of each ?difference? estimate would increase with
generation length of the life history (Neter et al., 1996).
3. Results
For each intraspecific analysis, I examined 9 models (i.e., the null, linear, and
nonlinear models) to identify how departures away from the stable population state affect
the initial net reproductive value of a population, and another 9 models to identify how
the initial net reproductive value affects transient dynamics. Transient dynamics
measured at the annual time scale did not exist in any of the simulations conducted for
the snowshoe hare life cycle without agestructured vital rates, meaning that asymptotic
dynamics always occurred. However, the results for all other life cycles and initial
population states had important ecological and conservation implications.
19
In all cases, I found that the signed Keyfitz ? between a population state vector
and the stable state vector (i.e., the departure distance) caused nonlinear changes in .
For species that exhibited some senescence in survival and/or fertility (i.e., blue tit and
red deer), I detected negative relationships between and the Keyfitz ? but found
positive relationships in all other species (Fig. 1). The resulting changes in caused by
departure in initial conditions away from the stable state largely determined the direction
and magnitude of transient population growth rate and its sensitivity to changes in the
vital rates. In all but two cases, the relationships between and the transient population
growth rates were positive. Across species, the slope of the versus GR relationships
tended to increase with the generation length of the species examined; however, this
generalization is not perfect (see Manatee results in Fig.2). Furthermore, withinspecies
slopes were greatest for the c and 5YRGR relationship (Fig.2), indicating that initial
conditions had an additive effect on population size over 5 years.
1
c
1
c
1
c
1
c
c
1
1
Similar to the GR results for each species, I detected strong linear or nonlinear
relationships between c and the variation in each of the transient sensitivities. I found a
positive relationship between and transient fertility and adult survival sensitivities for
the slower species, but found relatively flat relationships for the fastest species (Figs. 3
and 4). Because subadult survival contributed less to population growth when a smaller
segment of the population was comprised of subadults (e.g., population states with large
signed Keyfitz ?s), I found a negative relationship between c and transient subadult
survival sensitivities for 3 of 4 species. However, I detected interesting curvilinear
relationships between c and all vitalrate sensitivities for red deer, which may be related
1
1
1
c
1
20
to delayed maturity and senescence present in their life cycle (Figs. 3 and 4). Of further
significance, rankings of the transient sensitivities shifted across initial conditions in 5 of
6 species, and the rankintersection point shifted with the time step (Figs. 3 and 4).
Moreover, slope of the relationships between and the transient sensitivities was much
smaller for among fast species (Figs. 3 and 4).
1
c
Across species, mean and variance of the ?difference? measures between transient
and asymptotic dynamics (see Methods) increased and were highly correlated with
generation length (P < 0.10 except in 3 cases; Fig. 5). At the extremes (global max and
min), early transient growth rates (GR2 and GR5) were as much as 19%, 9%, 9%, 18%,
3%, and 1% different than asymptotic growth rate for wandering albatross, manatee,
snow goose, red deer, blue tit, and snowshoe hare (with agestructured vital rates)
respectively. More strikingly, effects of initial conditions were strongest over the
culmination of the five years of projection. 5YRGR differed from
( by as much as
59%, 21%, 31%, 55%, 11%, and 8% for the corresponding list of species just mentioned.
At time step 1to2, extreme transient sensitivities were as much as 248%, 335%, 155%,
249%, 5%, and 4% different from asymptotic estimates for the same list of species
mentioned above. While transient estimates are expected to approach asymptotic
estimates over time, differences at the extreme values were alarmingly large after 5 time
steps, 134%, 200%, 69%, 82%, 3%, and 2% for the corresponding list of species.
)
5
1
?
4. Discussion
Asymptotic demographic analysis has had a long history of use in population
ecology; however, my results indicated that the stable population state should not be
assumed unless empirically justified. Multistate capturemarkrecapture analysis can be
21
used to estimate a population?s state (Nichols et al. 1994, Williams et al. 2002) but
measuring the population state will be a challenging task in most studies. Still, my
approach can elucidate the potential importance of transient dynamics relative to
asymptotic dynamics for populations that could have unstable states at some point in
time. I discuss several biological underpinnings of transient dynamics so that more
biologists can comprehend and use transient dynamics in future population ecology and
management studies.
4.1. Intraspecific Patterns in Transient Dynamics
Damping ratios (Tuljapurkar, 1986; Law and Edley, 1990), asymptotic
convergence times (Taylor, 1979; DeAngelis et al., 1980; Hastings and Higgins, 1994),
Argand diagrams (Horst, 1977; Rago and Goodyear, 1987), examination of oscillations in
statevector components (Tuljapurkar, 1983, 1985) and examination of unstable
equilibrium (Cushing et al., 1998) have all been used to present transient dynamics.
However, these approaches fail to explicitly incorporate measures of population growth
rate, which is one of the unifying parameters of ecology and evolution (Sibly et al.,
2002).
In my study, I found that transient population growth rates and sensitivities were
highly dependent upon initial conditions. Initial net reproductive values usually
increased with the signed Keyfitz ? because large ?s represented population states with
proportionately more breeding adults than the stable state, and adult age classes
frequently had higher reproductive value than subadult age classes. However, initial net
reproductive value decreased with the signed Keyfitz ? when vital rates declined with
age (e.g., blue tit and red deer; Fig. 1).
22
Across all of the tested initial population states for each species, transient fertility
and adult survival sensitivities generally increased with , whereas the transient sub
adult survival sensitivity decreased (Figs. 3 and 4, but see red deer results). Each
occurred because initial population states with low c values generally consisted of more
sub adults or partially senescent old individuals than the stable state (see Fig. 1).
Individuals must survive to maturity before they can contribute young to the population,
which explains why early transient growth rates are generally most sensitive to subadult
survival for initial population states with low values. Initial population states with
high c values largely consisted of individuals that were at or near their peak
reproductive value (Fig.1), thus the continued survival and fertility output of primeaged
adults contributed most to transient population growth rates under these conditions.
1
c
1
1
c
1
Initial net reproductive value ( c ) is an omnibus measure that can be used to
predict transient dynamics (e.g., GR, TS
1
ij
) across initial state conditions, through time,
and to examine shifts in rankorder of vital rate contributions to transient GR (see figs. 3
and 4). Thus, the net reproductive value of a population can help explain many
properties of transient dynamics (Templeton, 1980) and it will be important to study the
ecological forces that affect .
1
c
4.2. Interspecific Patterns in Transient Dynamics
Tuljapurkar (1985) found that damping ratios and periods of oscillation in the
population state vector increased with generation length. In 15 of my 18 interspecific
comparisons, I found that either mean or variance of estimated ?differences? between
transient and asymptotic dynamics increased with generation length as well.
23
Furthermore, slopes of the abovementioned intraspecific relationships were steepest
among slow and mediumslow species (Figs. 2, 3, and 4).
Population dynamics in fast species were resilient to departures from stable state;
however, slow species experienced early transient dynamics that were variable across the
tested initialstate conditions, very different than asymptotic dynamics, and the dynamics
changed slowly over time as they slowly converged to the asymptotic stable state.
Moreover, unstable population states sometimes produced net decreases or net increases
in population size, a phenomenon known as population momentum (Keyfitz, 1971; Lande
and Orzack, 1988; Koons et al. unpublished data).
Compared to fast species, slow species mature late and live long lives, increasing
the chances for high variability in survival rates and reproductive investment across age
classes. These life history characteristics lead to high disparity in reproductive value
across age classes (see rv:rv, appendix A). Long generation length increases the time
required for transient dynamics to change (Tuljapurkar 1985), and disparate reproductive
values can lead to large changes in net reproductive following changes in population
state. Collectively, I believe these factors make the dynamics of slow species more
responsive to changes in population state.
4.3. Ecological Implications
Transient population analysis can reveal the possible effects of initial age or stage
structure (Fox and Gurevitch, 2000; this study), colonization (Caswell and Werner,
1978), life history (DeAngelis et al., 1980; this study), harvest, and especially pulse
perturbations to the environment (e.g., catastrophic mortality) on population dynamics.
Of immediate concern, my results indicate that can be a biased estimate of shortterm
1
?
24
population growth rate when population state is unstable (e.g., 335% difference between
transient and asymptotic estimates), especially among slow and mediumslow species.
Popular methods for managing and conserving populations include release of
captivereared animals into the wild, relocation of wild individuals (e.g., Starling, 1991;
Wolf et al., 1996; Ostermann et al., 2001), and statespecific harvest management
(Larkin, 1977; Holt and Talbot, 1978). All of these methods will perturb population state
and produce transient dynamics. Attempts to identify the best animal propagation or
harvest program with asymptotic projection models could lead to incorrect conclusions
(Merrill et al., 2003) and even mismanagement of populations. Long ago, MacArthur
(1960) showed that management programs that favor individuals with high reproductive
value will lead to large net reproductive values, which in turn cause high population
growth rates and abundance. Programs that favor individuals with low reproductive
value will produce opposite results. Moreover, favoring few individuals of high
reproductive value or many individuals of low reproductive value can result in similar net
reproductive values and transient dynamics. Using my approach, I suggest that resource
managers place a strong emphasis on estimation of population state and reproductive
value to examine the consequences of their management actions on shortterm population
dynamics, which are often more relevant to agency goals than longterm dynamics. Such
studies will help reduce uncertainty in decisionmaking and the likelihood of deleterious
management in the future.
Furthermore, anthropogenic catastrophes (e.g., oil and toxin spills, nuclear
disasters, mining, war, bioterrorism) are common in today?s world and many biologists
try to understand the impacts of anthropogenic catastrophes on population dynamics
25
(Brockwell et al., 1983; Brockwell, 1985; Lande, 1993; Mangel and Tier, 1993, 1994).
However, the impact of catastrophes on population dynamics cannot be elucidated with
asymptotic methods alone because catastrophes could severely perturb population state. I
have shown that this can drastically alter shortterm population dynamics, and Koons et
al. (unpublished data) have shown that it can significantly affect longterm population
size. When catastrophes have the potential to perturb population state, I suggest that risk
assessments, such as population viability analyses (Gilpin and Soul?, 1986) and
population recovery analyses, pay closer attention to transient dynamics and to effects of
population state on extinction or recovery times and probabilities.
4.4. Caveats
The degree to which asymptotic dynamics are a poor proxy for actual dynamics
depends on the population state, time, and life history. Like many transient analyses, my
results are unique to the time scale and models under examination. Because the number
of unique eigenvalues and eigenvectors can change with matrix dimension, the chosen
matrix dimension may influence transient dynamics. Yet, I found that expanding small
dimension matrices (e.g., 3by3, etc.) into a largedimension matrix (29by29) resulted
in transient growth rates that were identical to four decimal places. Matrix dimension did
affect net reproductive value and time required to converge to the asymptotic stable state
(see Caswell 2001:97), however differences were ? 0.005 and ? 0.05 years, respectively
(Koons et al. unpublished data). Thus, in this study I saw little reason to use matrices that
were larger than necessary to incorporate the published agespecific differences in vital
rates. Furthermore, I purposefully used a simplistic approach to elucidate some of the
biological factors causing transient population growth and related dynamics in an
26
otherwise deterministic environment. Longterm population size, growth rate,
sensitivities, and extinction probability can be approximated in stochastic environments
for any population state if the degree of environmental variability is small to moderate
(Tuljapurkar, 1982; Lande and Orzack, 1988; Lande et al., 2003). In the real world, vital
rates, as well as age, stage, or size structure may vary substantially across time and space
(e.g., CluttonBrock and Coulson, 2002). Questions concerning transient dynamics in
highly stochastic and periodically catastrophic environments have not been examined,
and offer an arena for much needed research in the future.
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35
Appendix A. Agespecific values of average ageatmaturity (denoted by ?), fertility (F), and survival rates (P) in the standardized
matrices for wandering albatross (Weimerskirch, 1992; Weimerskirch et al., 1997), manatee (Eberhardt and O'Shea, 1995), snow
goose (Cooke et al., 1995; Cooch et al., 2001), red deer (Benton et al., 1995, Albon et al., 2000), blue tit (Dhondt, 1989 a, b), and
snowshoe hare (Meslow and Keith, 1968). Also, original lifecycle matrices parameterized with vital rates attained from the literature
were multiplied by standardizing constants (K). Generation lengths (T, the time required for the population to increase by a factor of
R
0
[the net reproductive rate]), and disparity of reproductive value across age classes (rv:rv, the ratio of the highest agespecific
reproductive value to the lowest) are also displayed.
36
Wandering
Albatross
Manatee Snow Goose Red Deer Blue Tit Snowshoe
Hare
a
Snowshoe
Hare
b
K 0.97 0.95 0.91 0.94 0.88 0.91 0.92
T
rv:rv
P P P P P P
21.29
3.10
14.34
1.32
5.63
1.77
7.73
2.92
1.52
1.15
1.24
1.09
1.23
1.00
?
F
i
P
i
? F
i i
? F
i i
? F
i i
? F
i i
? F
i i
? F
i i
Age
c
1 0.81 0.91 0.74 0.85 ? 0.65 0.34 ? 0.81 0.18 ? 0.82 0.18
2 0.81 0.91 ? 0.17 0.74 0.91 0.69 0.34 0.85 0.18 0.82 0.18
37
2 9
20 9
3 0.81 0.91 0.33 0.74 ? 0.08 0.90 0.69 0.34 0.89 0.18 0.82 0.18
4 0.81 ? 0.12 0.91 0.40 0.74 0.19 0.90 0.59 0.34
5 0.94 0.12 0.91 0.46 0.74 0.21 0.90 0.59 0.34
6 0.94 0.21 0.90
7 0.94 0.21 0.90
8 0.94 0.21 0.86
9 0.94 0.21 0.86
10 ? 0.22 0.94 0.21 0.86
11 0.22 0.94 0.21 0.86
12 0.22 0.94 0.21 0.86
13 0.22 0.94 0.16 0.65
# 0.0.4
19 0.22 0.94
20 0.20 0.94
# 0.0.4
38
28 0.20 0.94
29 0.20 0.90
a ? Snowshoe hare life cycle with agestructured vital rates.
b ? Snowshoe hare life cycle without agestructured vital rates.
c ?Pseudo age class.
? ? All numerical values in the table are rounded to the nearest 10
2
decimal place. In the projection analysis I used values with
precision to the 10
6
decimal place.
?? ? The division of the values in the table by K yields the original vital rate values.
Appendix B. An analytical solution to the sensitivity (TS
ij
) of transient population
growth rate (GR) to changes in a vital rate, with comparisons to Yearsley?s (2004)
solution.
Traditionally, populations have been modeled with matrix equations of the form
(B1)
t
t
=nAn
0
i
where n and
t 0
n
are vectors describing the population state (i.e., age, stage, or size
distribution) at times t and 0, respectively, and A is an nbyn (deterministic) matrix
whose entries are denoted by a
ij
(I denote matrices and vectors in bold type with upper
case and lowercase notation respectively). Alternatively, equation B1 can be
decomposed and expressed with the eigenvalues and eigenvectors of the A matrix,
t
ti
i
c ?=
?
nw (B2)
where the w ?s are the right eigenvectors of A, the ?s are the associated eigenvalues,
and the c
i i
?
i
?s are dependent on initial conditions and the complex conjugates of the left
eigenvectors of A (Caswell, 2001). The dominant right eigenvector ( ) and the
dominant scalar (c
1
w
1
) describe the asymptotic stable state and net reproductive value of the
initial population (Templeton, 1980), respectively. The biological definitions of the sub
dominant eigenvectors and scalars are less clear (Caswell, 2001). Still, B2 can provide a
deeper understanding of the dynamics of , but some may find it difficult to work with.
t
n
For these reasons, Yearsley (2004) begins with equation B2 to project and
appends Fox and Gurevitch?s (2000) pioneering work by deriving a complex but elegant
solution to the sensitivity of ?transient population growth rate? to infinitely small changes
t
n
39
in a vital rate. His method allows one to calculate sensitivity of average transient growth
rate for specific age or stage classes, or for entire populations. If one does not need
detailed information about classspecific dynamics, I derive a simpler solution to the
sensitivity of transient population growth rate of the entire population to infinitely small
changes in a vital rate that begins with equation B1 rather than B2.
As described in the text, the population growth rate of a population in any state
(not assuming the stable state) can be defined according to equation 2. Because n
t
in
equation 2 is derived from the A matrix and the initial state vector, my definition of
growth rate is quantitatively equivalent to Yearsley?s (2004) calculation that uses the
weighted average of the eigenvalue spectrum belonging to the A matrix (the individual
statevector components at time t1 in my equation 2 operate as weights). For
comparative purposes, population growth rates for the kth element of the population
state vector can simply be calculated by deleting summation symbols in my equation 2.
Furthermore, to estimate population growth rate over any time step m, the denominator of
equation 2 can be changed to n
tm, k
.
Nevertheless, my goal was to find a simple analytical solution to sensitivity of the
transient GR for the entire population to infinitesimally small changes in a vital rate
( , equation 3). To do this I begin with equation B1 to project . I note that A
ij
TS
t
n
0
is
defined as the identity matrix. I denoted e as the vector whose components are all equal
to 1 and as the nbyn matrix whose entry in the ith row and jth column is 1 and 0
everywhere else. I make special note that for t = 1, 2, . . .
ij
?
t 1
t
ij
ij
0
a
ll
l
t1? ?
=
?
=
?
?
? A ? A (B3)
40
where l simply operates as a dummy variable. I then use this definition of the partial
derivative of the A matrix with respect to one of its entries to derive the sensitivity of
transient growth rate to changes in the Amatrix entries. Thus, for t = 1
t, k t 1, k
'
'
ij 0
k k
0
''
ij ij
00
= =
aa
??
???
?
??
??
??
nn
e? n
eAn
en en
(B4)
and for t = 2, 3, . . .
()
t, k t 1, k
't
kk
0
't  1
ij ij
0
't 't  1 't 't  1
000 0
ij ij
2
't  1
0
=
aa

aa
=
??
???
?
??
??
??
?? ? ?
?? ? ?
??
nn
eA n
eA n
eAneAneAn eAn
eA n
()
t 1 t 2
't1't  1't't2
ij 0 0 0 ij 0
00
2
't  1
0

=
ll ll?? ??
==
??
eA? A n eA n eA n eA? An
eA n
(B5)
where the last expression can also be written in the form,
()
()
t  2
' t 2 ' ' t  1 ' t  1 ' t  1
ij 0 0 0 ij 0 0
0
2
't  1
0
+
ll
l
??
=
?
?
eA? AAneneAAneA? neA n
eA n
(B6)
Therefore, given equations B5 and B6,
()
()
'
ij 0
'
0
t  2
' t 2 ' ' t  1 ' t  1 ' t  1
ij
ij 0 0 0 ij 0 0
0
2
't  1
0
for t 1
TS
+
for t 2, 3, . . .
ll
l
??
=
?
?
=
?
?
?
=
?
?
?
?
=
?
?
?
?
e? n
en
eA? A Ane neAA n eA ? neA n
eA n
(B7)
41
Figure 1. The top performing models of as a function of the signed Keyfitz ?
(mapped to the region [0, 2]) for wandering albatross (R
1
c
2
= 0.75; the initial dip in the
fitted line is an artifact of the model fit to sparse data at the lower tail of Keyfitz ? values;
the raw data do not indicate an initial dip), manatee (R
2
= 0.65), snow goose (R
2
= 0.76),
red deer (R
2
= 0.81), blue tit (R
2
= 0.44), and snowshoe hare (with agestructured vital
rates, R
2
= 0.87). Generation lengths for these species were 21.29, 14.34, 5.63, 7.73,
1.52, and 1.24 years, respectively (Appendix A). Data points were omitted to permit
viewing of the predicted relationship.
42
Wandering Albatross
Keyfitz Delta
0.00.51.01.52.0
c
1
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Snow Goose
Keyfitz Delta
0.0 0.5 1.0 1.5 2.0
c
1
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
Blue Tit
Keyfitz Delta
0.0 0.5 1.0 1.5 2.0
c
1
0.16
0.17
0.18
0.19
0.20
0.21
0.22
Manatee
Keyfitz Delta
0.0 0.5 1.0 1.5 2.0
c
1
0.190
0.195
0.200
0.205
0.210
0.215
0.220
0.225
Red Deer
Keyfitz Delta
0.0 0.5 1.0 1.5 2.0
c
1
0.04
0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Snowshoe Hare (with age structure)
Keyfitz Delta
0.0 0.5 1.0 1.5 2.0
c
1
0.31
0.32
0.33
0.34
0.35
0.36
0.37
43
Figure 2. The top performing models of the transient growth rates at time steps 1to2
(????GR2), 4to5 (GR5), and 0to5 (? 5YRGR) as functions of c for wandering
albatross (R
1
2
values listed in order for GR2, GR5, and 5YRGR; R
2
= 0.90, 0.86, 0.92)
manatee (R
2
= 0.87, 0.91, 1.00), snow goose (R
2
= 0.95, 0.95, 1.00), red deer (R
2
= 0.89,
0.87, 0.98), blue tit (R
2
= 0.74, 0.68, 1.00), and snowshoe hare (with agestructured vital
rates; R
2
= 0.87, 0.96, 1.00). Data points were omitted to permit viewing of predicted
relationships. For comparison, ? = 1 in all cases.
1
44
Wandering Albatross
c
1
0.01 0.02 0.03 0.04
GR
0.2
0.4
0.6
0.8
1.0
1.2
Snow Goose
c
1
0.12 0.15 0.18 0.21 0.24 0.27
GR
0.2
0.4
0.6
0.8
1.0
1.2
Blue Tit
c
1
0.18 0.19 0.20 0.21 0.22
GR
0.2
0.4
0.6
0.8
1.0
1.2
Manatee
c
1
0.16 0.18 0.20 0.22 0.24
GR
0.2
0.4
0.6
0.8
1.0
1.2
Red Deer
c
1
0.02 0.04 0.06 0.08 0.10 0.12
GR
0.2
0.4
0.6
0.8
1.0
1.2
Snowshoe Hare (with age structure)
c
1
0.31 0.32 0.33 0.34 0.35
GR
0.2
0.4
0.6
0.8
1.0
1.2
45
Figure 3. The top performing models of the transient sensitivities [(????) fertility, ()
subadult survival, and (? ) adult survival] at time step 1to2 as functions of for
wandering albatross (R
1
c
2
values listed in order for fertility, subadult survival, and adult
survival; R
2
= 0.75, 0.73, 0.74), manatee (R
2
= 0.97, 0.97, 0.97), snow goose (R
2
= 0.63,
0.86, 0.76), red deer (R
2
= 0.71, 0.67, 0.66), blue tit (R
2
= 0.93, , 0.38), and snowshoe
hare (with agestructured vital rates; R
2
= 1.00, , 0.86). Data points were omitted to
permit viewing of predicted relationships. For comparison, ? = asymptotic fertility
sensitivity, ? = asymptotic subadult survival sensitivity, ? = asymptotic adult survival
sensitivity.
46
Wandering Albatross
c
1
0.010 0.020 0.030 0.040
T
r
ansi
ent S
ensi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Snow Goose
c
1
0.12 0.15 0.18 0.21 0.24 0.27
T
r
ansient Sensi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Blue Tit
c
1
0.18 0.19 0.20 0.21 0.22
T
r
an
si
ent S
ensi
t
iv
i
t
y
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Manatee
c
1
0.16 0.18 0.20 0.22 0.24
T
r
ansient Se
nsi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Red Deer
c
1
0.02 0.04 0.06 0.08 0.10 0.12
T
r
ansient Se
nsi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Snowshoe Hare (with age structure)
c
1
0.31 0.32 0.33 0.34 0.35
T
r
ansient Se
nsi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
47
Figure 4. The top performing models of the transient sensitivities [(????) fertility, ()
subadult survival, and (? ) adult survival] at time step 4to5 as functions of for
wandering albatross (R
1
c
2
values listed in order for fertility, subadult survival, and adult
survival; R
2
= 0.88, 0.86, 0.86), manatee (R
2
= 1.00, 1.00, 1.00), snow goose (R
2
= 0.64,
0.87, 0.72), red deer (R
2
= 0.93, 0.64, 0.62), blue tit (R
2
= 0.97, , 0.55), and snowshoe
hare (with agestructured vital rates; R
2
= 1.00, , 0.96). Data points were omitted to
permit viewing of predicted relationships. For comparison, ? = asymptotic fertility
sensitivity, ? = asymptotic subadult survival sensitivity, ? = asymptotic adult survival
sensitivity.
48
Wandering Albatross
c
1
0.01 0.02 0.03 0.04
T
r
ansi
ent S
ensi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Snow Goose
c
1
0.12 0.15 0.18 0.21 0.24 0.27
T
r
ansient Se
nsi
t
iv
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Blue Tit
c
1
0.18 0.19 0.20 0.21 0.22
T
r
ansient S
ensiti
v
ity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Manatee
c
1
0.16 0.18 0.20 0.22 0.24
T
r
ansi
e
nt Sensiti
v
i
t
y
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Red Deer
c
1
0.02 0.04 0.06 0.08 0.10 0.12
T
r
ansi
e
nt Sensiti
v
i
t
y
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Snowshoe Hare (with age structure)
c
1
0.31 0.32 0.33 0.34 0.35
T
r
ansi
e
nt Sensiti
v
i
t
y
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
49
Figure 5. The interspecific relationship between generation length and estimated
differences between transient and asymptotic population dynamics (across all simulations
for each species). (a) Correlations between generation length and speciesspecific means
of, GR2  ?
1
(????) (R
2
= 0.80, n = 7, P = 0.006), GR5  ?
1
() (R
2
= 0.64, n = 7, P =
0.080), and 5YRGR  (? ) (R
()
5
1
?
2
= 0.71, n = 7, P = 0.018). (b) Correlations between
generation length and speciesspecific means of, TFS2  FS (????) (R
2
= 0.95, n = 7, P <
0.001), TSASS2  SASS () (R
2
= 0.18, n = 4, P = 0.57), and TASS2  ASS (? ) (R
2
=
0.71, n = 7, P = 0.045). (c) Correlations between generation length and speciesspecific
means of, TFS5  FS (????) (R
2
= 0.86, n = 7, P = 0.003), TSASS5  SASS () (R
2
=
0.92, n = 4, P = 0.043), and TASS5  ASS (? ) (R
2
= 0.92, n = 7, P = 0.001). (d)
Correlations between generation length and speciesspecific variances of, GR2  ?
1
(R
2
=
0.77, n = 7, P = 0.010), GR5  ?
1
(R
2
= 0.54, n = 7, P = 0.267), and 5YRGR  (R()
5
1
?
2
=
0.67, n = 7, P = 0.028). (e) Correlations between generation length and speciesspecific
variances of, TFS2  FS (R
2
= 0.73, n = 7, P = 0.025), TSASS2  SASS (R
2
= 0.46, n = 4,
P = 0.321), and TASS2  ASS (R
2
= 0.81, n = 7, P = 0.026). (f) Correlations between
generation length and speciesspecific variances of, TFS5  FS (R
2
= 0.76, n = 7, P =
0.011), TSASS5  SASS (R
2
= 0.99, n = 4, P = 0.007), and TASS5  ASS (R
2
= 0.90, n =
7, P = 0.001). Data points were omitted to permit viewing of predicted relationships.
50
51
a)
Generation Length
0 5 10 15 20 25
Me
an
D
i
ffere
n
c
e
G
r
o
w
th
R
a
te
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
b)
Generation Length
0 5 10 15 20 25
Me
an
D
i
ffe
ren
ce S
e
n
s
it
iv
i
t
y
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
c)
Generation Length
0 5 10 15 20 25
M
e
a
n
D
i
ffe
re
n
c
e
S
e
n
s
itiv
ity
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
d)
Generation Length
0 5 10 15 20 25
V
a
ri
an
ce
Di
ffere
n
c
e
G
r
o
w
th
R
a
te
0.000
0.005
0.010
0.015
0.020
e)
Generation Length
0 5 10 15 20 25
V
a
rian
ce D
i
ffe
ren
ce S
e
n
s
it
iv
i
t
y
0.00
0.01
0.02
0.03
0.04
0.05
f)
Generation Length
0 5 10 15 20 25
V
a
rian
c
e
D
i
ffe
re
n
c
e
S
e
n
s
itiv
ity
0.01
0.00
0.01
0.02
0.03
0.04
0.05
III. POPULATION MOMENTUM: IMPLICATIONS FOR WILDLIFE
MANAGEMENT
Abstract: Maintenance of sustainable wildlife populations is one of the primary purposes
of wildlife management. Thus, it is important to monitor and manage population growth
over time. Sensitivity analysis of longterm (i.e., asymptotic) population growth rate to
changes in vital rates is commonly used in management to identify the vital rates that
contribute most to population growth. Yet, dynamics associated with longterm
population growth rate only pertain to the special case when there is a stable age (or
stage) distribution of individuals in the population. Frequently, this assumption is
necessary because age structure is rarely estimated. However, management actions
themselves have strong potential to change the age distribution of a population. For
initially growing and declining populations, I instituted hypothetical management
targeted at halting the growth or decline of the population, and measured the effects of a
changing age structure on population dynamics. When I changed vital rates, age
structure became unstable and population momentum caused populations to grow in a
much different fashion than that predicted from the longterm population growth rate.
Interestingly, changes in fertility actually reversed the direction of shortterm population
growth. Population momentum can significantly affect population dynamics, and will be
important to consider in the use of population models for management.
52
INTRODUCTION
In wildlife management, agency and stakeholder goals are often centered on the
population, its size, and the change in size over time. To meet these goals, managers
often direct management at population growth rate because it can be manipulated to
increase population size in the case of conservation (e.g., Fujiwara and Caswell 2001) or
decrease size in the case of control (e.g., Rockwell et al. 1997, Merrill et al. 2003).
Studies of population growth rate sensitivity to changes in underlying vital rates
(e.g., fecundity, survival, growth, maturation, recruitment, movement) has become
popular in wildlife and conservation to prioritize management actions aimed at producing
change in population growth rate (e.g., Brault and Caswell 1993, Crowder et al. 1994,
Doak et al 1994, Heppell 1998). Sensitivity analysis can help answer important life
history and ecology questions as well (e.g., papers within Heppell et al. 2000a, Dobson
and Oli 2001, Oli and Dobson 2003).
Nonetheless, predictions made from analytical sensitivity analyses can be poor when
vital rates change simultaneously by different amounts, as they do in the real world (Citta
and Mills 1999, Mills et al. 1999, 2001). Furthermore, analytical sensitivity analyses
inherently assume existence of a stable age distribution, which means that calculations
depend on the longterm (i.e., asymptotic) population dynamics (de Kroon et al. 2000,
Caswell 2001, Ehrl?n et al. 2001, Mills and Lindberg 2002). Methods that are used to
examine contributions of stochastically fluctuating vital rates to population growth rate
typically focus on asymptotic dynamics as well (e.g., life table response experiments
[Horvitz et al. 1997], and life stage simulation analysis [Wisdom et al. 2000]). Simulation
based sensitivity analyses do not need to assume a stable age distribution but often do
53
because age distributions are rarely known. When changing a vital rate in a sensitivity
analysis, we assume that the change would not perturb the population out of a stable age
distribution (Mills and Lindberg 2002). It remains to be seen how robust this assumption is
in wildlife populations (Citta and Mills 1999, Mills and Lindberg 2002).
Management actions themselves have strong potential to change vital rates by
amounts large enough to disrupt the age structure. Before the age distribution begins its
approach to the stable age distribution after a perturbation, the population size can change
rapidly (Neubert et al. 2002). These dynamical responses to an unstable age structure are
known as transient dynamics (see Fig.1), and do occur in nature. Cessation of red deer
(Cervus elaphus) culling on the Isle of Rum, Scotland caused large changes in the red deer
age structure, with transient population dynamics persisting since culling stopped (Clutton
Brock and Coulson 2002, Coulson et al. 2004). In addition, when examining the potential
benefit of turtleexcluder devices on shrimp trawls to loggerhead sea turtle (Caretta caretta)
populations, Crowder et al. (1994) found that sudden improvements in survival rates caused
instability in the age structure. The ensuing transient dynamics resulted in a population size
much different than that predicted by asymptotic projections, which essentially is the
phenomenon that is better known as ?population momentum? (Keyfitz 1971) (see Fig.1).
Population momentum could occur in wildlife populations when management or large
environmental perturbations (e.g., hurricanes, floods, fires, epidemics) cause any vital rate to
change by an amount large enough to alter the age structure, but it has not been explicitly
examined.
If wildlife populations do experience population momentum, estimating the
effects of momentum on population size should be taken into account when developing
54
conservation and management plans. Here, I used computer simulation to examine
control and conservation of hypothetical wildlife populations. To simulate management
practices focused on population growth rate, I halted population growth or decline by
changing survival or fertility rates which often have very different elasticity values
(Heppell et al. 2000b, S?ther and Bakke 2000). It is often suggested that management of
the vital rate with the highest elasticity will produce the best returns in population growth
(Caswell 2000). Yet, I show that shortterm population growth and eventual size
following some virtual management experiments are so different from the expectations of
asymptotic analysis that my findings could influence the way we manage populations for
control and conservation.
METHODS
Data Simulation
To examine population momentum following management actions, I created 3 life
histories with stationary asymptotic growth rates (?
1
= 1) that mature at 1, 2, and 3 year(s)
of age. I designed life histories where survival rates to age i (P
i
) increase with age of
maturity (?) and fecundity (m, average number of daughters born to a reproductively
mature female) decreases which is the pattern observed across birds and mammals
(S?ther 1988, Gaillard et al. 1989, Promislow and Harvey 1990).
In my population model I calculated fertility (F) assuming a prebreeding census
(F = P
1
*m), with birth occurring at one time of the year. Fertility and the agespecific
survival rates for each life history were parameterized into a population projection matrix
(A).
55
2
3+ 3+
FF F
P00
0P P
? ?
? ?
=
? ?
? ?
? ?
A (1)
This model assumes geographic closure of a population and density independent growth.
Although density dependence is ultimately a necessity for all populations, my focus was
on rapidly growing or declining populations (see below) where density feedback on
survival or fertility is often negligible.
To set up virtual management experiments, I used projection matrices with
stationary growth:
123
???
0.555 0.555 0.555 0 0.286 0.286 0 0 0.139
0.4 0 0 ; 0.7 0 0 ; 0.8 0 0
0 0.5 0.5 0 0.8 0.8 0 0.9 0.9
?????
?????
===
?????
AAA
?
?
?
(fertility values in the top row of each matrix are rounded to the 3
rd
decimal place), and
created growing (?
1
= 1.2) populations by multiplying the matrix parameters by 1.2. For
the ? = 3 life history, multiplication of matrix parameters by 1.2 yielded survival rates >
1; thus, fertility had to be increased by a greater amount than survival to achieve the
desired population growth rate. Hence, I multiplied survival rates by a constant C, and
fertility by C
x
, then solved for C (solution = 1.104) and x (solution = 9) simultaneously.
Similarly, for each life history I created declining (?
1
= 0.8) populations by multiplying
the matrix parameters by 0.8. For the ? = 3 life history, I also created populations that
were initially growing (?
1
= 1.35 and 1.05) or declining (?
1
= 0.65 and 0.95) by greater
and lesser amounts in ?supplemental experiments? using a similar application of
constants.
56
Virtual Management Experiments
Here, I considered populations that were growing so rapidly that they could cause
environmental damage and populations that are declining at a rate that could yield
extinction (i.e., the growing and declining populations defined above). For the ? = 3 life
history, I also consider populations that are growing and declining at rates within the
common bounds of longterm environmental variation (i.e., ?
1
= 1.05 and 0.95).
Depending on the bird or mammal life history, the functional contribution of
survival or fertility to asymptotic population growth will vary (Heppell et al. 2000b,
S?ther and Bakke 2000). I focused my experiments on changing adult survival or
fertility. In the first set of experiments for each life history, I started with a growing
population as the initial condition and then decremented survival rate of adult age classes
by the amount necessary to attain stationary asymptotic growth (i.e., the amount required
to change ?
1
from 1.2 to 1). I performed a similar experiment by decrementing fertility.
In the second set of experiments for each life history, I started with a declining population
as the initial condition and then augmented survival rate of adult age classes by the
amount necessary to achieve stationary asymptotic growth, and then performed similar
experiments by augmenting fertility.
To elucidate the effects of managing the asymptotic population growth rate, I
assumed that populations initially had a stable age distribution. I then projected the
actual dynamics caused by an unstable age structure following perturbations to a vital
rate. Population momentum (M) was calculated according to Keyfitz (1971):
0
M
lim
t
t??
=
n
n
(2)
57
where n is the total population size. This is simply the ratio of the ultimate population
size following a perturbation to population size immediately before the perturbation. To
attain exact analytical estimates of population momentum, equation 2 was adjusted to
birthpulse vector form:
( )
T
T
(new) (initial) (new)
11 1
(initial)
1
*
M =
*
ev w w
ew
(3)
where e is a vector of ones, is the dominant left eigenvector of the projection matrix A
(i.e., the nonscaled agespecific reproductive values), is the dominant right
eigenvector of the projection matrix (i.e., the nonscaled stable age distribution), ?initial?
refers to the projection matrix for the initial conditions (growing or declining population),
and ?new? refers to the projection matrix following changes in adult survival rates or
fertility (Caswell 2001:104). Population momentum is measured on a percentage scale
centered on 1, thus a momentum of 1.25 means that the population will grow by 25%
before eventually reaching the new stable age distribution. Further, a momentum of 1 has
zero magnitude. To evaluate the relative importance of momentum, I compared
momentum across management experiments, and across three life histories. Because it
was necessary to change vital rates by different amounts to attain the fixed management
goal (?
1
v
1
w
1
= 1) in my experiments, I also examined the influence of proportional change in
a vital rate on population momentum.
To compare results from my experiments to predictions from an asymptotic
elasticity analysis, I first calculated elasticities (Elasticity = rate vitalloglog
1
?? ? ; de
Kroon et al. 1986) for the initially growing and declining populations. Elasticities may
provide a better means for comparing the functional contributions of different vital rates
58
to ?
1
than sensitivities (Sensitivity = rate vital
1
??? ; Caswell 1978) because elasticities
are measured on a relative scale (de Kroon et al. 1986, Benton and Grant 1999, but see
Link and Doherty 2002). I then used elasticities to predict the impact of each
management experiment on asymptotic population growth rate for the time step
immediately following the vital rate perturbation (tt) (Heppell 1998, Caswell
2001).
1?+
rate growth
onal change
growth rat
Predicted proportional change in ?
1
? Proportional change in vital rate ? Elasticity (4)
Next, I measured the observed proportional change in the actual population growth rate
(
1t+
N
t
N ) for the same time step in a way similar to Mills et al. (1999); however, they
measured the observed proportional change in ?
1
, rather than the observed proportional
change in
1tt+
NN.
1t
t
1t
t
t
1t
N
N
N
N
N
N
?
?
?
?
?
?
?
=
+
actual in change alproportion Observed (5)
I then calculated the bias of the elasticity prediction, which assumes a stable age
distribution following a vital rate perturbation.
1
 Bias of elasticity prediction = Predicted proporti in Observed proportional
change in actual e
?
(6)
Finally, I examined the influence of proportional change in a vital rate on bias of the
elasticity prediction.
RESULTS
Using Keyfitz?s measurement of population momentum, I estimated the actual
effect of instantaneous vital rate changes on age structure, shortterm population growth,
59
and longterm population size. The initially increasing populations with ? = 1,2, and 3
life histories had 52%, 17%, and 10% more adults in the last age class, respectively,
relative to the stable age distribution following the decrement in adult survival. The
initially decreasing populations had 30%, 12%, and 5% fewer adults in the last age class
relative to the stable age distribution following augmentation of adult survival. As a
result, the direction of population momentum for each life history was consistent with the
historical pattern of growth (Fig. 2), and was small in magnitude (absolute magnitude ?
0.045). Moreover, the actual population growth rate rapidly stabilized to an asymptotic
stationary rate (e.g., see Fig. 2) for life histories that had high adult survival elasticities
(Table 1). I also detected a small negative relationship between age at maturity and
magnitude of population momentum for the increasing population experiment, and a
small positive relationship for the decreasing population experiment (Fig. 3).
Effects of changing fertility contrasted with the findings for changes in adult
survival (Figs. 2 and 3). Surprisingly, initially increasing populations with ? = 1,2, and 3
life histories all had 32% fewer adults in the last age class relative to the stable age
distribution following the decrement in fertility. The initially decreasing populations all
had 60% more adults in the last age class relative to the stable age distribution following
augmentation of fertility. As a result, the initially growing populations decreased and the
initially declining populations increased (Fig. 2). Thus, changes in fertility, which had
the lowest asymptotic elasticity for the ? = 2 and 3 life histories (Table 1), caused
population size to change in a direction that would be considered more desirable for
control and conservation purposes. Furthermore, in experiments with decreasing
populations, the magnitude of momentum resulting from changes in fertility increased
60
with age at maturity (Fig. 3). Population size responded more to changes in fertility than
to changes in adult survival rates. As a result, magnitudes of momentum in the
experiments were much larger (as much as 0.17) than those produced by changes in adult
survival (Fig. 3), except in the increasing population experiment for the ? = 1 life history.
Because fertility often exhibited a small elasticity value, it had to be changed by
large amounts to achieve stationary population growth. Large changes in a vital rate
cause large perturbations to the age structure (see above), which then causes the
magnitude of population momentum to increase. This explains why the magnitude of
momentum was larger following changes in fertility versus changes in adult survival
(Fig. 4).
When I considered the supplemental population experiments for the ? = 3 life
history, magnitude of population momentum increased as the difference between the
initial population growth rate and the management objective (?
1
= 1.00) increased. This
relationship was most pronounced when fertility was decremented in the decreasing
population experiments (Fig. 5). Once again, these results were influenced by
proportional change in a vital rate that was required to achieve the management objective
(?
1
= 1.00) (Fig. 4).
Additionally, I detected substantial bias when using elasticity values to predict
proportional change in population growth rate following a given proportional change in a
vital rate. As the proportional change in adult survival rate increased, the absolute values
of the bias in the elasticity prediction also increased. Furthermore, the bias was high for
increases in adult survival and low for decreases in adult survival (Fig. 6). Thus, the
direction of bias was consistent with the direction of change in adult survival. When the
61
proportional change in fertility increased, the absolute value of the bias in the elasticity
prediction also increased, similar to the results for changes in adult survival. However,
the bias was low for decreases in fertility and high for increases in fertility (Fig. 6), which
occurred because the unstable age structure caused the direction of population growth to
change following perturbations to fertility.
DISCUSSION
Maintenance of sustainable wildlife populations is one of the primary purposes of
wildlife management. Thus, understanding the true nature of management actions on
population dynamics should be of fundamental importance to wildlife biologists.
Asymptotic sensitivity analysis has become a popular tool in wildlife biology because it
has the potential to elucidate efficient means of managing vital rates for optimal returns
in population growth rate and size (e.g., Doak et al. 1994, Rockwell et al. 1997, Wisdom
and Mills 1997, Hoekman et al. 2002). However, empirical evidence indicates that
asymptotic measures do not always capture the actual dynamics that occur in nature
(Bierzychudek 1999, CluttonBrock and Coulson 2002, Coulson et al. 2004, Franklin et
al. 2004).
Management actions themselves can affect a population?s age structure (Crowder
et al. 1994, Coulson et al. 2004). In my population experiments, changes in vital rates
did perturb populations away from the stable age distribution. As a result, I found that
asymptotic elasticity values did not accurately predict the proportional change in
population growth rate following a given proportional change in a vital rate (Fig. 6).
Thus, I conclude that predictions made from asymptotic elasticities are not robust to the
inherent assumption of a stable age distribution. Analytical solutions for examining the
62
sensitivity of the transient population growth rate (Yearsley 2004, Koons et al.
unpublished data) and size (Fox and Gurevitch 2000) now exist for cases when age
structure is unstable, but known. Age structure is rarely known, but if it is, my results
support use of these methods to address ecological questions and to guide management
practices.
For stable agestructured populations that were growing or declining, I used
matrix models to demonstrate that suddenly changing a vital rate can result in population
momentum. Similarly, management actions that attempt to change the direction of long
term population growth (e.g., changing ?
1
from 1.2 to 0.97, or from 0.8 to 1.03) result in
population momentum values that are quite similar to those presented in my paper (<
0.016 units of difference; Koons unpublished data). Population momentum has lasting
effects on population size in ways that are not revealed by asymptotic analyses, and I
show that momentum varies with life history, depends on the specific vital rate that is
changed, and depends on the proportional change that is made to a vital rate.
Population momentum is a complicated process that proximately depends on how
changes in a vital rate affect the age structure (Eqn. 3). In the virtual management
experiments, the number of adults in the initial population relative to the stable age
distribution that was eventually attained depended on the vital rate that was changed (i.e.,
adult survival or fertility). Populations that initially had a surplus of older adults
experienced positive population growth and positive (> 1) population momentum, and
populations that initially had deficits of older adults experienced negative population
growth and negative (< 1) population momentum. Caswell (2001:106) also found that
the direction of population momentum depended on initial surplus or deficit of adults in a
63
stagestructured model. Thus, it seems clear that initial age (or stage) structure relative to
that following changes in a vital rate, will dictate the direction of shortterm population
growth and momentum.
While the direction of population momentum is determined by the specific vital
rate that is changed, the magnitude of population momentum must be influenced by the
magnitude of change in the vital rate. Large changes in a vital rate will produce large
changes in the age structure and thus the equilibrium population size.
Still, population momentum was related to life history (Fig. 3). Perturbations to
the age distribution of populations with short generation lengths are assuaged quickly
because of rapid population turnover, and the converse is true for animals with long
generation lengths (Tuljapurkar 1985). Still, oscillations in the age structure (see Fig. 1)
may take a long time to converge to the stable age distribution, but cause little change in
the net population size. Recently, Koons et al. (unpublished data) have found that
transient population growth rates amongst latematuring, longlived birds and mammals
are initially very reactive to direct changes in age structure, but less so for earlymaturing,
shortlived species. This initial reaction to a perturbation can have a lasting effect on the
longterm population size (Figs. 1 and 2; Neubert et al. 2002). Thus, I believe that the
?reactivity? of a population to perturbations is the primary factor driving patterns of
population momentum across life histories, and that the time required for a population to
return to a stable age distribution plays a secondary role. Nevertheless, more work will
be required to quantitatively establish the connection between life history and population
momentum.
64
My studies considered population momentum following onetime perturbations to
life histories with exponential growth, in otherwise constant environments. In addition,
my goal was to examine population dynamics following vital rate perturbations in a
fashion that is one step closer to reality than that assumed under asymptotic sensitivity
analysis. The virtual management experiments were not meant to represent reality.
Fertility in my models was the product of fecundity and survival to the first birthday.
Thus, it would be interesting to examine the response of population momentum to
changes in these lowerlevel life history parameters as well as the seasonal components
of survival (e.g., survival of young to fledging, winter survival, etc.). In some
populations, density dependence may cause vital rates to change more slowly over time
than the instantaneous changes considered here. Li and Tuljapurkar (1999) found that
momentum increases with the time it takes to attain the vital rate goal. Thus, momentum
could have strong effects on populations regulated by density dependence and future
management studies should build on my findings by examining population momentum in
these environments and others (e.g., stochastic and periodic environments).
RESEARCH AND MANAGEMENT IMPLICATIONS
Population momentum could push populations far past an environmental carrying
capacity or even to extinction, depending on the direction of momentum. By affecting
the longterm population trajectory, momentum could shorten or lengthen the time for a
population to go extinct or recover from a perturbation, relative to that projected with an
asymptotic analysis (e.g., Merrill et al. 2003). Thus, population momentum should be
considered in population viability analyses (sensu Gilpin and Soul? 1986).
65
These matters, as well as others, should be of concern to wildlife managers when
the management goals are centered on both population size and growth rate (e.g., North
American Waterfowl Management Plan). If population size is already too large or too
small, managing the asymptotic population growth rate may be inefficient. For example,
I might prefer a management action that can initially reduce or increase the population
size, and at the same time produce the desired population growth rate in the future. It is
conceivable that managers could actually use population momentum to their advantage in
order to achieve such a goal. For example, even if fertility has a low elasticity value, it
may be more feasible to change fertility than adult survival. My results indicate that
changing fertility produces shortterm population growth and momentum that is
consistently in a ?desirable? direction for population control or conservation, but changing
adult survival does not. In my prebreeding census models, fertility was the product of
fecundity and survival to the first birthday. Depending on the organism, managers may
find it easier to manipulate fecundity (e.g., releasing hatcheryreared young) or survival
to the first birthday (e.g., improving fawn survival or chick survival).
Animal harvest can also affect vital rates like survival (e.g., additive harvest), and
it can alter the structure of a population (e.g., slot limits in fisheries, trophy management
of ungulates). Thus, it seems highly probable that harvest management could cause
population momentum, and in turn affect population size and the sustainability of harvest
itself. Yet, population dynamics operating under the pressures of harvest are thought to
follow densitydependent processes. My results only pertain to densityindependent
population growth. Thus, more work will be required to elucidate the possible effects of
harvest on population momentum in densitydependent systems.
66
In order to more accurately examine the actual effects of vital rate management
on the population dynamics, it will be necessary for field biologists to collect data on
population age distributions (e.g., Coulson et al. 2004). Knowledge of age distributions
could alleviate the reliance of management actions based on asymptotic population
analysis. If age distributions can be estimated, one would be able to determine if wild
populations are close to their stable age distributions. If they are, then asymptotic
sensitivity analysis might be very useful to wildlife managers; however, if they are not,
the tools presented here and by others (Fox and Gurevitch 2000, Yearsley 2004, Koons et
al. unpublished data) will be more useful. Lastly, incorporation of age distribution
estimates into population models will allow me to more accurately examine past and
future population dynamics. For example, it is largely believed that fluctuations in
population growth rate and size in wild populations are caused by stochastic fluctuations
in the vital rates. However, observed fluctuations in population growth rate and size are
partially caused deterministically by unstable age structures (e.g., the transient dynamics
and population momentum). Understanding the actual influence of age structure on
population dynamics in the wild will improve our knowledge about population dynamics
as a whole (CluttonBrock and Coulson 2002, Coulson et al. 2004).
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of matrix population models. Ecological Modelling 177:245258.
72
Table 1. Vital rate elasticity values for populations that were initially growing (?
1
= 1.2)
and declining (?
1
=0.8) across the three life histories (? = 1, 2, 3).
Elasticity Values
a
Growing Populations Declining Populations
? Fertility Subadult
Survival
Adult
Survival
Fertility Subadult
Survival
Adult
Survival
1 0.53 0.47 0.53 0.47
2 0.16 0.17 0.67 0.16 0.17 0.67
3 0.13 0.25 0.62 0.08 0.17 0.75
3
b
0.17 0.34 0.49 0.08 0.17 0.75
3
c
0.08 0.17 0.75 0.08 0.17 0.75
a ? Elasticities are summed across the relevant age classes and rounded to two decimal
places.
b ? Additional results for populations of the ? = 3 life history that were initially growing
(?
1
= 1.35) or declining (?
1
= 0.65) at a greater rate than in the standard experiments.
Whenever a matrix is multiplied by a constant (C), the eigenvalues are simply scaled by
the constant and the eigenvector structure does not change. Thus, elasticity values do not
change.
c ? Additional results for populations of the ? = 3 life history that were initially growing
(?
1
= 1.05) or declining (?
1
= 0.95) at a lesser rate than in the standard experiments.
73
Figure 1. An example of a population with 3 age classes growing at the rate of ?
1
= 1.2
until the second time step, when survival rate is changed to produce stationary asymptotic
growth (?
1
= 1). In (a) I show continual asymptotic growth following the changed
survival rate, which can only happen if the population somehow transitions automatically
to the new stable age distribution. In (b) I show the oscillating transient dynamics that
would actually occur (barring direct manipulation of the age distribution by managers)
after the survival rate is changed. In (c) the age classes are summed together to depict
the projected net population size for each scenario (a and b), and I show how the
population initially reacts (reactivity R) to the changed survival rate, how long it takes the
population to reach a stable age distribution (convergence ?), and the net increase in size
caused by population momentum (momentum M).
74
75
c) Net Asymptotic and Transient Population Dynamics
Time
02468101214
N
e
t
Population Siz
e
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Asymptotic Dynamics
Transient Dynamics
a) Asymptotic Population Dynamics
02468101214
Popu
lat
i
on
Siz
e
0.0
0.2
0.4
0.6
0.8
1.0
age class 1
age class 2
age class 3
b) Transient Population Dynamics
02468101214
Population Siz
e
0.0
0.2
0.4
0.6
0.8
1.0
1.2
age class 1
age class 2
age class 3
R
?
M
? = convergence to stable age distribution
M = net growth caused by momentum
R = reactivity to change in survival rate
Figure 2. Population projections of the ? = 3 life history for the standard increasing (a; ?
1
= 1.2) and decreasing (b; ?
1
= 0.8) population experiments. Across experiments, either
adult survival rate or fertility was changed at the second time step to achieve stationary
asymptotic growth (?
1
= 1.0). Results were similar for the other life histories except the
transient dynamics and resulting population momentum values were generally of lesser
magnitude.
76
77
b) Decreasing Experiments
Time
02468
P
o
pulation S
i
ze
0.6
0.7
0.8
0.9
1.0
Trajectory before changing a vital rate
Expected trajectory with asymptotic growth
Trajectory after augmenting adult survival
Trajectory after augmenting fertility
a) Increasing Experiments
Time
02468
P
opulat
ion S
i
ze
1.0
1.1
1.2
1.3
1.4
1.5
Trajectory before changing a vital rate
Expected trajectory with asymptotic growth
Trajectory after decrementing adult survival
Trajectory after decrementing fertility
10
10
Figure 3. Population momentum values across all life histories (? = 1, 2, 3) for the
standard increasing (a; ?
1
= 1.2) and decreasing (b; ?
1
= 0.8) population experiments
where adult survival or fertility was manipulated. The scaling of the yaxis differs
between (a) and (b).
78
b) Decreasing Experiments
?
123
P
opulat
ion M
o
ment
um
0.8
0.9
1.0
1.1
1.2
a) Increasing Experiments
?
123
P
opulat
ion
M
o
ment
um
0.96
0.98
1.00
1.02
1.04
Management of adult survival
Management of fertility
Reference line for momentum = 1
79
Figure 4. The magnitude of population momentum as it relates to the proportional
change made to a vital rate across all the population experiments and life histories.
80
Proportional Change in Rate
202468101214
P
o
pulat
i
on M
o
ment
um
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Management of adult survival
Management of fertility
Reference line for momentum = 1
81
Figure 5. Population momentum values across all population experiments for the ? = 3
life history where adult survival or fertility was manipulated.
82
83
Initial ?
1
0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35
P
opulat
ion M
o
mentum
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Management of adult survival
Management of fertilty
Reference line for momentum = 1
Figure 6. The relationship between the bias produced by the predicted proportional
change in population growth rate (based on elasticity of each vital rate in the initial
condition matrices; Table 1) relative to the observed proportional change in population
growth rate (yaxis), and the proportional change made to each vital rate (xaxis) across
all the population experiments and life histories. The xaxis is measured on a
proportional scale; thus, a value of 4 indicates a fourfold increase. The yaxis measures
the difference between two proportional measures.
84
85
Proportional Change in Vital Rate
2 0 2 4 6 8 10 12 14
B
i
as of
Elast
i
cit
y
P
r
edict
ion
0.5
0.4
0.3
0.2
0.1
0.0
0.1
0.2
Management of adult survival
Management of fertility
Reference lines
IV. POPULATION MOMENTUM ACROSS VERTEBRATE
LIFE HISTORY STRATEGIES
Abstract. Population momentum is a measure of the relative change in ultimate
population size following change in a vital rate or population structure. Population
dynamics vary across vertebrates, but population momentum across this group of
organisms has not been examined. To help fill this void, I simulated iteroparous and
semelparous life histories that represent a wide variety of vertebrates. In a series of
theoretical experiments, I demonstrated that population momentum is related to age at
maturity in different ways across vertebrates. During gradual vital rate changes (e.g., a
change that takes 5 years to complete), I controlled for the effects of per capita vital rates
on population growth and found that population momentum was generally similar to that
following an instantaneous change. In many cases, population momentum significantly
affected shortterm population growth and longterm population size; thus, momentum
will be important to consider in natural resource management as well as studies of life
history evolution.
86
INTRODUCTION
A central aspect of biology is the population, its size and changes in size over
time. Population size and growth rate are commonly modeled with tools that inherently
assume stability of population structure (i.e., age, stage, or size structure) through time
(e.g., see papers within Heppell et al. 2000, Sibly et al. 2002). Yet, ecologists realize that
this assumption may rarely be met in nature (Bierzychudek 1999, CluttonBrock and
Coulson 2002, Nichols and Hines 2002, Hastings 2004). An unstable population
structure can have a strong residual effect on future population size, which is known as
population momentum. In demography, studies of population momentum have
influenced international policy (e.g., Bos et al. 1992, Bos et al. 1994, United Nations
2003); however, population ecologists have paid very little attention to population
momentum.
Some ecologists might wonder how population momentum occurs. As an
example, if per capita fertility rates were high, creating an abundance of young
individuals with high likelihood of surviving to maturity, and then suddenly dropped to
the stationary level (i.e., ?
1
= 1, the level of lifetime individual replacement), a population
would keep growing because overabundance of young individuals would ensure high net
fertility rates long after the transition to stationary per capita fertility (sensu Keyfitz
1971). In general, momentum could occur whenever any vital rate changes, or when
activities like commercial fishing alter population structure (Hall 1999). Understanding
population momentum and its effect on population dynamics could increase the efficacy
of conservation, natural resource management, and pest control practices. Yet, only one
study of population momentum has been conducted on nonhuman populations (Koons et
87
al. 2005), and theory describing the behavior of population momentum across species is
lacking.
To partially fill this void, I used conventional methods to calculate population
momentum, and used computer simulation and theoretical experiments (i.e., thought
experiments) to examine population momentum across vertebrate life history strategies.
In a structured population, changes in vital rates will alter population structure, causing
transient dynamics (i.e., shortterm dynamics of a population with unstable population
structure), and ultimately, population momentum (Caswell 2001). Because transient
dynamics of longlived, slow reproducing vertebrates with delayed maturity are more
reactive to changes in population structure than they are for shortlived, fast reproducing
vertebrates with early maturity (Koons et al. 2005b), I predicted that the magnitude of
population momentum would increase with age at maturity.
METHODS
Data Simulation
To find a theoretical relationship between population momentum and life history,
I explored 3 strategies where reproduction can occur several times within a lifetime (i.e.,
iteroparous), and 2 where reproduction occurs once and is followed by death (i.e.,
semelparous). Life histories were based on theory (Stearns 1992, Charnov 1993) and
generally representative of iteroparous 1) birds and mammals, 2) turtles, lizards, and
snakes, 3) bony fishes, and semelparous 4) largebodied bony fishes, 5) smallbodied
bony fishes. To generate life histories for 13, I simulated age at maturity (?), fecundity
(m; average number of daughters born to a mature female), and survival for ageclasses 2
and older (P
2+
) according to the equations and corresponding curves shown in Appendix
88
1. Then, I set longterm growth rate (i.e., ?
1
: the asymptotic growth rate) to 1 and
numerically solved for ageclass 1 survival (P
1
). To generate life histories for each
semelparous strategy, I simulated ? and m according to equations and corresponding
curves shown in Appendix 1, set asymptotic growth rate to 1, assumed P
2+
to be 36 times
greater than P
1
(Kareiva et al. 2000, Wilson 2003)
,
and numerically solved for P
1
.
Population momentum was examined at regular intervals across the range of simulated ?
values for each life history strategy.
I parameterized the life history variables into pseudo agestructured (A),
(Lefkovitch 1965) and agestructured (B) projection matrices (Leslie 1945, 1948) for
iteroparous and semelparous strategies, respectively. I assumed birthpulse reproduction
and used a prebirth census, where fertility equaled the product of fecundity and ageclass
1 survival (F = m * P
1
) (Caswell 2001), and ? represents the age at maturity.
1
2+
2+
2+ 2+
00 FF
P0 00
0P 0 0
00 0 0
00 P P
??+
? ?
? ?
? ?
? ?=
? ?
? ?
? ?
? ?
A
"
"
"
%
"
2+
2+
2+
00 F0
P0 0
0P 00
00 00
00 P0
?
0
? ?
? ?
? ?
? ?=
? ?
? ?
? ?
? ?
"
"
"
%
"
B
I used projection matrices with stationary asymptotic growth (?
1
= 1) as the
starting point in my theoretical experiments. Because quickly growing populations are
often caused by high fertility rates, I created growing (?
1
= 1.1) populations by
numerically changing fertility. Because quickly declining populations are often caused
by suppressed survival rates, I created declining (?
1
= 0.9) populations by changing
survival rates for adult age classes and the age class immediately preceding maturity. For
simplicity, I limited my study to these special cases of growing and declining
89
populations, but acknowledge that fertility or survival could cause populations to change
in either direction.
Theoretical experiments
Here, I consider populations that are growing so rapidly that they could cause
environmental damage and populations that are declining at a rate that could yield
extinction (i.e., the growing and declining populations defined above). My objective was
to examine population size following changes in the vital rates that would, all else being
equal, eventually halt the growth or decline. Hypothetically, changes in vital rates could
be caused by the environment or anthropogenic manipulation. In the first group of
experiments, I started with a growing population (?
1
= 1.1) as the initial condition and
then decremented fertility by the necessary amount to attain stationary growth (?
1
= 1). I
considered an instantaneous decrement in fertility and a gradual change of equal amounts
over 5 years. A 5year transition was used because our empirical understanding of
population dynamics often comes from ecological studies and experiments that are short
term, and because management plans and objectives are often short term as well.
In the second group of experiments, I started with a declining population (?
1
=
0.9) as the initial condition and then augmented survival rates for adult age classes and
the age class immediately preceding maturity by the amount necessary to attain stationary
growth. I considered both an instantaneous augmentation of survival and a gradual
change of equal amounts over 5 years.
I measured population momentum (M) following instantaneous changes in vital
rates according Caswell?s (2001:104) discretetime formula:
90
( )
T
T
(new) (initial) (new)
11 1
(initial)
1
*
M =
*
ev w w
ew
. (1)
Here, e is a vector of ones, is the dominant left eigenvector of the projection matrix A
or B that describes reproductive value (it is not normalized into a unit vector), is the
dominant right eigenvector of the projection matrix and describes the stable population
structure (here, it is not normalized into a unit vector), ?initial? refers to the projection
matrix for the initial conditions (growing or declining population), and ?new? refers to the
projection matrix following changes in adult survival rates or fertility. The left and right
eigenvectors of a projection matrix satisfy
1
v
1
w
,1
ii
=vw and ,0 for
ij
i=vw j? where
is the scalar product. Equation 1 describes the ultimate population size that is
actually attained following a vitalrate change, relative to the size of an otherwise
equivalent population that experiences the same vitalrate change but always grows
according to a stable population structure (i.e., the Stable Equivalent Population).
Because the term ?population momentum? is reserved by some demographers for the
special case where per capita fertility transitions to the stationary level, Tuljapurkar and
Lee (1997) called the ratio of the ultimate population size to that in a Stable Equivalent
Population, the Stable Equivalent Ratio. Yet, population momentum can be measured
following change in any vital rate (Li and Tuljapurkar 1999).
Because iteroparous organisms can successfully reproduce in successive time
steps, positive fertilities exist in ? 2 adjacent columns in the projection matrix. Thus,
projection matrices for iteroparous life histories are primitive, meaning that there exists
one dominant eigenvalue and the asymptotic dynamics approach a stable point.
91
However, semelparous organisms have positive fertility in only one age class; therefore,
projection matrices for semelparous life histories are imprimitive and have codominant
eigenvalues. Thus, unless the population begins in a stable population structure, the
asymptotic population size and (st)age distribution of semelparous organisms are cyclic
with a period equal to the number of eigenvalues that share the largest size (d) (Caswell
2001). Still, a running average of the (st)age distribution over d converges to and
grows at the rate ?
1
w
1
(Cull and Vogt 1973). For semelparous life histories, I measured
population momentum with the limit of these running average values.
To measure population momentum for gradual changes in fertility or survival, I
began with Keyfitz?s original formula (1971):
0
M
lim
t
key
t??
=
n
n
(2)
where
i
i
n=
?
n is the total population size. This is simply the ratio of the ultimate
population size following a transition to the stationary level, to that immediately before
the transition. Then, I calculated the numerator of equation 2 with a Markov chain of
gradually changing vital rates:
12 10ttt?? 0
=nAA AAn" , (3)
and denoted this measure of population momentum as M
2
(e.g., Schoen and Kim 1998, Li
and Tuljapurkar 1999, 2000, Goldstein 2002). Here, is the timespecific projection
matrix of vital rates, and
.
, which is the dominant right eigenvector of
normalized to 1. When changes in a vital rate occur gradually, two factors cause
population momentum. First, the population structure that acts on each new set of vital
rates is not stable, which can produce a residual effect on population size. Second,
t
A
00,nor
=nw
0
A
92
gradually changing vital rates will continue to cause population growth or decline
regardless of population structure (Tuljapurkar and Lee 1997, Schoen and Jonsson 2003).
This measurement of net population growth following gradual changes is not directly
comparable to that following an instantaneous change, in which only the first factor
causes momentum (Keyfitz 1971). To explicitly understand population momentum for
gradual vital rate changes, I also calculated a measure of population momentum that
controls for the effects of gradually changing percapita vital rates on the population size
M
3
:
3
,
M
lim
t
t t asymptotic??
=
n
n
(4)
Here, n was projected with equation 3 and was projected with the following
Markov chain:
t ,t asymptotic
n
, 1 1, . 2 2, . 1 1, . 0 0, .t asymptotic t t nor t t nor nor nor?? ??
=nAwAwAwAw" (5)
where is the dominant right eigenvector of normalized to 1 (i.e., the stable age
distribution). This is simply the ratio of the ultimate population size that is attained
following a gradual vital rate change to the size that would be expected if population
structure were stable throughout time with respect to AA (i.e., the Stable
Equivalent Ratio; Lee and Tuljapurkar 1997). Hence, equation 4 controls for changes in
population size directly caused by vital rates during the gradual change.
,t nor
w
.
1
t
A
01
, ,...,
t?
A
All measures of population momentum are centered on 1. Values of momentum
above 1 indicate that the population will grow beyond that expected under asymptotic
conditions, and values below 1 indicate that the population size will be less than that
expected under asymptotic conditions.
93
I characterized each life history with the age at maturity. Age at maturity is a life
history invariant that has a large influence on mean fitness (Cole 1954, Lewontin 1965),
is highly correlated with other life history variables (Stearns 1992, Charnov 1993), and
provides a consistent means for comparison across iteroparous and semelparous
vertebrates.
RESULTS
For the first time in population biology, I calculate and make interspecific
comparisons of population momentum for iteroparous (n = 30) and semelparous (n = 10)
life histories. For each experiment, all 3 measures of population momentum (M, M
2
, M
3
,
eqns. 15) exhibited the same pattern across life histories within a vertebrate life history
strategy. However, the relationship between population momentum and ageatmaturity
differed among the 5 life history strategies that I examined (Figs. 1 and 2).
Following instantaneous decrements in fertility to growing populations, the
historical population structure reversed the direction of population growth (M < 1, and
M
3
also < 1; Fig. 1, solid and open circles). Population momentum generally decreased
with age at maturity, and the distance between both M and M
3
and 1, generally increased.
However, the relationship was concave for bird and mammal life histories (Fig. 1 a),
indicating that the earliest and latest maturing life histories were more resistant to forces
of population momentum. As age at maturity increased, gradual transitions in fertility
had a greater effect on population momentum (M
3
) relative to instantaneous transitions
(M) for bird and mammal, as well as turtle, lizard, and snake life history strategies, but
not for other strategies (Fig. 1, comparison of solid and open circles). Despite the effects
of population structure (M
3
< 1), net increases in population size usually occurred (M
2
>
94
1; Fig. 1, triangles). Nevertheless, actual population structure restricted populations from
growing as large as they would have if population structure were stable throughout time,
especially for life histories with delayed maturation (intermediate age at maturity in birds
and mammals). In fact, the effect of population structure was so strong (indicated by M
3
)
for iteroparous bony fishes that it caused a net decline in population size (M
2
< 1) for life
histories that mature at ? 10 years of age (Fig. 1 c).
After augmenting the survival rate (for age classes described in the methods) in
declining populations, historical population structure had a small effect on population
momentum for all bird and mammal, as well as turtle, lizard, and snake life histories (Fig.
2 ab, solid and open circles). As a result, a net reduction in population size occurred (M
and M
2
< 1), which was largely dictated by the low per capita survival rates during the
gradual change (Fig. 2 ab, triangles). To the contrary, bony fish life histories were not
resistant to the effects of historical population structure following changes in survival.
Historical population structure usually reversed the direction of population growth
following instantaneous changes in survival (M > 1). In fact, M and M
3
increased with
age at maturity for all bony fish life history strategies (Fig. 3 ce, solid and open circles).
However, gradual changes in survival still resulted in net reductions in population size
(M
2
< 1) for all bony fish life histories (Fig. 3 ce, triangles). Nonetheless, population
size did not decline as much as it would have had population structure remained stable
throughout time, especially for bony fish life histories with delayed maturity.
Because the functional contributions of fertility and survival to ?
1
(measured with
elasticities) vary with life history, I generally had to perturb the vital rates by different
proportionate amounts across life histories in order to achieve stationary population
95
growth in my experiments (Figs. 1 and 2, right axes). The population structure of the
growing populations (?
1
= 1.1) had a surplus of young immature individuals, and a deficit
of mature adults, relative to the asymptotic population structure for stationary vital rates
(?
1
= 1). The decrease in fertility also shifted some of the reproductive value toward
younger age classes (Fig. 3). As a result, populations for each life history experienced
transient dynamics with greater net mortality and lesser net fertility than would have
occurred in a Stable Equivalent Population. This caused a reversal in the direction of
population growth following instantaneous change in fertility (M < 1, Fig. 1; M
3
also <
1).
To varying degrees across the 5 life history strategies, population structures of
declining populations (?
1
= 0.9) had a deficit of mature adults and individuals in the
youngest immature age classes, but a surplus of the middleaged immature individuals,
relative to the asymptotic population structure for stationary vital rates. The increase in
survival shifted bird and mammal, as well as turtle, lizard, and snake reproductive values
towards the oldest age classes but shifted bony fish reproductive values towards the older
subadults rather than adults (Fig. 4). This led to transient population dynamics that had
little effect on longterm population size in birds and mammals, as well as turtles, lizards,
and snakes, but often led to a reversal in the direction of population growth amongst bony
fish life histories (M and M
3
> 1; Fig. 2 ce).
DISCUSSION
Our understanding of population momentum is limited because it has not been
examined for most animals. Recently, Koons et al. (2005) examined three animal life
histories and found that population momentum varied according to the vital rate that was
96
changed, the magnitude of that change, and life history. I used theoretically derived life
histories to examine this latter finding in greater depth, and describe how population
momentum varies across vertebrate life histories for instantaneous and gradual changes in
vital rates.
Li and Tuljapurkar?s (1999) measure of M
2
following gradual changes in fertility
increased exponentially with the time over which the gradual transition occurred and was
generally much larger than Keyfitz?s momentum (M). However, M
2
is not comparable to
M because M
2
can be caused by historical population structure as well as gradually
changing per capita vital rates, whereas M is only caused by historical population
structure (Bongaarts and Bulatao 1999, Schoen and Jonsson 2003). By controlling for
effects of nonstationary vital rates during a gradual transition (M
3
), I generally found
little difference in population momentum between instantaneous and 5year changes.
The only differences I detected were for longerlived bird and mammal, as well as turtle,
lizard, and snake life histories (Figs. 1 and 2). Furthermore, patterns of net population
growth following 5year vital rate changes were caused by historical population structure
(M
3
), rather than nonstationary vital rates. Thus, I focus my discussion on the causes
and consequences of life history variation in population momentum caused by the
historical population structure (M and M
3
).
Following changes in fertility or survival, M and M
3
generally moved further
away from 1 as age at maturity increased, supporting my hypothesis. However, there
were notable exceptions among the bird and mammal, as well as turtle, lizard, and snake
life history strategies. For these two strategies, M and M
3
were negligible after
augmentation of survival in the declining populations, suggesting that these strategies
97
were resistant to population momentum (Fig. 2 ab). Further, the earliest and latest
maturing bird and mammal life histories, as well as the turtle, lizard, and snake life
histories with intermediate age at maturity, exhibited signs of resistance to population
momentum following decremented fertility in the increasing populations (Fig. 1 ab). To
the contrary, population momentum in bony fish life histories was very reactive to all
vitalrate changes, especially for life histories with delayed maturity (Figs. 1 and 2, ce).
Demographically, these patterns can be explained by population structure and
reproductive value, which directly affect population momentum (e.g., see equation 1). In
my experiments, vital rates of latematuring life histories generally had to be changed by
larger proportionate amounts than for early maturing life histories, especially amongst the
bony fishes (Figs. 1 and 2). Larger changes in a vital rate will cause larger changes in
asymptotic population structure, meaning that a historical population structure that acts
on newly changed vital rates will have either a surplus or deficit of breeding adults. This
produces transient population dynamics with greater or lesser net reproduction,
recruitment, or both, than would occur under asymptotic conditions. Thus, variation in
population momentum across life histories can often be explained by the amount that a
vital rate is changed (Fig. 1 be, Fig. 2 ce). However, this relationship is not universal.
Changes in reproductive value of multiple age classes could potentially offset the
abovementioned effect of historical age structure (Fig. 1a), sometimes resulting in little
population momentum (Fig. 2 ab).
Furthermore, the pattern between population momentum and age at maturity
differed across life history strategies because stable population structures and the
allocation of reproductive value across age classes are inherently different among
98
vertebrate life history strategies. For example, the stable population structure in bony
fishes is heavily skewed towards young and the distribution of reproductive value is
heavily skewed toward adults. Among birds, mammals, turtles, lizards, and snakes these
distributions are less skewed (e.g., Figs. 3 and 4), which makes these organisms more
resistant to population momentum following changes in survival. Yet, to better
understand the demographic connection between population momentum and population
structure, reproductive value, and life history vital rates, a method for measuring the
sensitivity of population momentum to equal unit or proportional changes in vital rates or
initial population structure will be needed (presented in Chapter 5).
Population size is important in ecology, conservation, pest control, and harvest
management. Thus, ecologists and resource managers should consider population
momentum in population projections that are used to make management decisions or
when quantifying the ecological causes of historical population dynamics (Caswell
2001). My results indicated that population momentum will have the strongest effect on
population size of longlived vertebrates with delayed maturity, especially amongst
iteroparous bony fishes, which are often harvested for commercial or sport purposes.
Recently, Hauser, Cooch, and Lebreton found that population momentum could limit the
ability of managers to regulate populations with harvest techniques (unpublished data).
Thus, I highly recommend implementation of population momentum into harvest
management models to better assess the effects of harvest on population dynamics.
The direction of population momentum (> 1 or < 1) will also be important to
consider in pest control and conservation. Demographers have always found population
momentum to follow the direction of historical population growth (e.g., Fischer and
99
Heilig 1997). Yet, population momentum for a stagestructured population of Calathea
ovandensis was in the opposite direction of historical growth (Caswell 2001:106), and
Koons et al. (2005) found similar results for pseudo agestructured animal populations.
In Physics, momentum of an object that changes direction is known as an ?impulse?
(Buckwalter and Riban 1987). Similar to the aforementioned studies, impulses occurred
in my study, causing momentum in the opposite direction of historical population growth
following decrements in fertility for increasing populations and following augmentations
of survival in decreasing bony fish populations. On the other hand, population
momentum following augmentation of survival in decreasing bird, mammal, turtle, lizard,
and snake populations was small and sometimes in the same direction of historical
population growth. Depending on the life history and vital rate that is changed,
population momentum could be beneficial or detrimental for pest control and
conservation (Koons et al. 2005). My results should provide guidance to population
ecologists and resource managers that will have to carefully assess how population
momentum could affect population size of the animals they monitor.
To conclude, I note that single, temporally isolated vitalrate changes seldom
occur in nature. I purposefully ignored continual stochastic changes to clarify the effects
of population momentum on population size. My theoretical findings should be used as a
starting point to guide empirical studies, which will be needed to test my theoretical
predictions and to gain an understanding of the ubiquity of population momentum in
nature. Lastly, I recommend that studies of population viability (sensu Gilpin and Soul?
1986), and other subject matter of conservation and evolutionary concern, consider the
contribution of population momentum to projected dynamics.
100
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105
Appendix 1. Theoretical generating functions and corresponding curves for fecundity
(m), and survival rate for age classes ? 2 (P
2+
; iteroparous life histories only) across the
age at maturity (?) of vertebrate life histories. Figure a represents iteroparous birds and
mammals where me
0.3*
13*
?
= ,
0.25*
2+
P1e
??
=?
1.1m
, and empirical estimates of fecundity
(?) and adult survival (?) are presented as examples for (from left to right on all figures):
snowshoe hare (Lepus americanus), wandering albatross (Diomedea exulans), orca
(Orcinus orca), and the U.S. human population (Homo sapiens). Figure b represents
iteroparous turtles, lizards, and snakes where *?= ,
()
0.39*
2+
P0.91e
??
=? , and
empirical estimates of fecundity (?) and adult survival (?) are presented for Iguanidae
lizards, painted turtle (Chrysemys picta), snapping turtle (Chelydra serpentina), and
loggerhead sea turtle (Caretta caretta). Figure c represents iteroparous bony fishes
where
0.15*
7830*me7830
?
=? + ,
0.1*
2+
P1e
??
=? , and empirical estimates of
fecundity (?) and adult survival (?) are presented for threespine stickleback (Gasterosteus
aculeatus), hake (Merluccius merluccius), and charr (Salvelinus alpinus). Figure d
represents semelparous largebodied bony fishes where 650*m ?= , and empirical
estimates of fecundity (?) are presented for pink (Oncorhynchus gorbuscha), coho
(Oncorhynchus kisutch), chum (Oncorhynchus keta), Chinook (Oncorhynchus
tshawytscha), and sockeye (Oncorhynchus nerka) salmons. Figure e represents
semelparous smallbodied bony fishes where m
0.55*
9360*e9360
?
=? + , and empirical
estimates of fecundity (?) are presented for eulachon (Thaleichthys pacificu) and alewife
(Alosa pseudoharengus).
106
a) Iteroparous birds and mammals
1 4 7 10131619222528
Fec
u
n
d
ity
0
2
4
6
8
10
P
2+
0.0
0.2
0.4
0.6
0.8
1.0
= Fecundity
x column 20 vs y column 20
alpha vs adult surv.
alpha vs female m
b) Iteroparous turtles, lizards, and snakes
0 4 8 12162024283236
Fe
cu
nd
ity
0
10
20
30
40
P
2+
0.0
0.2
0.4
0.6
0.8
1.0
c) Iteroparous bony fishes
?
024681012141618
F
e
cu
ndit
y
0
20000
40000
60000
80000
100000
120000
P
2+
0.0
0.2
0.4
0.6
0.8
1.0
= Theoretical Fecundity
= Theoretical P
2+
Survival
= Empirical Adult Survival
= Empirical Fecundity
d) Semelparous largebodied bony fishes
0123456
0
1000
2000
3000
4000
5000
6000
e) Semelparous smallbodied bony fishes
?
0123456
0
50000
100000
150000
200000
250000
300000
107
Figure 1. Plots of population momentum (left axis), evaluated at regular intervals of ?, for
the growing population experiments (?
1
= 1.1) where fertility was decremented to the
stationary level across iteroparous a) birds and mammals, b) turtles, lizards, and snakes,
c) bony fishes, and semelparous d) largebodied bony fishes, e) smallbodied bony fishes.
The right axis shows proportional change in fertility that was required to achieve the
stationary level.
108
a) Iteroparous birds and mammals
?
1 4 7 10 13 16 19 22 25 28
Popu
lation M
o
mentum
0.6
0.8
1.0
1.2
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
b) Iteroparous turtles, lizards, and snakes
?
0 4 8 12 16 20 24 28 32 36
Pop
u
lation
M
o
mentum
0.6
0.8
1.0
1.2
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
c) Iteroparous bony fishes
?
0 2 4 6 8 1012141618
P
o
pulatio
n Momen
t
um
0.6
0.8
1.0
1.2
1.4
Pro
por
t
i
onal
C
h
a
n
g
e
in Fe
rtil
ity
1.2
1.0
0.8
0.6
0.4
0.2
0.0
d) Semelparous largebodied bony fishes
?
1234567
0.6
0.8
1.0
1.2
1.4
P
r
op
or
t
i
onal C
h
a
n
g
e
in Fe
rtili
t
y
1.2
1.0
0.8
0.6
0.4
0.2
0.0
M
alpha vs M3
alpha vs M5
Plot 3 Upper specification
alpha vs Prop. change in fertility
e) Semelparous smallbodied bony fishes
?
1234567
0.6
0.8
1.0
1.2
1.4
Pro
por
tional
C
h
ang
e in F
e
r
t
il
ity
1.2
1.0
0.8
0.6
0.4
0.2
0.0
M
M
2
M
3
Reference line for momentum = 1
Proportional Change in Fertility
109
Figure 2. Plots of population momentum (left axis), evaluated at regular intervals of ?,
for the declining population experiments (?
1
= 0.9) where survival rates for adult age
classes and the age class immediately preceding maturity were augmented to the
stationary level across iteroparous a) birds and mammals, b) turtles, lizards, and snakes,
c) bony fishes, and semelparous d) largebodied bony fishes, e) smallbodied bony fishes.
The right axis shows proportional change in survival that was required to achieve the
stationary level (a > 8fold increase in survival was required for the iteroparous bony fish
life history with an age at maturity of 1).
110
c) Iteroparous bony fishes
?
0 2 4 6 8 1012141618
Pop
u
l
a
tio
n
M
o
mentu
m
0.6
0.8
1.0
1.2
1.4
P
r
op
ort
i
o
n
a
l
Ch
an
g
e
i
n
S
u
rv
iv
al
0
2
4
6
8
d) Semelparous largebodied bony fishes
?
1234567
0.6
0.8
1.0
1.2
1.4
P
r
o
p
o
r
t
i
on
al
Ch
a
nge
in
S
u
rv
iv
a
l
0.0
0.2
0.4
0.6
0.8
1.0
b) Iteroparous turtles, lizards, and snakes
?
0 4 8 12162024283236
P
o
pu
la
t
i
o
n
Mo
me
nt
um
0.6
0.8
1.0
1.2
1.4
0.0
0.2
0.4
0.6
0.8
1.0
e) Semelparous smallbodied bony fishes
?
1234567
0.6
0.8
1.0
1.2
1.4
P
r
op
ort
i
o
n
a
l
Ch
an
g
e
i
n
S
u
rv
iv
a
l
0.0
0.2
0.4
0.6
0.8
1.0
M
M
2
M
3
Reference line for momentum = 1
Proportional Change in Survival
a) Iteroparous birds and mammals
?
1 4 7 10131619222528
P
o
pu
la
t
i
on
Mo
me
n
t
um
0.6
0.8
1.0
1.2
1.4
0.0
0.2
0.4
0.6
0.8
1.0
111
Figure 3. Example plots of the asymptotic stable age distributions (sad; left bars), and
reproductive values (rv; right bars) for the experimental growing and stationary
populations of selected life histories with an age at maturity of 16 for iteroparous
strategies and 6 for semelparous strategies.
112
c) Iteroparous bony fish
? = 16
% Total Population
80 60 40 20 0 20 40 60 80
Age Cl
ass
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17+
a) Iteroparous bird and mammal
? = 16
80 60 40 20 0 20 40 60 80
Age Cl
ass
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17+
b) Iteroparous turtle, lizard, and snake
? = 16
80 60 40 20 0 20 40 60 80
Age Cl
a
s
s
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17+
d) Semelparous largebodied bony fish
? = 6
80 60 40 20 0 20 40 60 80
Age Cl
ass
1
2
3
4
5
6
7
e) Semelparous smallbodied bony fish
? = 6
% Total Population
80 60 40 20 0 20 40 60 80
Age Cl
ass
0
1
2
3
4
5
6
7
sad ?
1
= 1.1
sad ?
1
= 1
rv ?
1
= 1.1
rv ?
1
= 1
113
Figure 4. Example plots of the asymptotic stable age distributions (sad; left bars), and
reproductive values (rv; right bars) for the experimental declining and stationary
populations of selected life histories with an age at maturity of 16 for iteroparous
strategies and 6 for semelparous strategies.
114
e) Semelparous smallbodied bony fish
? = 6
% Total Population
80 60 40 20 0 20 40 60 80
Age Cl
ass
0
1
2
3
4
5
6
7
d) Semelparous largebodied bony fish
? = 6
80 60 40 20 0 20 40 60 80
Age
Cl
ass
0
1
2
3
4
5
6
7
c) Iteroparous bony fish
? = 16
% Total Population
80 60 40 20 0 20 40 60 80
Age C
l
a
s
s
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17+
b) Iteroparous turtle, lizard, and snake
? = 16
80 60 40 20 0 20 40 60 80
Ag
e Cl
ass
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17+
a) Iteroparous bird and mammal
? = 16
80 60 40 20 0 20 40 60 80
Age C
l
ass
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17+
sad ?
1
= 0.9
sad ?
1
= 1
rv ?
1
= 0.9
rv ?
1
= 1
115
V. GENERAL FORMULAS FOR THE SENSITIVITY OF POPULATION
MOMENTUM TO CHANGES IN POPULATION VITAL RATES AND INITIAL
CONDITIONS
Abstract. Population structure can produce strong residual effects on population
size, commonly known as population momentum. Population momentum has a long
history of study in demography, but has only recently received attention in the ecological
literature. Prior to now, there has not been a general way to examine change in
population momentum resulting from unit changes in population vital rates (e.g., birth
rates, death rates, etc.) or population structure. I derive general formulas for the
sensitivity of population momentum to changes in population vital rates or population
structure. The sensitivity for changes in vital rates is similar to population growth rate
sensitivity that is widely used in demography, ecology, evolution, conservation, and
management. My formulas are readily computable, and I provide examples and
applications that illustrate their use in studies of life history, ecology, and applied
management.
116
1. Introduction
Population size is central to the fields of demography and population biology.
Demographers often study population size because it can affect economies, policy, social
dynamics, and even natural resource supplies (Bos et al. 1994, Fischer and Heilig 1997,
United Nations 2003). Biologists pay special attention to population size when trying to
keep small populations from going extinct, controlling pest populations, and in
management of populations that provide hunting, fishing, and viewing opportunities
(Caughley 1977). Additionally, the change in population size over time (i.e., population
growth rate) describes the average fitness and performance of the population (Fisher
1930, Sibly et al. 2002), and is used in many theoretical studies of life history. Thus,
population size is an important parameter for many reasons.
When population structure (i.e., the distribution of abundance across age, stage,
size, or sex classes) is stable, population size is solely determined by the vital rates (i.e.,
fecundity, survival, age at maturity, immigration, emigration). However, it is
fundamental to understand that when population structure is not stable, population
structure can actually influence the longterm trajectory of population size (Tuljapurkar
and Lee 1997). For example, if a population initially has an ?over abundance? of mature
adults it will experience transient dynamics and might reach a larger population size than
a population that initially has a stable population structure and grows according to the
same vital rates. Alternatively, a population may have a stable population structure, but
experience a systematic change in one or several vital rates, such that the new set of vital
rates act on the old population structure. In either case, it is the historical population
structure that affects the physical behavior of momentum in population size (Tuljapurkar
117
and Lee 1997), which is commonly known as ?population momentum? (sensu Keyfitz
1971a).
Because the study of population momentum can help us better understand
increases and decreases in population size, it has been studied extensively in demography
(e.g., Keyfitz 1971a, Frauenthal 1975, Mitra 1976, Wachter 1988, Fischer and Heilig
1997, Kim and Schoen 1997, Schoen and Kim 1998, Bongaarts and Bulatao 1999, Li and
Tuljapurkar 1999, 2000, Schoen and Jonsson 2003), and has recently been examined in
wildlife management (Koons et al. 2005; Hauser, Cooch, and Lebreton In Review) and
population ecology (Koons et al. 2005b, Chapter 4).
I know of no general formulae that relate unit changes in vital rates or population
structure to changes in population momentum. Here, I present analytical formulae for the
sensitivity of population momentum to unit changes in any vital rate or initial conditions.
I compute population momentum sensitivities across a wide variety of life histories, in a
simple sourcesink metapopulation, and for cases where applied management could
directly affect population structure.
2. Derivation
2.1. Population model
I use boldtype capital letters to denote matrices and boldtype lower case letters
to denote vectors. Italicized letters denote the entries of vectors and matrices. I use x to
denote the conjugate of x, x
T
to denote the transpose of the vector x and x* to denote the
complex conjugate transpose.
Most studies of population momentum are conducted with continuous time
models. However, the underlying mechanisms of population momentum are more easily
118
seen in discrete form (Schoen and Jonsson 2003). Thus, my derivation is based on a
linear, discrete, timeinvariant population model
t+1 t
=nAn
i
. (1)
Here, n is an ndimensional vector with describing the number of individuals in
the ith stage at time t. A is an n ? n matrix with (i, j)entry a
t
( )
i
nt
ij
equal to the transition rate
from the jth stage to the ith stage. Alternatively, the population vector at any time t can
be expressed as:
t
ti
i
c ?=
?
nw
i
, (2)
where the w ?s are right eigenvectors of A, the ?s are the associated eigenvalues, and
the c
i i
?
i
?s are dependent on the initial conditions and left eigenvectors (Caswell 2001). The
ith eigenvalues and corresponding right w and left eigenvectors of A satisfy
i
?
i i
v
ii
?=Aw w (3)
ii
?
? ?
=vA v. (4)
The indexing is chosen in such a way that is the eigenvalue with largest modulus. For
large t, this eigenvalue and its eigenvector dominate the expression for given in
equation 2, and so eventually, the population grows approximately geometrically at the
rate ? . The dominant right and left eigenvectors describe the asymptotic
population structure and reproductive values, respectively (Goodman 1968). Throughout,
I assume that the eigenvectors have been scaled so that
1
?
1
v
t
n
1 1
w
,1
ii
=vw and
,0 for
ij
i= j?vw . I further assume that the eigenvalues of A are distinct.
119
2.2. Measurement of population momentum
Keyfitz (1971a) originally calculated population momentum for a stable
population that was originally growing at some rate g that underwent an instantaneous
transition to stationary growth (i.e., g = 1; the level of lifetime individual replacement).
He defined population momentum M as the ultimate size of the population relative to that
immediately before the instantaneous change in the vital rate(s):
0
M
lim
t
t??
=
n
n
, (5)
where
t
()
i
i
nt=
?
n is the total population size. However, physical momentum in
population size is not restricted to the special case described by Keyfitz. I seek a model
that is general enough to calculate the phenomenon of momentum in populations for 1)
Keyfitz?s original thought experiment, 2) populations that are initially unstable and
eventually grow according to an asymptotic growth rate g (not necessarily the stationary
rate), 3) populations that initially have a stable structure and grow at a rate g, that
undergo instantaneous change in population structure resulting in population momentum,
and 4) populations that initially have a stable structure and grow at a rate g
1
, that undergo
instantaneous change in a vital rate resulting in a new asymptotic growth rate g
2
(not
necessarily the stationary rate, e.g., a population that undergoes a vitalrate change that
causes asymptotic growth rate to change from 1.05 to 0.98). Caswell?s (2001:104)
discrete formula for population momentum M happens to satisfy all of these conditions:
( )
T
T
10 1
0
M =
?
evnw
en
, (6)
120
where e is a vector of ones. In condition 2 above, is a vector describing the initially
unstable distribution of population size across stage classes, and all else being equal, the
population eventually grows at the rate g = . In condition 3 above, is a vector
describing the initially stable population structure and the population grows at the rate g =
, then is perturbed and the population undergoes transient dynamics, which can
cause population momentum. All else being equal, the population will eventually grow
at the same rate g = . In conditions 1 and 4 above, describes the stable distribution
of population size before the vital rate(s) changes (change), and and are the left
and right eigenvectors of the transition matrix following change in the vital rate(s) (A
0
n
n
1
?
0
n
w
1
?
0
n
1
?
0
1
v
1
new
).
For projections from initial conditions and for instantaneous changes in vital rates or
population structure, equation 6 essentially describes the ultimate population size of a
population growing at any rate (g or g
2
above) as determined by the actual population
structure, relative to the size of an otherwise equivalent population that always grows
according to a stable population structure (i.e., the Stable Equivalent Population).
Because the term ?population momentum? is reserved by some demographers for the
special case where per capita fertility transitions to the stationary level, Tuljapurkar and
Lee (1997) call the ratio of the ultimate population size to that in the Stable Equivalent
Population, the Stable Equivalent Ratio. Here, I do not restrict my use of ?population
momentum? to Keyfitz?s special case (1971a) because momentum is a general physical
phenomenon.
121
2.3. General formulas for the sensitivity of population momentum to unit changes in
population vital rates
I wish to measure the sensitivity of population momentum to infinitesimal unit
changes in the underlying vital rates (a
ij
) of the transition matrix A. Perturbations of the
vital rates will change the right and left eigenvectors; thus, sensitivities of these
eigenvectors to change in a vital rate will also be needed. To develop general formulas, I
used the product rule to differentiate equation 6 with respect to a single vital rate a
ij
:
( )
()
() ()
()
T*
10 1
T
0
T*
10 1
T
0
T* *
1
10 10 1
T
0
*
T*
11
10 0
T
0
M
=
1
= 0
1
=
1
=
ij ij
ij
ij ij
ij ij
aa
a
aa
aa
??
??
??????
??
??
??
??
?
??
+??
??
??
?
??
??
? ?
? ?
??
? ?
? ?
? ?+??
??
? ?
? ?
??
? ?
? ?
??
??
+??
??
??
evnw
en
evnw
en
w
evn vnw
en
wv
evn n
en
1
? ?
??
? ?
??
? ?
??
? ?
w
. (7)
Caswell (1980, 2001), developed solutions to the derivatives of the right and left
eigenvectors with respect to change in a vital rate a
1
w
1
v
ij
()
(1)
1
1
1
m
s
i
mj
ij m
m
v
w
a ??
?
?
=
??
?
w
w . (8)
()
(1)
1
1
1
m
s
j
mi
m
ij
m
w
v
a ??
?
?
=
? ?
?
v
v . (9)
122
where is the jth entry of . By incorporating Caswell?s solutions (eq. 8 and 9)
into equation 7, my general solution to the sensitivity of population momentum can be
seen as
()m
j
w
m
w
()
*
()
()
(1) (1)T*
10 0 1
T
1
1
110
M1
m
m
ss
j
i
mmji
m
ij m
mm
w
v
wv
a ?? ??
??
? ?
??
??
??
? ?
????
?
??
=+? ?
???
? ?
??
??
? ?
??
? ?
??
evn w v nw
en
.(10)
Sometimes matrixlevel entries are computed from multiple lowerlevel vital rates. For
example, fertilities are the product of fecundity and some component of survival. To
calculate the sensitivity of population momentum to lowerlevel parameters x, just use the
chain rule
,
MM
kl
kl
kl
a
x ax
?? ?
=
? ??
?
. (11)
The current definition of population momentum is asymptotic, and measured in
the direction of the dominant left and right eigenvectors. Thus, equation 6 could
also be written as
1
v
1
w
()( )
T1,1
10 1
M =
?
vn w en
T
0
e , where the superscripts on M indicate
direction with respect to (any k,lth pair of) the left and right eigenvectors. From
equations 8 and 9, I can define
()(1)
1
b
m
j
i
m
m
w
v
? ?
=
?
, and (12)
(1) ()
1
c
m
j
i
m
m
w
v
? ?
=
?
. (13)
Using these definitions, patterns in the abovementioned equations reveal that
123
(
1,1
1, ,1
1
M
bM cM
s
m
mm
ij
m
a
?
?
=+
?
? )
m
. (14)
Momentum is conserved in a system, and in the short term, portions of the population
vector lie along each eigenvector (eq. 2). Thus, we could measure population momentum
at any time t, in the direction of any k,lth pair of left and right eigenvectors. Additionally,
equation 14 can be used to calculate the sensitivity of population momentum in the k,l
direction to changes in a vital rate:
,
M
kl
ij
a
?
?
.
Population momentum sensitivities can also be calculated numerically
MM
M
post pre
ij ij
a
?
?
??
. (15)
Here, and are the measurements of population momentum before and after
the change in the vital rate . Although I have chosen to focus on the sensitivity of
population momentum to unit changes in vital rates, the elasticity of population
momentum to proportional changes in vital rates can easily be calculated from the
analytical sensitivity or numerically,
M
pre
M
post
ij
?
ij
a
log M M
log M
ij
ij ij
a
aa
??
=
??
(16)
MM
log M 1
log M
post pre
ij pre ij
ap
?
?
?
(17)
where
ij
p is the proportional change in the vital rate . Population momentum is not a
linear function of the a , thus the elasticities do not sum to unity. Unfortunately, the
elasticities do not quantify the contribution of the a to M like they do for the geometric
ij
a
ij
ij
124
population growth rate (de Kroon et al. 1986). Nevertheless, elasticities are still useful
for measuring the effect of relative change in a vital rate on population momentum.
2.4. General formulas for the sensitivity of population momentum to unit changes in
population structure
Population momentum also depends on the initial population vector n . Thus, I
also seek to measure the sensitivity of population momentum to changes in the initial
population structure. To develop general formulas for this sensitivity, I first used the
quotient rule to differentiate equation 6 with respect to a single entry of the initial
population vector :
0
(0)
i
n
()
()
( )
()
*
T 10
1
T
0
T**
01010
T
1
2
T
0
(1)
T*
010
T
1
2
T
0
M
=
(0) (0)
(0) (0)
=
=
ii
ii
i
nn
nn
v
??
??
??
??
??
T
0
? ?
?
? ?
??
? ?
? ?
? ?
? ?
??
?
??
??
vn
ew
en
e n vn vn e n
ew
en
en vn
ew
en
. (18)
I note that the ?initial? point in time can be defined as the point in time from which the
population will be studied forward. If one assumes that
TT
11 00
1, 1= ==w en n =ew ,
and is real, then
1
v
(1) *
10
M
= 
(0)
i
i
v
n
?
?
vn . (19)
In particular, if one examines the special case where
01
=nw, then
125
1
(1)
M
= 1
(0)
i
i
v
n
?
?
w
. (20)
Thus, knowledge of only the reproductive value and the initial population structure are
needed to measure the sensitivity of population momentum to change in a single entry of
the initial population structure. If it is safe to assume that the population structure is
initially stable, then only reproductive value is needed, which is easily computed from A.
Now I consider perturbations that could affect multiple stage classes. To do this,
let u be an arbitrary unit vector (i.e.,
[
T
1
,,
n
uu= "
]
1=u ) to be regarded as a
perturbation vector applied to n . Furthermore, the directional derivative of M in the
direction u is denoted , and is given by the formula
0
MD
u
, (21)
T
M = MD
u
u?
where
[
T
1
M = M (0), , M (0)
n
n?? ??"
T
M0=x
]
1
? is the gradient of M. The set of all vectors x
for which ? form a hyperplane in nspace. The directional derivative is
positive, negative, or zero when u is on the positive side of the hyperplane, the negative
side, or in the hyperplane itself, respectively. Furthermore, the directional derivative is
maximized when u lies in the same direction as the gradient of M.
Considering the special case where
0
=nw (with each being a unit vector),
()
1
T
*
1
M = M
=
D
?
u
w
u
veu
?
. (22)
When population structure is initially stable, population momentum equals 1. Therefore,
the new value of M after perturbation is > 1, < 1, or = 1 when the directional derivative is
126
> 0, < 0, or = 0, respectively. If one only perturbs the ith entry of (i.e.,
), then the sign of the ith entry of
1
w
[
T
0, ,0,1,0, ,0=u ""
]
1
?ve dictates the direction that
M moves away from 1. In this case
1
1
(1)
M
M = = 1
(0)
i
i
D
n
?
?
u
w
w
v. (23)
Thus, equation 20 is just a special case of the directional derivative (eq. 23). In addition,
if u , then the distribution of individuals among stage classes does not change
following the perturbation (u), so M is expected to remain at 1. This is indeed the case,
since
1
= w
()
1
*
*T *
1111111
M1=? =?=?=
w
vew vwew vw
1
D
w
0.
3. Applications and examples
3.1. Sensitivity of population momentum across bird and mammal life histories
Here, I visit my study in Chapter 4. For increasing ( = 1.1) bird and mammal
populations, I found that instantaneous decrements in fertility to the stationary level
always decreased the net population size (M < 1), which contradicted patterns found in
human populations (Fischer and Heilig 1997). Although of lesser magnitude,
instantaneous augmentation of adult survival in decreasing ( = 0.9) bird and mammal
populations also produced M < 1, and populations always stabilized at smaller sizes than
the Stable Equivalent Population. Both experiments produced interesting patterns in M
across life histories (Fig. 1).
1
?
1
?
Yet, I did not control for the size of the vitalrate perturbation in Chapter 4. Life
history characteristics determine how much each vital rate has to be changed in order to
achieve stationary growth (Fig. 1, right axis), which is problematic because the
127
magnitude of change in a vital rate affects the magnitude of change in asymptotic
population structure (eq. 8), and sometimes population momentum (eq. 10). I concluded
that because the magnitude of population momentum did not always increase with the
magnitude of change in a vital rate (Fig. 1, both axes), the earliest and latest maturing life
histories must be more resistant to the forces of population momentum than others.
Examining the sensitivity of population momentum to equal changes in vital rates
across life histories avoids the aforementioned problem in Chapter 4, and could shed
more light on these findings. Using the data set in Chapter 4, I used equation 10 to
examine patterns in population momentum sensitivities across bird and mammal life
histories. Across populations that were initially increasing ( = 1.1), I found that the
sensitivity of population momentum to unit changes in fertility (summed across relevant
age classes) varied with age at maturity in a similar way (Fig. 2) as the actual values of
population momentum in Chapter 4 (Fig. 1). The positive sensitivities of population
momentum to unit changes in fertility indicate that increased fertility will increase
population momentum. Likewise, decrements in fertility, like those applied in Chapter 4,
will decrease population momentum, which explains why I observed M < 1 and net
losses in population size in Chapter 4. Here, I found that different initial population
growth rates (e.g., stationary growth) affected stable age structures and population
momentum sensitivity values, but not the general pattern in sensitivities across life
histories. Thus, both Chapters 4 and 5 provide evidence that early and late maturing
birds and mammals are more resistant to the forces of population momentum following
changes in fertility than those with intermediate age at maturity.
1
?
128
Yet, across populations that were initially decreasing ( = 0.9), the relationship
between age at maturity and sensitivity of population momentum to unit changes in adult
survival was different (Fig. 3) than that for actual values of population momentum in
Chapter 4 (Fig. 1). Life history patterns in the sensitivities were similar to my findings in
Chapter 4 up until ? = 10. However, the actual sign of the sensitivities changed from
negative to positive for ? ? 16 (Fig. 3), indicating that small unit increases in adult
survival can increase population momentum.
1
?
To examine why infinitesimally small increases in adult survival led to an
increase in population momentum for certain life histories while large increases in adult
survival led to a decrease in momentum, I used numerical simulation to examine how
population momentum changed with increasing size of the adult survival perturbation.
As predicted by the positive population momentum sensitivity for the ? = 16 life history,
I found that small increases in adult survival produced M > 1, but that increases > 0.0687
produced M < 1, as found in Chapter 4. Interestingly, population momentum was
maximized at a perturbation of 0.0384 (Fig. 4a). For the ? = 28 life history, population
momentum was maximized (1.0255) at a perturbation of 0.0409. Still, perturbations >
0.0951 produced M < 1, as found in Chapter 4 (Fig. 4b).
Compared to the initially declining populations ( = 0.9), asymptotic population
structure and reproductive values of latematuring life histories (? ? 16) changed
substantially with increasing size of the perturbation applied to adult survival. The
asymptotic distribution of abundance in the oldest adult age class and youngest subadult
age classes increased with the size of perturbation applied to adult survival, while
abundance in all other age classes decreased. Furthermore, the slopes of the relationship
1
?
129
between adult survival perturbation size and asymptotic population structure were
steepest in the oldest age classes (e.g., see Fig. 5). Conversely, reproductive value of the
older subadult and adult age classes increased with the size of perturbation applied to
adult survival, while that of the younger subadult age classes decreased. The slopes of
the relationship between adult survival perturbation size and reproductive value were
steepest in the oldest and youngest age classes (e.g., see Fig. 6). Longlived, late
maturing life histories live a substantial amount of life as a subadult, which can create a
complex population structure. Because of these life history properties of latematuring
birds and mammals, the nature of how a perturbation affects the disparity between actual
population structure and the new asymptotic population structure and reproductive values
following the perturbation can have a substantial impact on both the quantitative and
qualitative aspects of population momentum (e.g., see Fig. 4).
3.2. Population momentum in a sourcesink system
Here, I provide an example of population momentum and its sensitivity to
changes in the dispersal of young produced in a source habitat that move to a sink habitat
(Pulliam 1988). To illustrate my example, I use the following sourcesink system
11
22
33
44
55
66
1
002000
0.25 0 0 0 0 0
0 0.50 0.75 0 0 0
0 0 0.10 0 0 1
0000.2500
0 0 0 0 0.50 0.75
tt
nn
nn
+
? ?????
? ?????
?
?
=
?
?
?
? ??
? ??
??
?
? ???
.
In my example, stages 13 belong to the subpopulation in the source habitat, stages 46
belong to the subpopulation in the sink habitat and represent the same stage classes as 1
3. All matrix entries represent local fertility or survival probabilities except , which
4,3
a
130
represents the number of offspring produced in the source habitat that successfully
disperse and survive to the next census in the sink habitat (i.e., dispersants). The
population in the source habitat is locally stationary ( = 1) and that in the sink habitat
does not have selfsustaining fertility levels and would rapidly decline ( = 0.9) to
extinction without connection to the source habitat. The asymptotic growth rate of the
overall sourcesink population is stationary ( = 1) because it is dictated by that in the
source habitat. The sourcesink population has asymptotic stage structure
1
?
1
?
1
?
[ ]
T
1
0.87 0.22 0.43 0.09 0.02 0.04=w
ij
0 0
0.85 0
0 0.35
0 0
00
0 0
?
=MS
, and a small number of successful dispersants
(0.10). I used equation 10 to calculate the matrix of population momentum sensitivities
(MS) to each nonzero a , assuming stable stage structure:
0.07
0
0.71
0
0.91
0.5
0 0 0
0 0 0
0 0 0
0 0 0.09
0 200
0 0.23 0.47
??
??
??
??
??
.
It is clear that population momentum of this particular sourcesink system is highly
sensitive to unit changes in the number of dispersants. Furthermore, the sign of this
particular sensitivity value indicates that an increase in the number of dispersants should
increase population momentum while a decrease should decrease population momentum.
Using this information, I examined how the population dynamics of a sourcesink
system would respond to an increase in the number of successful dispersants from 0.1 to
1. Such an event could occur if resources became abundant in the source habitat,
131
allowing high fertility, but competition forced the extra offspring to disperse to the sink
habitat.
Because the increase in dispersant offspring did not affect the population growth
rate in the source habitat, asymptotic growth of the sourcesink system remained
stationary ( still = 1). However, increasing the number of successful dispersants did
increase shortterm growth of the sourcesink system, which caused large population
momentum (M = 1.82). All of the net growth in population size occurred in the sink
habitat (
1
?
[ ]
T
0.87 0.22 0.43 0.87 0.22 0.43
?
n = . Thus, change in the number of
dispersants did not change the asymptotic growth of the sourcesink system, but it did
cause population momentum that equilibrated the number of individuals in each stage
class across habitats. Depending on the organism, this result might be beneficial because
it could provide more hunting and viewing opportunities, buffer the overall population
from extinction in the event of stochastic events, or it could be deleterious if the organism
is considered a pest. Although this is just one simple example, population momentum
sensitivities could prove to be very useful for studying the dynamics of spatially
structured populations.
3.3. Effects of changing population structure
In many cases, managers might want to consider how different management
strategies could change population momentum in their favor to keep population size
within reasonable limits (Hauser, Cooch, and Lebreton In Review), or to decrease risks of
extinction. Plant and animal release and relocation programs provide managers a variety
of ways to directly ?add? individuals to specific age or stage classes of a population, while
harvest, livetrapping, and other removal techniques allow managers to directly decrease
132
abundance in specific age or stage classes. All of these management practices could
change population momentum (eq. 6). Thus, I provide an example that illustrates how
population momentum is affected by perturbations that add or remove individuals from
specific age classes of a population. I use the following matrix A, which describes the
mean vital rates of the lesser snow goose (Chen caerulescens) population at La Perouse
Bay, Manitoba from 1973 to 1990 (Cooch et al. 2001).
0 0.12 0.26 0.38 0.41
0.83 0 0 0 0
00.830 0 0
0 0 0.83 0 0
0 0 0 0.83 0.83
? ?
? ?
? ?
? ?=
? ?
? ?
? ?
? ?
A
In A, fertilities are represented on the top row and survival probabilities are on the sub
diagonal and bottomright corner of the matrix. The 1
st
age class represents young and all
other age classes represent aging adults.
To begin, I calculated the left and right
eigenvectors of A. To simplify my scenario, I
assumed that population structure was initially stable (i.e.,
[]
T
1
0.34 0.44 0.52 0.56 0.57=v
]
T
01
[
1
0.46 0.36 0.28 0.21 0.73=w
=n ), indicating M
initially = 1. I also normalized into a unit vector (i.e.,
w
1
w
1, .
1
nor
=w ), and normalized
, such that
1
v
1, . 1 1nor
=vvw, which allowed me to use equations 22 and 23 to easily
address my scenario. In addition, the normalized right and left eigenvectors are still
eigenvectors of A, and the condition ,
ii
1 =vw is maintained. Rounded to the second
133
decimal, w and
.
[]
T
1, .
0.23 0.17 0.14 0.10 0.36
nor
=
T
0.70 0.90 1.06 1.15 1.17
1, .nor
1, .nor
=v
Next, I used the directional derivative to measure the sensitivity of population
momentum to a variety of perturbations to initial population structure (Table 1). It is
readily seen that if only the ith entry of initial population structure is perturbed, then the
direction M moves away from 1 is dictated by the sign of the ith entry of or .
For example, a unit increase in the 1
M?
(1)
1
i
v
st
age class decreases population momentum (M < 1),
a unit increase in an older adult age class increases population momentum (M > 1), and
unit decreases produce opposite results (Table 1). Furthermore, a perturbation in the
same direction as the gradient vector maximizes the response of population momentum to
a unit change in the population structure, while a perturbation in the opposite direction of
the gradient vector produces the exact opposite response. Perturbations equal to the
stable age distribution ( w ) do not change population momentum at all (M still = 1;
Table 1).
Compared to the stable age structure of A, removing young (e.g., through clutch
removal), releasing adults, or both, would allow a goose manager to quickly increase
population momentum because each action shifts age structure towards reproducing
adults. On the other hand, releasing young, removing adults (via harvest or live
trapping), or both, would allow the manager to quickly decrease population momentum
because these practices shift age structure towards offspring. However, it is important to
remember that these results pertain only to the effects of directly changing population
structure, not the effects of perturbing vital rates, which are presented above.
134
4. Discussion
The sensitivity of population growth rate to changes in population vital rates has a
long history of use in demography, theoretical ecology, and evolution (Lewontin 1965,
Hamilton 1966, Demetrius 1969, Emlen 1970, Goodman 1971, Keyfitz 1971b, and Mertz
1971). However, Caswell?s (1978) discreteform sensitivity formula, that simply requires
the vectors describing reproductive value and stable population structure , has
made calculating this metric relatively simple. As a result, population growth rate
sensitivity is now widely used in conservation, management, and applied ecology (e.g.,
van Groenendael et al. 1988, Horvitz et al. 1997, Benton and Grant 1999, papers within
Heppell et al. 2000). Caswell also developed formulas for the sensitivities of
reproductive value and stable population structure to examine the equivalence of
maximizing reproductive value versus maximizing fitness (1980). These formulas have
not been widely used; however, I have drawn upon this theory to develop my general
formula for population momentum sensitivity.
1
v
1
w
Population momentum is a measure related to population size rather than
population growth rate. All else being equal, population growth rate is ergodic, meaning
that it forgets initial conditions. In contrast, population size is very responsive to initial
conditions (Lee and Tuljapurkar 1997). Thus, I caution against confusing sensitivities
calculated for the asymptotic population growth rate (e.g., ) with those calculated for
population momentum. Increased survival and fertility will always increase longterm
growth rate, but this is not necessarily the case for population momentum.
1
?
The sensitivity of population size to changes in population vital rates or initial
population structure can also be measured from eigenvectors of A (Fox and Gurevitch
135
2000, Caswell 2001). However, population momentum is the ratio of the population size
that is ultimately attained following transient dynamics to the size of a hypothetical
population that is always in an asymptotic stable state. Thus, those that are used to
thinking about asymptotic population dynamics may find my momentum sensitivities
very useful.
I have shown that unit increases in adult survival probabilities can often decrease
population momentum, meaning that if population momentum is initially = 1, increased
survival could decrease ultimate population size (Figs. 1 and 3). Interestingly,
simultaneous changes in fertility and survival would have large effects on population
growth rate; however, with regard to population momentum, simultaneous changes could
offset each other and not change population momentum at all. Because reproductive
value often depends on age or developmental stage, the nature of how perturbed vital
rates affect age structure will dictate the direction and magnitude of population
momentum. By studying the sensitivity of population momentum to direct changes in
stable population structure, I was able to explicitly reveal the dependence of population
momentum on reproductive value and population structure (Table 1). My simple
formulas for the sensitivity of population momentum to perturbations away from the
stable population structure (eq. 22, 23) can easily be used by demographers, managers,
and conservation biologists to understand how changing population structure can affect
population momentum.
Because the world is moving toward largescale conservation planning, it is
important to seek general patterns in lifecycle properties across species that could help
guide conservation and management. Across several vertebrate groups, there are strong
136
relationships between the population growth rate sensitivities (and elasticities) and simple
life history characteristics (Heppell 1998, Heppell et al. 2000b, S?ther and Bakke 2000).
In the bird, mammal, turtle, lizard, snake, and bony fish taxa, I have shown that
population momentum also varies in predictable ways with life history (Chapter 4). In
my first example, I used equation 10 to provide further evidence that particular life
histories do indeed respond more to the forces of population momentum than others.
General patterns in lifecycle properties and population dynamics do exist across
vertebrate species. Thus, it may be possible to use life history characteristics of
organisms to develop very general, but widespread conservation and management plans
for many species.
For populations with stable or unstable population structures, my family of
population momentum sensitivities can be used to examine how unit or proportional
changes in vital rates or initial population structure will affect population momentum.
This will be important in studies of demography, life history evolution, metapopulation
and multiregion dynamics, conservation, pest control, and natural resource management.
Lastly, when comparing the results from small vitalrate perturbations to the large
perturbations made in Chapter 4, I made an interesting discovery. Population momentum
(M) can be maximized, and the size of a vitalrate perturbation can actually reverse the
direction of population momentum in latematuring organisms. These results seem to be
related to the intricate differences between actual population structure and the asymptotic
population structure associated with the changed set of vital rates, as well as the
corresponding reproductive values (Figs. 56). I encourage further exploration of the
behavior of population momentum across life history and perturbation space in linear and
137
nonlinear systems. Hopefully, my discovery regarding the maximization of population
momentum will generate new and exciting theoretical questions about the potential
behavior of population momentum in nature.
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142
Table 1. The sensitivity of population momentum to unit changes u in the initial
population structure of the lesser snow goose population at La Perouse Bay, Manitoba,
indicated by the directional derivative
1
MD
u
w
T
, and the effect of the perturbation on
population momentum M. Here, I used the normalized left eigenvector
, and the following gradient of M:
[
1, .
0.70 0.90 1.06 1.15 1.17
nor
=v
[
]
]
T
.17M 0.30 0.10 0.06 0.15 0=? ?? , each rounded to the second decimal.
u
a
1
MD
u
w
Value of M after perturbation
[1 0 0 0 0]
T
0.30 < 1
[0 0 1 0 0]
T
0.06 > 1
[0 0 0 0 1]
T
0.17 > 1
[1 0 0 0 0]
T
0.30 > 1
[0 0 1 0 0]
T
0.06 < 1
[0 0 0 0 1]
T
0.17 < 1
[0.2 0.2 0.2 0.2 0.2]
T
0.003 < 1
[0.2 0.2 0.2 0.2 0.2]
T
0.003 > 1
[18.57 6.16 3.74 9.40 10.59]
T
9.80 >> 1
[18.57 6.16 3.74 9.40 10.59]
T
9.80 << 1
[0.23 0.17 0.14 0.10 0.36]
T
b
0 1
a ? Rounded to the second decimal
b ? The stable age distribution
143
Figure 1. Plots of population momentum (left axis), evaluated at regular intervals of age
at maturity (?) for the thought experiments described by Koons and Grand (In Review)
where fertility of growing populations ( = 1.1) was decremented to the stationary level
across hypothetical bird and mammal life histories (?), and where adult survival of
declining populations ( = 0.9) was augmented to the stationary level (?). The right
axis shows proportional change in each vital rate (open symbols) that was required to
achieve the stationary level.
1
?
1
?
144
145
Iteroparous birds and mammals
?
1 4 7 10 13 16 19 22 25 28
P
opulation M
o
m
entum
0.8
0.9
1.0
1.1
1.2
P
r
oportional C
hang
e in Vital Rate
1.0
0.5
0.0
0.5
1.0
M
Fertility Decrease
M
Survival Increase
Reference line for momentum = 1
Change in fertility
Change in survival
Figure 2. The sensitivity of population momentum to infinitesimal unit changes in
fertility (?) for increasing ( = 1.1) populations of bird and mammal life histories.
1
?
146
147
Iteroparous birds and mammals
?
1 4 7 10131619222528
S
ens
it
iv
it
y
of
Populat
ion M
o
men
t
um
0.2
0.1
0.0
0.1
0.2
0.3
0.4
Reference line for 0 sensitivity
Figure 3. The sensitivity of population momentum to infinitesimal unit changes in adult
survival (?) for decreasing ( = 0.9) populations of bird and mammal life histories.
1
?
148
Iteroparous birds and mammals
?
1 4 7 10131619222528
S
ens
it
iv
it
y
of
Populat
ion M
o
men
t
um
0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Reference line for 0 sensitivity
149
Figure 4. The relationship between perturbation size of adult survival (xaxis) and
population momentum (yaxis) for the ? = 16 life history (a), and ? = 28 life history (b).
150
151
Figure 5. The relationship between perturbation size of adult survival (xaxis) and the
resulting asymptotic population structure (yaxis) for the ? = 16 life history. Distribution
of abundance in the oldest age classes is indicated by black and dark gray lines while
distribution of abundance in the youngest age classes is indicated with light gray lines.
152
153
Figure 6. The relationship between perturbation size of adult survival (xaxis) and the
resulting relative reproductive values (i.e., reproductive value across age classes
normalized to sum to 1; yaxis) for the ? = 16 life history. Relative reproductive value of
the oldest age classes is indicated by black and dark gray lines while relative reproductive
value of the youngest age classes is indicated with light gray lines.
154
155
VI. GENERAL CONCLUSIONS
Asymptotic demographic analysis has a long history of use in population ecology;
however, empirical studies suggest that conditions in nature might not justify this use in
some cases (Bierzychudek 1999, CluttonBrock and Coulson 2002, Coulson et al. 2004).
For example, management actions can dramatically affect population structure (Crowder
et al. 1994, Coulson et al. 2004). Therefore, the time has come for ecologists to pay
stronger attention to population structure and its effect on population, community, and
evolutionary dynamics (Hastings 2004).
Analysis of transient population dynamics and population momentum can reveal
the possible effects of initial population structure (Fox and Gurevitch 2000, this study),
colonization (Caswell and Werner 1978), life history (DeAngelis et al. 1980, this study),
harvest, release of captivereared animals into the wild, relocation of wild individuals,
and environmental pulse perturbations (e.g., epidemics, catastrophic episodes) on
population dynamics. If population structure is perturbed away from the stable state,
asymptotic population analysis might not reveal the actual dynamics that could occur,
and it could be misleading. For example, shortterm transient growth can be highly
erratic and set the population on a completely different longterm trajectory (i.e.,
population momentum). In effect, population momentum could push populations far past
an environmental carrying capacity or even to extinction, depending on the direction of
momentum. Relative to predictions from asymptotic dynamics, population momentum
156
could shorten or lengthen the time it takes for a population to go extinct, recover from a
perturbation, or explode to levels that become a nuisance (e.g., Merrill et al. 2003).
I suggest that resource managers place a stronger emphasis on estimation of
population structure and reproductive value to examine the consequences of their actions
on shortterm population growth, and longterm population size. Such studies will help
reduce uncertainty in decisionmaking and the likelihood of deleterious management in
the future.
My dissertation is a theoretical starting point that could direct empirical studies
and natural resource management. I largely used thought experiments in an attempt to
elucidate patterns in transient population dynamics and population momentum across
vertebrate life history strategies. By relaxing the classic assumption of stable population
structure, I was able to explicitly explore the effect of unstable population structure on
vertebrate population dynamics.
Many of my theoretical findings were novel. For example, I found that net
reproductive value of the initial population could explain much of the intraspecific
variation in transient population dynamics. Across species, long lifespan of slow
reproducing species increases the chances for variability in somatic and reproductive
investment across age classes (Charlesworth 1994). For this reason, transient dynamics
of slow reproducing species were very responsive to changes in population structure, and
slowly converged to the asymptotic stable state when compared to fast reproducing
species.
Although seldom conveyed in the literature, transient dynamics dictate the
behavior of longterm population momentum (Lande and Orzack 1988). Contrary to
157
158
findings in human demography (Fischer and Heilig 1997), I found that large experimental
changes in vital rates sometimes reversed the direction of population growth, resulting in
large amounts of population momentum. In addition, population momentum varied with
age at maturity in bird, mammal, turtle, lizard, snake, and bony fish life histories, but
changed most abruptly with increasing maturity in bony fishes (Chapter 4, Figs 1 and 2)
because of the underlying shortterm transient dynamics. By using pyramid graphs to
examine the distribution of abundance and reproductive values across (st)age classes of
different life history strategies, it seemed that relative skew of each distribution
determined how responsive populations in each vertebrate taxa were to the forces of
population momentum.
However, it was not clear how changes in population structure, reproductive
value, or both would affect population momentum. Much of my exploration into
transient population dynamics and population momentum consisted of developing
hypotheses about the effects of unstable population structure on transient population
dynamics and momentum, and then testing these hypotheses with thought experiments.
Yet, like empirical studies, thought experiments do not provide a general foundation for
science to build upon. I needed to synthesize the myriad of ideas produced by the work
in Chapters 2 ? 4 with a general analytical tool that explicitly linked changes in vital rates
to changes in population structure and reproductive value to change in population
momentum. Therefore, in Chapter 5 I developed analytical formulas relating the
sensitivity of population momentum to equal unit or proportional (elasticities) changes in
vital rates or initial population structure. These formulas provide a consistent means for
comparison of momentum across populations, life histories, initial conditions, or
159
management practices Perhaps more importantly, the sensitivities presented in my
dissertation provide a general theory relating transient population growth rate (Chapter 2)
and population momentum (Chapter 5) to the underlying demographic parameters for
scientists to build upon or reject outright.
My studies considered transient dynamics and population momentum following
onetime perturbations to populations with densityindependent population growth. The
theoretical experiments were not meant to represent reality, but were developed to
improve understanding of population dynamics in a way that is one step closer to reality
than asymptotic analysis. Indeed, population density and speciesinteractions regulate
many populations in nonlinear ways (Hixon et al. 2002). Under nonlinear population
regulation, population dynamics can be very sensitive to initial conditions, and small
changes in the initial conditions can even produce longterm chaos (May 1974, 1976,
Cushing et al. 2002). Thus, transient dynamics and the physical force of momentum will
be important to consider in nonlinear population models (Hastings 2004).
The frequent occurrence of longlasting transient dynamics (e.g., thousands of
years) in structured community models and spatially structured population models
suggests that transients may be the norm over ecological times scales, rather than ?short
term? (Hastings and Higgins 1994, Hastings 2001, Chen and Cohen 2001). Not only
should population ecologists pay more attention to transient dynamics and population
momentum, but evolutionary biologists should as well. If transient dynamics can occur
over long periods of time in nature, then natural selection could be operating on transient
measures of fitness rather than asymptotic ? or r. Ultimately, if agespecific densities
affect the fitness of a phenotype, then the physical forces of momentum could influence
160
the permeation rate of specific newborn phenotypes into adulthood. Similarly, if
nonlinear species interactions (i.e., competition, parasitism, mutualism, commensalism,
amensalism, and predation) affect the fitness of a phenotype, momentum could influence
the invasion rate and success of underlying mutant genotypes into wild types.
Studies of transient dynamics, chaos, and population momentum are slowly
becoming more popular in ecology (e.g., Fox and Gurevitch 2000, Hastings 2004).
Perhaps future studies on these topics will reveal that many components of variation in
demographic parameters that we currently believe to be random noise are actually caused
by the deterministic forces of transient growth, physical momentum, and chaos.
161
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