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## Spatial Spread and Front Propagation Dynamics of Nonlocal Monostable Equations in Periodic Habitats

##### Date

2011-07-25##### Author

Zhang, Aijun

##### Type of Degree

dissertation##### Department

Mathematics and Statistics##### Metadata

Show full item record##### Abstract

This dissertation is concerned with spatial spread and front propagation dynamics of
monostable equations with nonlocal dispersal in spatially periodic habitats. Such equations
arise in modeling the population dynamics of many species which exhibit nonlocal internal
interactions and live in spatially periodic habitats. The main results of the dissertation
consist of the following four parts.
Firstly, we establish a general principal eigenvalue theory for spatially periodic nonlocal
dispersal operators. Some su cient conditions are provided for the existence of principal
eigenvalue and its associated positive eigenvector for such dispersal operators. It shows that a
spatially periodic nonlocal dispersal operator has a principal eigenvalue for the following three
special but important cases: (i) the nonlocal dispersal is nearly local; (ii) the periodic habitat
is nearly globally homogeneous or (iii) it is nearly homogeneous in a region where it is most
conducive to population growth. It also provides an example which shows that in general, a
spatially periodic nonlocal dispersal operator may not possess a principal eigenvalue, which
reveals some essential di erence between random dispersal and nonlocal dispersal. The
principal eigenvalue theory established in this dissertation provides an important tool for
the study of the dynamics of nonlocal monostable equations and is of also great importance
in its own.
Secondly, applying the principal eigenvalue theory for nonlocal dispersal operators and
comparison principle for sub- and super-solutions, we obtain one of the important features
for monostable equations, that is, the existence, uniqueness, and global stability of spatially
periodic positive stationary solutions to a general spatially periodic nonlocal monostable
equation. In spite of the use of the principal eigenvalue theory for nonlocal dispersal operators
in the proof, this feature is generic for nonlocal monostable equations in the sense it is
iiindependent of the existence of the principal eigenvalue of the linearized nonlocal dispersal
operator at the trivial solution of the monostable equation, which is of great biological
importance.
Thirdly, applying the principal eigenvalue theory for nonlocal dispersal operators and
comparison principle for sub- and super-solutions, we obtain another important feature for
monostable equations, that is, the existence of a spatial spreading speed of a general spatially
periodic nonlocal equation in any given direction, which characterizes the speed at which a
species invades into the region where there is no population initially in the given direction.
It is also seen that this feature is generic for nonlocal monostable equations in the same sense
as above. Moreover, it is shown that spatial variation of the habitat speeds up the spatial
spread of the population.
Finally, this dissertation also deals with front propagation feature for monostable equa-
tions with non-local dispersal in spatially periodic habitats. It is shown that a spatially
periodic nonlocal monostable equation has in any given direction a unique stable spatially
periodic traveling wave solution connecting its unique positive stationary solution and the
trivial solution with all propagating speeds greater than the spreading speed in that direc-
tion for the special but important cases mentioned above, that is, (i) the nonlocal dispersal
is nearly local; (ii) the periodic habitat is nearly globally homogeneous or (iii) it is nearly
homogeneous in a region where it is most conducive to population growth. It remains open
whether this feature is generic or not for spatially periodic nonlocal monostable equations.