Spatial Spread Dynamics of Monostable Equations in Spatially Locally Inhomogeneous Media with Temporal Periodicity by Liang Kong A dissertation submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 03, 2013 Keywords: Monostable equation, random dispersal, nonlocal dispersal, discrete dispersal, spreading speed, principal eigenvalue/eigenfunction. Copyright 2013 by Liang Kong Approved by Wenxian Shen, Chair, Professor of Mathematics and Statistics Georg Hetzer, Professor of Mathematics and Statistics Xiaoying Han, Associate Professor of Mathematics and Statistics Bertram Zinner, Associate Professor of Mathematics and Statistics Abstract This dissertation is devoted to the study of semilinear dispersal evolution equations of the form ut(t;x) = (Au)(t;x) +u(t;x)f(t;x;u(t;x)); x2H; where H = RN or ZN, A is a random dispersal operator or nonlocal dispersal operator in the case H= RN and is a discrete dispersal operator in the case H= ZN, and f is periodic in t, asymptotically periodic in x (i.e. f(t;x;u) f0(t;x;u) converges to 0 as kxk!1 for some time and space periodic function f0(t;x;u)), and is of KPP type in u. These type of equations are called as Monostable or KPP type equations, which arise in modeling the population dynamics of many species which exhibit local, nonlocal and discrete internal interactions and live in locally spatially inhomogeneous media with temporal periodicity. The following main results are proved in the dissertation. Firstly, it is proved that Liouville type property holds for such equations, that is, time periodic strictly positive solutions are unique. It is proved that if time periodic strictly positive solutions (if exists) are globally stable with respect to strictly positive perturbations. Moreover, it is proved that if the trivial solution u = 0 of the limit equation of such an equation is linearly unstable, then the equation has a time periodic strictly positive solution. Secondly, spatial spreading speeds of such equations is investigated. It is also proved that if u 0 is a linearly unstable solution to the time and space periodic limit equation of such an equation, then the original equation has a spatial spreading speed in every direction. Moreover, it is proved that the localized spatial inhomogeneity neither slows down nor speeds up the spatial spreading speeds. In addition, in the time dependent case, various spreading features of the spreading speeds are obtained. ii Finally, the e ects of temporal and spatial variations on the uniform persistence and spatial spreading speeds of such equations are considered. As in the periodic media case, it is shown that temporal and spatial variations favor the population?s persistence and do not reduce the spatial spreading speeds. iii Acknowledgments First and foremost I want to thank my advisor Dr. Wenxian Shen. It has been an honor to be her Ph.D. student. I will forever be thankful to all her contributions of time, ideas, and funding to make my Ph.D. experience stimulating and productive. The positivity and enthusiasm she has for her research was contagious and motivational for me. The patience and caring she gave inspire me not only to be a better mathematician, but also to be a better person. I would like to show my gratitude to my committee members, Drs. Georg Hetzer, Xiaoying Han and Bertram Zinner, for their time and attention during busy semesters. I would also like to thank my parents for their unconditional love and support during the long years of my education. I am truly indebted and thankful to my wife, Li Cheng, for her sacri ce and support to ensure me stay focus on my graduate study. In addition, I would like thank the NSF for the nancial support (NSF-DMS-0907752). iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Notations, Hypotheses, De nitions, and Main Results . . . . . . . . . . . . . . 6 2.1 Notions, hypotheses and de nitions . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 Comparison principle and global existence . . . . . . . . . . . . . . . . . . . 16 3.2 Convergence on compact subsets and strip type subsets . . . . . . . . . . . . 19 3.3 Part metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Principal eigenvalue theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4.1 Principal eigenvalues of time periodic dispersal operators . . . . . . . 26 3.4.2 Principal eigenvalues of spatially periodic dispersal operators . . . . . 28 3.5 Positive solutions and spreading speeds of KPP equations . . . . . . . . . . . 30 3.5.1 Time periodic positive solutions and spreading speeds of KPP equa- tions in periodic media . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.5.2 KPP equations in spatially periodic media . . . . . . . . . . . . . . . 32 4 Existence, Uniqueness, and Stability of Time Periodic Strictly Positive Solutions 34 4.1 Uniqueness and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Tail property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Spatial Spreading Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6 Another Method to Show the Time Independent Case . . . . . . . . . . . . . . . 56 v 6.1 Another method to prove Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . 56 6.2 Spatial Spreading Speeds and Proofs of Theorems 2.2 and 2.3 . . . . . . . . 66 7 E ects of Temporal and Spatial Variations . . . . . . . . . . . . . . . . . . . . . 79 8 Concluding Remarks and Future Plan . . . . . . . . . . . . . . . . . . . . . . . 81 8.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2 Future plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.2.1 Single-species population model . . . . . . . . . . . . . . . . . . . . . 82 8.2.2 Multi-species population model . . . . . . . . . . . . . . . . . . . . . 83 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vi Chapter 1 Introduction In this dissertation, we investigate the Liouville type property and spatial spreading dy- namics of monostable evolution equations in locally spatially inhomogeneous periodic media, in particular, we consider the existence, uniqueness, and stability of time periodic positive solutions and spatial spreading speeds of monostable type dispersal evolution equations in periodic media with localized spatial inhomogeneity. We also study the in uence of the inhomogeneity of the underlying media on the spatial spreading speeds of the monostable equations. Our model equations are of the form, ut(t;x) =Au+uf0(t;x;u); x2H; (1.1) where H= RN or ZN; in the case H= RN, Au = u or (Au)(t;x) = Z RN (y x)u(t;y)dy u(t;x) ( ( ) is a smooth non-negative convolution kernel supported on a ball centered at the origin and RRN (z)dz = 1), and in the case H= ZN, (Au)(t;j) = X k2K ak(u(t;j +k) u(t;j)) 1 (ak > 0 and K = fk 2Hjkkk = 1g); and f0(t + T;x;u) = f0(t;x + piei;u) = f0(t;x;u) (T 2R and pi2H are given constants), and @uf0(t;x;u) < 0 for u 0, f0(t;x;u) < 0 for u 1. Among others, equation (1.1) is used to model the evolution of population density of a species. The case that H = RN and Au = u indicates that the environment of the underlying model problem is not patchy and the internal interaction of the organisms is random and local (i.e. the organisms move randomly between the adjacent spatial locations, such A is referred to as a random dispersal operator) (see [1], [2], [9], [21], [23], [24], [45], [56], [77], [79], [81], [82], [86], etc., for the application in this case). If the environment of the underlying model problem is not patchy and the internal interaction of the organisms is nonlocal, (Au)(t;x) = RRN (y x)u(t;y)dy u(t;x) is often adopted (such A is referred to as a nonlocal dispersal operator) (see [3], [10], [16], [22], [26], [41], etc.). The case that H = ZN and (Au)(t;j) = Pk2Kak(u(t;j + k) u(t;j)) (which is referred to as a discrete dispersal operator) arises when modeling the population dynamics of species living in patchy environments (see [21], [55], [56], [77], [78], [81], [82], [83], etc.). The periodicity of f0(t;x;u) in t and x re ects the periodicity of the environment. In literature, equation (1.1) is called of Fisher or KPP type due to the pioneering works of Fisher [24] and Kolmogorov, Petrowsky, Piscunov [45] on the following special case of (1.1), ut = uxx +u(1 u); x2R: (1.2) Central problems about (1.1) include the existence, uniqueness, and stability of time and space periodic positive solutions and spatial spreading speeds. Such problems have been extensively studied (see [1]-[8], [11]-[13], [18], [20], [19], [25], [27]-[30], [35], [39], [40], [44], [49]-[52], [54], [57]-[62], [65]-[76], [80]-[82], [84], etc.). It is known that time and space periodic positive solutions of (1.1) (if exist) are unique, which is referred to as the Liouville type property for (1.1). If u 0 is linearly unstable with respect to spatially periodic 2 perturbations, then (1.1) has a unique stable time and space periodic positive solutionu 0(t;x) and for any 2RN withk k= 1, (1.1) has a spreading speed c 0( ) in the direction of (see section 2.4 for detail). In reality, the underlying media of many biological problems is non-periodically inho- mogeneous. It is therefore of great importance to investigate the dynamics of monostable evolution equations in various types of non-periodically inhomogeneous media, for example, in almost periodic media, in periodic media with locally spatial perturbations, etc.. There are many works on various extensions of the spatial spreading dynamics of monostable evolution equations in periodic media, see, for example, [4], [31], [37], [59], [68]-[72], etc.. The aim of the current dissertation is to investigate the dynamics of KPP type equations in periodic media with spatially localized inhomogeneity, in particular, to deal with the extensions of the above results for (1.1) to KPP type equations in periodic media with spatially localized inhomogeneity. We hence consider ut =Au+uf(t;x;u); x2H; (1.3) where A and H are as in (1.1), @uf(t;x;u) < 0 for u 0, f(t;x;u) < 0 for u 1, f(t + T;x;u) = f(t;x;u), and jf(t;x;u) f0(t;x;u)j! 0 as kxk!1 uniformly in (t;u) on bounded sets (f0(t;x;u) is as in (1.1)) (See (H1) in Chapter 2 for detail). We show that localized inhomogeneity does not destroy the existence and uniqueness of time periodic positive solutions and it neither slow down nor speed up the spatial spreading speeds. We also show that temporal and spatial inhomogeneity does not slow down the spatial spreading speeds. More precisely, we prove (Liouville type property or uniqueness of time periodic strictly positive solutions) Time periodic strictly positive solutions of (1.3) (if exist) are unique (see Theorem 2.1(1)). (Stability of time periodic strictly positive solutions) If (1.3) has a time periodic strictly positive solution u (t;x), then it is asymptotically stable (see Theorem 2.1(2)). 3 (Existence of time periodic strictly positive solutions) If u = 0 is a linearly unstable solution of (1.1) with respect to periodic perturbations, then (1.3) has a time periodic strictly positive solution u (t;x) (see Theorem 2.1(3)). (Tail property of time periodic strictly positive solutions) If u = 0 is a linearly unstable solution of (1.1) with respect to periodic perturbations, then u (t;x) u 0(t;x)!0 askxk! 1 uniformly in t (see Theorem 2.1(4)). (Spatial spreading speeds) If u = 0 is a linearly unstable solution of (1.1) with respect to periodic perturbations, then for each 2RN with k k = 1, c 0( ) is the spreading speed of (1.3) in the direction of (see Theorem 2.2). (E ect of temporal variation) If u = 0 is a linearly unstable solution of ut(t;x) =Au+u^f0(x;u); x2H; (1.4) whereH andAis as in (1.1) and ^f0 is the time average of f0(t;x;u) (see (2.19)), then (1.3) has a time periodic strictly positive solution u (t;x) and for each 2RN with k k= 1, c 0( ) is the spreading speed of (1.3) in the direction of . Moreover, c 0( ) ^c 0( ); where ^c 0( ) is the spatial spreading speed of (1.4) in the direction of (see Theorem 2.4 (1)). (E ect of spatial variation) If u = 0 is a linearly unstable solution of ut(t;x) =Au+u^^f0(u); x2H; (1.5) where H and A is as in (1.1) and ^^f0(u) is the spatial average of ^f0(x;u) (see (2.21)), then (1.3) has a time periodic strictly positive solution u (t;x) and for each 2RN with k k= 1, 4 c 0( ) is the spreading speed of (1.3) in the direction of . Moreover, c 0( ) ^c 0( ) ^^c 0( ); where ^^c 0( ) is the spatial spreading speed of (1.5) in the direction of (see Theorem 2.4 (2)). We remark that, in the case thatH= RN andA= , Liouvile type property of (1.3) is discussed in [8] and the methods used in [8] quite rely on the special properties of parabolic equations. The current thesis recovers the results obtained in [8] by di erent methods, which apply to all three di erent type dispersal operators. The rest of the dissertation is organized as follows. In chapter 2, we introduce the standing notions, hypotheses, and de nitions, and state the main results of the dissertation. In chapter 3, we present some preliminary materials to be used in the proofs of the main results. We study the existence, uniqueness, and stability of time periodic positive solutions of (1.3) in chapter 4. In chapter 5, we explore the spreading speeds of (1.3). We give another elegant method working on time independent case in chapter 6. In chapter 7, we consider the temporal and spatial variations on the spatial spreading dynamics of monostable stable equations. In chapter 8, We will address some remarks and open problems. 5 Chapter 2 Notations, Hypotheses, De nitions, and Main Results In this chapter, we rst introduce some standing notations, hypotheses, and de nitions. We then state the main results of the paper. 2.1 Notions, hypotheses and de nitions In this section, we introduce standing notions, hypotheses, and de nitions. Throughout this subsection, H= RN or ZN and pi2H with pi > 0 ( i = 1;2; ;N ) (2.1) Let X = Cbunif(H;R) :=fu2C(H;R)ju is uniformly continuous and bounded on Hg (2.2) with norm kuk= supx2Hju(x)j, X+ =fu2Xju(x) 0 8x2Hg; (2.3) and X++ =fu2X+j inf x2H u(x) > 0g: (2.4) Let Xp =fu2Xju( +piei) = u( )g; (2.5) X+p = X+\Xp; (2.6) 6 and X++p = X++\Xp: (2.7) For given u;v2X, we de ne u v (u v) if v u2X+ (u v2X+) (2.8) and u v (u v) if v u2X++ (u v2X++): (2.9) Let Hi and Ai :D(Ai) X!X (i = 1;2;3) be de ned by H1 = RN; (A1u)(x) = u(x) 8u2D(A1); (2.10) where D(A1) =fu2Xj@xju( );@2xjxku( )2X; 1 j;k Ng; H2 = RN; (A2u)(x) = Z RN (y x)u(y)dy u(x) 8u2D(A2) = X; (2.11) and H3 = ZN; (A3u)(j) = X k2K ak(u(j +k) u(j)) 8u2D(A3) = X: (2.12) Let Xp =fu2C(R H;R)ju(t+T;x+piei) = u(t;x)g (2.13) 7 with norm kuk= maxt2R;x2Hju(t;x)j. For given 2SN 1 and 2R, let A ; :D(A ; ) Xp!Xp be de ned by (A ; u)(t;x) = 8 >>>> >>> >>>> >>> < >>> >>> >>> >>>> >: u(t;x) 2 ru(t;x) + 2u(t;x) if H=H1; A=A1 R RN e (y x) (y x)u(t;y)dy u(t;x) if H=H2; A=A2 P k2Kak(e k u(t;j +k) u(t;j)) if H=H3; A=A3 (2.14) for u2D(A ; ). Observe that A ;0 =A 8 2SN 1: For any given a2Xp, 2SN 1, and 2R, let ( @t+A ; +a( ; )I) be the spectrum of the operator @t +A ; +a( ; )I :D( @t +A ; +a( ; )I) Xp!Xp, ( @ t +A ; +a( ; )I)u (t;x) = u t(t;x) + (A ; u)(t;x) +a(t;x)u(t;x): Let ; (a) be de ned by ; (a) = supfRe j 2 ( @t +A ; +a( ; )I)g: (2.15) We call ; (a) the principal spectrum point of @t +A ; + a( ; )I. It equals the principal eigenvalue of @t +A ; +a( ; )I if it exists (see De nition 3.1 for the de nition of principal eigenvalue). It is clear that ;0(a) is independent of 2SN 1 and we may put (a) = ;0(a): 8 Observe that ( @t+A ; +a( ; )I) is the spectrum of the following eigenvalue problem, @tu+A ; u+a( ; )Iu = u; u2Xp: (2.16) When a(t;x) a(x) is independent of t, (2.16) reduces to A ; u+a( )Iu = u; u2Xp: (2.17) In the following, to indicate the dependence of X, Xp, Xp on the media, we may put Xi = X = Cbunif(Hi;R) in the case H=Hi; Xi;p = Xp =fu2Xiju( +piei) = u( )g in the case H=Hi; etc.. Consider (1.3). We introduce the following standing hypotheses. (H0) f(t + T;x;u) = f(t;x;u) for (t;x;u) 2R H R (T > 0 is a given positive num- ber); f(t;x;u) is C1 in t;u and f(t;x;u), ft(t;x;u), fu(t;x;u) are uniformly continuous in (t;x;u) 2R H E (E is any bounded subset of R); f(t;x;u) < 0 for all t2R, x2H, and u M0 (M0 > 0 is some given constant); and inft2R;x2Hfu(t;x;u) < 0 for all u 0. (H1) f0(t;x;u) satis es (H0), f0(t;x + piei;u) = f0(t;x;u) for (t;x;u) 2R H R, and jf(t;x;u) f0(t;x;u)j!0 as kxk!1 uniformly in (t;u) on bounded sets. (H1)0 f0(u) satis es (H0), f(t;x;u) f(x;u), and f(x;u) f0(u)!0 as kxk!1. (H2) (f0( ; ;0)) > 0. Observe that f(t;x;u) = 1 u is a typical example which satis es (H0) and (H0) is referred to as Fisher or KPP type condition. Throughout this section, we assume (H0). By general semigroup theory (see [32], [64]), for any u0 2X, (1.3) has a unique (local) solution 9 u(t; ;u0) with u(0; ;u0) = u0( ). Furthermore, if f(t;x + piei;u) = f(t;x;u) and u0 2Xp, then u(t; ;u0)2Xp. To indicate the dependence of u(t;x;u0) on f, we may write u(t;x;u0) as u(t;x;u0;f). Observe also that assumption (H1) re ects the localized spatial inhomogeneity of the media. (H1)0 is a special case of (H1). Assumption (H2) is the linear instability condition of the trivial solution of (1.3) in the case f(t;x;u) = f0(t;x;u). If f0(t;x;u) f0(u), then (H2) becomes f0(0) > 0. Let ^f(x;u), ^f0(x;u), ^^f(u), and ^^f0(u) be de ned as follows. ^f(x;u) = 1 T Z T 0 f(t;x;u)dt; (2.18) ^f0(x;u) = 1 T Z T 0 f0(t;x;u)dt; (2.19) and ^^f(u) = lim R!1 1 jB(0;R)j Z B(0;R) ^f(x;u)dx; (2.20) ^^f 0(u) = limR!1 1 jB(0;R)j Z B(0;R) ^f0(x;u)dx; (2.21) where B(0;R) =fx2Hjjxij R; i = 1;2; ;Ng and jB(0;R)j is the Lebesgue measure of B(0;R) in the case H = RN and jB(0;R) is the cardinality of B(0;R) in the case H= ZN. Observe that if f(t;x;u) satis es (H0), then so are ^f(x;u) and ^^f(u). If f(t;x;u) and f0(t;x;u) satisfy (H0) and (H1), then so are ^f(x;u) and ^^f(u). If ^f(x;u) and ^f0(x;u) satisfy (H0) and (H1), then so are ^^f(u) and ^^f0(u). Let SN 1 =f 2RNjk k= 1g: (2.22) 10 For given 2SN 1 and u2X+, we de ne lim inf x ! 1 u(x) = lim infr! 1 inf x2H;x r u(x): For given u : [0;1) H!R and c> 0, we de ne lim inf x ct;t!1 u(t;x) = lim inft!1 inf x2H;x ct u(t;x); lim sup x ct;t!1 u(t;x) = lim sup t!1 sup x2H;x ct u(t;x): The notions lim sup jx j ct;t!1 u(t;x), lim sup jx j ct;t!1 u(t;x), lim sup kxk ct;t!1 u(t;x), and lim sup kxk ct;t!1 u(t;x) are de ned similarly. We de ne X+( ) by X+( ) =fu2X+j lim inf x ! 1 u(x) > 0; u(x) = 0 for x 1g: (2.23) De nition 2.1 (Time periodic strictly positive solution). A solution u(t;x) of (1.3) on t2R is called a time periodic strictly positive solution if u(t + T;x) = u(t;x) for (t;x)2R H and inf(t;x)2R Hu(t;x) > 0. De nition 2.2 (Spatial spreading speed). For given 2SN 1, a real number c ( ) is called the spatial spreading speed of (1.3) in the direction of if for any u0 2X+( ), lim inf x ct;t!1 u(t;x;u0) > 0 8cc ( ): 2.2 Main Results In this section, we state the main results of this dissertation. The rst theorem is about time periodic strictly positive solutions. 11 Theorem 2.1 (Time periodic strictly positive solutions). Consider (1.3) and assume (H0). (1) (Liouville type property or uniqueness) If (1.3) has a time periodic strictly positive solution, then it is unique. (2) (Stability) Assume that u (t;x) is a time periodic strictly positive solution of (1.3). Then it is stable and for any u0 2X++, limt!1ku(t; ;u0;f( + ; ; )) u (t+ ; )kXi = 0 uniformly in 2R. (3) (Existence) Assume also (H1) and (H2). Then (1.3) has a unique time periodic strictly positive solution u (t;x). (4) (Tail property) Assume also (H1) and (H2). Then u (t;x) u 0(t;x)!0 as kxk!1 uniformly in t2R, where u (t;x) is as in (3) and u 0(t;x) is the unique time and space periodic positive solution of (1.1) (see Proposition 3.8 for the existence and uniqueness of u 0(t;x)). Remark 2.1. (1) Theorem 2.1 indicates that localized spatial inhomogeneity does not de- stroy the Liouville type property of (1.3), in particular, it does not destroy the existence of time periodic positive solution. Moreover, it shows that localized spatial inhomogene- ity does not a ect the behavior of the time periodic positive solution as the space variable goes to 1. (2) Assume (H0) and (H1)0. Then u 0(t;x) is a positive constant, denoted by u0, such that f0(u0) = 0. (3) Biologically, Theorem 2.1 implies that if u = 0 is a linearly unstable solution of the limit equation of (1.3), then the population will persist and is eventually time periodic. The second theorem is about spatial spreading speeds. 12 Theorem 2.2 (Existence of spreading speeds). Consider (1.3) and assume (H0)-(H2). Then for any given 2SN 1, (1.3) has a spreading speed c ( ) in the direction of . Moreover, for any u0 2X+( ), lim sup x ct;t!1 ju(t;x;u0) u (t;x)j= 0 8c0 f0(0) + 2 = 2 p f0(0) in the case H=H1; A=A1; c 0( ) = inf >0 R RN e z (z)dz 1 +f0(0) in the case H=H2; A=A2 and c 0( ) = inf >0 P k2Kak(e k 1) +f0(0) in the case H=H3; A=A3: For time independent case, we have the following additional result regarding the spread- ing speeds. 13 Theorem 2.3 (Spreading features of spreading speeds). Assume (H2) and (H1)0 and f0(0) > 0. Then for any given 2SN 1, the following hold. (1) For each u0 2X+ satisfying that u0(x) = 0 for x2H with jx j 1, lim sup jx j ct;t!1 u(t;x;u0) = 0 8c> maxfc 0( );c 0( )g: (2) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for x2Hwithjx j r, lim sup jx j ct;t!1 ju(t;x;u0) u (x)j= 0 80 sup 2SN 1 c 0( ): (4) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for kxk r, lim sup kxk ct;t!1 ju0(t;x;u0) u (x)j= 0 80 0, then (1.3) has a unique time periodic strictly positive solution u (t;x) and has a spreading speed c ( ) in the direction of . Moreover, for any u0 2X+( ), lim sup x ct;t!1 ju(t;x;u0) u (t;x)j= 0 8c 0, then (1.3) has a unique time periodic strictly positive solution u (t;x) and has a spreading speed c ( ) in the direction of . Moreover, for any u0 2X+( ), lim sup x ct;t!1 ju(t;x;u0) u (t;x)j= 0 8c 0 at which both u(t; ;u01) and u(t; ;u02) exist. Moreover, if u01 6= u02, then u(t;x;u01) < u(t;x;u02) for all x2H and t> 0 at which both u(t; ;u01) and u(t; ;u02) exist. (3) If u01;u02 2X and u01 u02, then u(t; ;u01) u(t; ;u02) for t > 0 at which both u(t; ;u01) and u(t; ;u02) exist. Proof. (1) The case that H = H1(= RN) and A = H1(= ) follows from comparison principle for parabolic equations. We prove the case thatH=H2(= RN) andA=A2. The case that H=H3(= ZN) and A=A3 can be proved similarly. Observe that for any t2[0; ), Z RN (y x)u1(t;y)dy = Z RNn (y x)u1(t;y)dy + Z (y x)u1(t;y)dy Z RNn (y x)u2(t;y)dy + Z (y x)u1(t;y)dy: (3.2) Let v(t;x) = u2(t;x) u1(t;x). By (3.2), vt(t;x) Z (y x)v(t;y)dy v(t;x) +u2(t;x)f(t;x;u2(t;x)) u1(t;x)f(t;x;u1(t;x)) = Z (y x)v(t;y)dy v(t;x) +a(t;x)v(t;x); x2 ; t2(0; ); where a(t;x) = f(t;x;u2(t;x)) +u1(t;x) Z 1 0 @uf(t;x;su2(t;x) + (1 s)u1(t;x))ds: The rest of the proof follows from the arguments of [43, Proposition 2.4]. (2) It follows from (1) with u1(t;x) = u(t;x; ;u01), u2(t;) = u(t;x;u02), and =H. (3) We provide a proof for the case that H = H2 and A = A2. Other cases can be proved similarly. Take any > 0 such that both u(t; ;u01) and u(t; ;u02) exist on [0; ]. It su ces to prove that u(t; ;u02) u(t; ;u01) for t 2 [0; ]. To this end, let 17 w(t;x) = u(t;x;u02) u(t;x;u01). Then w(t;x) satis es the following equation, wt(t;x) = Z RN (y x)w(t;y)dy w(t;x) +a(t;x)w(t;x); where a(t;x) = f(x;u(t;x;u02)) +u(t;x;u01) Z 1 0 @uf(x;su(t;x;u02) + (1 s)u(t;x;u01))ds: Let M > 0 be such that M supx2RN;t2[0; ](1 a(t;x)) and ~w(t;x) = eMtw(t;x). Then ~w(t;x) satis es ~wt(t;x) = Z RN (y x) ~w(t;y)dy + [M 1 +a(t;x)] ~w(t;x): Let K: X!X be de ned by (Ku)(x) = Z RN (y x)u(y)dy for u2X: (3.3) Then K generates an analytic semigroup on X and ~w(t; ) = eKt(u02 u01) + Z t 0 eK(t )(M 1 +a( ; )) ~w( ; )d : Observe that eKtu0 0 for any u0 2 X+ and t 0 and eKtu0 0 for any u0 2 X++ and t 0. Observe also that u02 u01 2X++2 . By (2), ~w( ; ) 0 and hence (M 1 + a( ; )) ~w( ; ) 0 for 2 [0;T]. It then follows that ~w(t; ) 0 and then w(t; ) 0 (i.e. u(t; ;u02) u(t; ;u01)) for t2[0; ]. Proposition 3.2 (Global existence). For any given u0 2X+, u(t; ;u0) exists for all t 0. 18 Proof. Let u0 2X+ be given. There is M 1 such that 0 u0(x) M and f(t;x;M) < 0 for all x2H. Then by Proposition 3.1, 0 u(t; ;u0) M for any t > 0 at which u(t; ;u0) exists. It is then not di cult to prove that for any > 0 such that u(t; ;u0) exists on (0; ), limt! u(t; ;u0) exists in X. This implies that u(t; ;u0) exists and u(t; ;u0) 0 for all t 0. 3.2 Convergence on compact subsets and strip type subsets In this section, we explore the convergence property of solutions of (1.3) on compact subsets and strip type subsets. As mentioned before, to indicate the dependence of solutions of (1.3) on the nonlinearity, we may write u(t; ;u0) as u(t; ;u0;f). Proposition 3.3. Suppose that u0n;u0 2X+ (n = 1;2; ) withfku0nkgbeing bounded, and fn, gn (n = 1;2 ) satisfy (H0) with fn(t;x;u), gn(t;x;u), and @ufn(t;x;u) being bounded uniformly in x2H and (t;u) on bounded subsets. (1) (Convergence on compact subsets) If u0n(x) !u0(x) as n!1 uniformly in x on bounded sets and fn(t;x;u) gn(t;x;u) as n!1 uniformly in (t;x;u) on bounded sets, then for each t> 0, u(t;x;u0n;fn) u(t;x;u0;gn)!0 as n!1 uniformly in x on bounded sets. (2) (Convergence on strip type subsets) If u0n(x) !u0(x) as n!1 uniformly in x on any set E with fx jx2Eg being a bounded set of R and fn(t;x;u) gn(t;x;u)!0 as n!1 uniformly in (t;x;u) on any set E with f(t;x ;u)j(t;x;u)2Eg being a bounded set of R3, then for each t> 0, u(t;x;u0n;fn) u(t;x;u0;gn)!0 as n!1 uniformly in x on any set E with fx jx2Eg being a bounded set of R. Proof. (1) We prove the case that H=H2 and A2. Other cases can be proved similarly. 19 Let vn(t;x) = u(t;x;u0n;fn) u(t;x;u0;gn). Then vn(t;x) satis es vnt (t;x) = Z RN (y x)vn(t;y)dy vn(t;x) +an(t;x)vn(t;x) +bn(t;x); where an(t;x) =fn(t;x;u(t;x;u0n;fn)) +u(t;x;u0;fn) Z 1 0 @ufn(t;x;su(t;x;u0n;fn) + (1 s)u(t;x;u0;gn))ds and bn(t;x) = u(t;x;u0;gn) fn(t;x;u(t;x;u0;gn)) gn(t;x;u(t;x;u0;gn)) : Observe thatfan(t;x)gis uniformly bounded and continuous in t and x and bn(t;x)!0 as n!1 uniformly in (t;x) on bounded sets of [0;1) RN. Take a > 0. Let X( ) =fu2C(RN;R)ju( )e k k2Xg with norm kuk = ku( )e k kk. Note that K : X( ) ! X( ) also generates an analytic semigroup, where K is as in (3.3), and there are M > 0 and !> 0 such that ke(K I)tkX( ) Me!t 8t 0; where I is the identity map on X( ). Hence vn(t; ) =e(K I)tvn(0; ) + Z t 0 e(K I)(t )an( ; )vn( ; )d + Z t 0 e(K I)(t )bn( ; )d 20 and then kvn(t; )kX( ) Me!tkvn(0; )kX( ) +M sup 2[0;t];x2RN jan( ;x)j Z t 0 e!(t )kvn( ; )kX( )d +M Z t 0 e!(t )kbn( ; )kX( )d Me!tkvn(0; )kX( ) +M sup 2[0;t];x2RN jan( ;x)j Z t 0 e!(t )kvn( ; )kX( )d + M! sup 2[0;t] kbn( ; )kX( )e!t: By Gronwall?s inequality, kvn(t; )kX( ) e(!+M sup 2[0;t];x2RNjan( ;x)j)t Mkvn(0; )kX( ) + M! sup 2[0;t] kbn( ; )kX( ) : Note that kvn(0; )kX( ) !0 and sup 2[0;t] kbn( ; )kX( ) !0 as n!1: It then follows that kvn(t; )kX( ) !0 as n!1 and then u(t;x;u0n;fn) u(t;x;u0;gn) as n!1 uniformly in x on bounded sets. (2) It can be proved by similar arguments as in (1) with X( ) being replaced by X ( ), where X ( ) =fu2C(H;R)ju ; 2Xg; with norm kukX ( ) =ku ; kX, where u ; (x) = e jx ju(x). 21 3.3 Part metric In this section, we investigate the decreasing property of the so called part metric between two positive solutions of (1.3). Throughout this subsection, we also assume (H0). First, we introduce the notion of part metric. For given u;v2X++, de ne (u;v) = inffln j 1 u v u; 1g: Observe that (u;v) is well de ned and there is 1 such that (u;v) = ln . Moreover, (u;v) = (v;u) and (u;v) = 0 i u v: In literature, (u;v) is called the part metric between u and v. Proposition 3.4 (Strict decreasing of part metric). For any > 0, > 0, M > 0, and > 0 with < M and ln M , there is > 0 such that for any u0;v0 2 X++ with u0(x) M, v0(x) M for x2H and (u0;v0) , there holds (u( ; ;u0);u( ; ;v0)) (u0;v0) : Proof. We give a proof for the case that H = H1 and A = A1. Other cases can be proved similarly. Let > 0, > 0, M > 0, and > 0 be given and < M, < ln M . First, note that by Proposition 3.1, there are 1 > 0 and M1 > 0 such that for any u0 2 X++ with u0(x) M for x2RN, there holds 1 u(t;x;u0) M1 8t2[0; ]; x2RN: (3.4) 22 Let 1 = 21e (1 e ) sup t2[0; ];x2RN;u2[ 1;M1M= ] fu(t;x;u): (3.5) Then 1 > 0 and there is 0 < 1 such that 1 2 1 0. We prove that de ned in (3.9) satis es the property in the proposition. For any u0;v0 2 X++ with u0(x) M and v0(x) M for x 2 R and (u0;v0) , there is 1 such that 1(u0;v0) = ln and 1 u0 v0 u0: Note that e M . We rst show that (u(t; ;u0);u(t; ;v0)) is non-increasing in t> 0. By Proposition 3.1, u(t; ;v0) u(t; ; u0) for t> 0: 23 Let v(t;x) = u(t;x;u0): We then have vt(t;x) = v(t;x) +v(t;x)f(t;x;u(t;x;u0)) = v(t;x) +v(t;x)f(t;x;v(t;x)) +v(t;x)f(t;x;u(t;x;u0)) v(t;x)f(t;x;v(t;x)) v(t;x) +v(t;x)f(t;x;v(t;x)) 8t> 0: By Proposition 3.1 again, u(t; ; u0) u(t; ;u0) and hence u(t; ;v0) u(t; ;u0) for t> 0. Similarly, we can prove that 1 u(t; ;u0) u(t; ;v0) for t> 0. It then follows that (u(t; ;u0);u(t; ;v0)) (u0;v0) 8t 0 and then (u(t2; ;u0);u(t2; ;v0)) (u(t1; ;u0);u(t1; ;v0)) 8 0 t1 t2: 24 Next, we prove that (u( ; ;u0);u( ; ;v0)) (u0;v0) : Note that e M and vt(t;x) = v(t;x) +v(t;x)f(t;x;u(t;x;u0)) = v(t;x) +v(t;x)f(t;x;v(t;x)) +v(t;x)f(t;x;u(t;x;u0)) v(t;x)f(t;x;v(t;x)) v(t;x) +v(t;x)f(t;x;v(t;x)) + 1 80 0 for (t;x) 2R H and ( ; ) 2Xp) and for any 2 ( @t +A ; + a( ; )I), Re 0. We remark that ; (a)2 ( @t +A ; +a( ; )I) and if @t +A ; +a( ; )I admits a principal eigenvalue 0, then 0 = ; (a). We also remark that in the case thatH=Hi and A = Ai with i = 1 or 3, principal eigenvalue of @t +A ; + a( ; )I always exists. But in the case thatH=H2 andA=A2, @t +A ; +a( ; )I may not have a principal eigenvalue (see [17] and [74] for examples). 26 For given a2Xp, let ^a(x) = 1T Z T 0 a(t;x)dt: The following proposition is established in [65] regarding principal eigenvalues of time periodic nonlocal dispersal operators. Proposition 3.5. (1) If ^a( ) is CN and there is x0 2RN such that ^a(x0) = maxx2RN ^a(x0) and the partial derivatives of ^a(x) up to order N 1 at x0 are zero, then for any 2SN 1 and 2R, ; (a) is the principal eigenvalue of @t +A ; +a( ; )I. (2) Let a( ; ) 2Xp be given. For any > 0, there is a ( ; ) 2Xp such that ; (a ) are principal eigenvalues of @t +A ; +a ( ; )I, a (t;x) a(t;x) a+(x;) 8(t;x)2R H; and ; (a+) ; (a) ; (a ) + : Proof. We only need to prove the case that H=H2 and A=A2. (1) It follows from [65, Theorem B(1)]. (2) It follows from [65, Proposition 3.10, Lemma 4.1]. The following proposition shows that the temporal variation does not reduce the prin- cipal spectrum point of dispersal operators. Proposition 3.6. For any given 2SN 1, 2R, and a2Xp, ; (a) ; (^a): Proof. It follows from Theorem 6.5 in [73] (see also [42] for the case thatH=H1 andA=A1 and see [43] for the case that H=H2 and A=A2, 27 3.4.2 Principal eigenvalues of spatially periodic dispersal operators In this subsection, we present some special principal eigenvalue theories for time in- dependent but spatially periodic dispersal operators with random, nonlocal, and discrete dispersals. Recall that, if a(t;x) a(x), the eigenvalue problem (2.16) reduces to the eigenvalue problem (2.17). To be more precise, when H=H1 and A=A1, (2.16) reduces to 8 >>< >>: u(x) 2 ru(x) + (a(x) + 2)u(x) = u(x); x2RN u(x+piei) = u(x); x2RN: (3.10) When H=H2 and A=A2, (2.16) reduces to 8> >< >>: R RN e (y x) (y x)u(y)dy u(x) +a(x)u(x) = u(x); x2RN u(x+piei) = u(x); x2RN: (3.11) When H=H3 and A=A3, (2.16) reduces to 8 >>< >>: P k2Kak(e k u(j +k) u(j)) +a(j)u(j) = u(j); j2ZN u(j +piei) = u(j); j2ZN: (3.12) Observe that when = 0, (3.10), (3.11), and (3.12) are independent of . Observe also that if u(t;x) = e (x t) (x) is a solution of ut(t;x) = u(t;x) +a(x)u(t;x); x2RN (3.13) with ( )2X1;pnf0g, or a solution of ut(t;x) = Z RN k(y x)u(t;y)dy u(t;x) +a(x)u(t;x); x2RN (3.14) 28 with ( )2X2;pnf0g, or a solution of ut(t;j) = X k2K ak(u(t;x+j) u(t;j)) +a(j)u(t;j); j2ZN (3.15) with ( ) 2X3;pnf0g, then is an eigenvalue of (3.10) or (3.11) or (3.12) with ( ) being a corresponding eigenfunction. If a(x) = f(x;0), then (3.13) (resp. (3.14), (3.15)) is the linearized equation of (1.3) with f(t;x;u) = f(x;u) and H = H1 and A1 (resp. H = H2 and A=A2, H=H3 and A=A3) at u = 0. For given ai( )2Xi;p, let ^^ai be the space average of ai( ) (i = 1;2;3), that is, 8 >>>> >>< >>>> >>: ^^ai = 1 jDij R Diai(x)dx for i = 1;2 ^^a3 = 1 #D3 P j2D3 a3(j); (3.16) where Di = [0;p1] [0;p2] [0;pN]\Hi; i = 1;2;3 (3.17) and 8 >>> >>> < >>> >>> : jDij= p1 p2 pN for i = 1;2 #D3 = the cardinality of D3: (3.18) The following proposition shows a relation between ; (ai) and ; (^^ai) for ai2Xi;p. Proposition 3.7 (In uence of spatial variation). For given 1 i 3, 2R, 2SN 1, and ai2Xi;p, there holds ; (ai) ; (^^ai): Proof. The case i = 1 is well known. The cases i = 2 and 3 follow from [35, Theorem 2.1]. 29 We remark that ; (^^ai) (ai2Xi;p, i = 1;2;3) have the following explicit expressions, 8 >>> >>>> >>> >>>> < >>> >>> >>> >>>> >: ; (^^a1) = ^^a1 + 2 ; (^^a2) = RRN e z (z)dz 1 + ^^a2 ; (^^a3) = Pk2Kak(e k 1) + ^^a3: (3.19) 3.5 Positive solutions and spreading speeds of KPP equations In this section, we recall some existing results on the existence, uniqueness, and stability of time and space periodic positive solutions and spatial spreading speeds of (1.1). 3.5.1 Time periodic positive solutions and spreading speeds of KPP equations in periodic media A solution u(t;x) of (1.1) is called time and space periodic solution if it is a solution on t2R and u(t + T;x) = u(t;x + piei) = u(t;x) for t2R, x2H, and i = 1;2; ;N. It is called a positive solution if u(t;x) > 0 for all t in the existence interval and x2H. Proposition 3.8. Consider (1.1) and assume that f0 satis es (H0) and f0(t;x+piei;u) = f0(t;x;u) (i = 1;2; ;N). (1) (Uniqueness of periodic positive solutions) If (1.1) has a time and space periodic pos- itive solution, then it is unique. (2) (Stability of periodic positive solutions) If (1.1) has a time and space periodic positive solution u (t;x), then it is globally asymptotically stable with respect to perturbations in X+p nf0g. 30 (3) (Existence of periodic positive solutions) If (f0( ; ;0)) > 0, then (1.1) has a time and space periodic positive solution. Proof. The case that H = H1 and A = A1 follows from the results in [57]. The case that H = H2 and A = A2 follows from the results in [65]. The case that H = H3 and A = A3 can be proved by the similar arguments as in [65]. Corollary 3.1. (1) If ( ^f0( ;0)) > 0, then (1.1) has a time and space periodic positive solution. (2) If ( ^^f0(0)) > 0, then (1.1) has a time and space periodic positive solution. Proof. (1) follows from Propositions 3.6 and 3.8. (2) follows from Propositions 3.6, 3.7 and 3.8. Proposition 3.9. Consider (1.1). Assume that f0 satis es (H0), f0(t;x+piei;u) = f0(t;x;u) (i = 1;2; ;N), and (f0( ; ;0)) > 0. Then for any given 2SN 1, (1.1) has a spatial spreading speed c 0( ) in the direction of . Moreover, c 0( ) = inf >0 ; (f0( ; ;0)) (3.20) and for any c 0. Then for any given 2 SN 1, (1.1) has a spatial spreading speed c 0( ) in the direction of and c 0( ;f0) c 0( ^f0) c 0( ^^f0): Proof. First, by Propositions 3.6, 3.7, and 3.9, c 0( ;f0, c 0( ; ^f0), and c 0( ; ^^f0) exist. More- over, by Propositions 3.6 and 3.7 again, and by (3.20), c 0( ;f0) c 0( ^f0) c 0( ^^f0): 3.5.2 KPP equations in spatially periodic media In this subsection, we recall some additional spatial spreading dynamics of KPP equa- tions in spatially periodic media. Consider (1.1). Throughout this subsection, we assume that f0(t;x;u) f0(x;u). Proposition 3.10 (Spreading speeds). Consider (1.1). Assume that f0(t;x;u) = f0(x;u) satis es (H0), f0(x+piei;u) = f0(x;u) (i = 1;2; ;N), and (f0( ;0)) > 0. Then for any given 2SN 1, (1.1) has a spatial spreading speed c 0( ) in the direction of . Moreover, c 0( ) is of the following spreading features. (1) For each u0 2X+ satisfying that u0(x) = 0 for x2H with jx j 1, lim sup jx j ct;t!1 u(t;x;u0;f0( ; )) = 0 8c> maxfc 0( );c 0( )g: (2) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for x2Hwithjx j r, lim sup jx j ct;t!1 ju(t;x;u0;f0( ; )) u 0(x;f0( ; ))j= 0 32 for all 0 sup 2SN 1 c 0( ): (4) For each > 0, r> 0, and u0 2X+ satisfying that u0(x) for x2H withkxk r, lim sup kxk ct;t!1 ju(t;x;u0;f0( ; )) u 0(x)j= 0 80 0: It then su ces to prove that the limit in(4.2) exists for = 0. Let 0 1 be such that (u0;u (0; )) = ln 0 and 1 0u (0;x) u0(x) 0u (0;x) 8x2H: By Proposition 3.4, there is 1 1 such that limt!1 (u(t; ;u0);u (t; )) = ln 1: Moreover, by (4.1), (u(t; ;u0);u (t; )) (u0;u (0; )) = ln 0 and hence 1 0u (t;x) u0(t;x;u0) 0u (t;x) 8t> 0; x2H: (4.3) 35 If 1 = 1, then for any > 0, there is > 0 such that for t , (u(t; ;u0);u (t; )) ln(1 + ): This implies that 1 1 + u (t;x) u(t;x;u0) (1 + )u (t;x) 8t ; x2H: Hence ju(t;x;u0) u (t;x)j u (t;x) 8t ; x2H: It then follows that limt!1ku(t; ;u0) u (t; )k= 0: Assume 1> 1. By (4.3), there are > 0, M > 0, and > 0 such that u(t;x;u0) M; u (t;x) M 8t 0; x2H and (u(t; ;u0);u (t; )) 8t 0: By Proposition 3.4 again, there is > 0 such that for any n 1, (u(nT; ;u0);u (nT; )) (u((n 1)T; ;u0);u ((n 1)T; )) and hence (u(nT; ;u0);u (nT; )) (u0;u (0; )) n 8n 1: Let n!1, we have limn!1 (u(nT; ;u0);u (nT; )) = 1: 36 This is a contradiction. Therefore, we must have 1 = 1 and limt!1ku(t; ;u0) u (t; )k= 0: 4.2 Existence In this subsection, we show the existence of time periodic strictly positive solutions of (1.3), i.e., show Theorem 2.1(3). To this end, we rst prove some lemmas. Throughout this subsection, we assume the conditions in Theorem 2.1(3). Then by Proposition 3.8, (1.1) has a unique time and space periodic positive solution u 0(t;x). Let 0 > 0 be such that 0 < 0 < inf (t;x)2R H u 0(t;x): Let 0 : R!R+ be a non-increasing smooth function such that 0 0( ) 0; lim infr! 1 0(r) > 0; 0(r) = 0 8r 1: (4.4) Lemma 4.1. For give 2SN 1, let u0(x) = 0(x ) and un; (x) = u0(x + n ) (n2N). There are K 0 and n 0 such that u(KT; ;un ; ;f) un ; ( ): Proof. Let > 0 be such that 0 < inf (t;x)2R H u 0(t;x) : 37 Fix 0 0 such that u(t;x;un ; ) ~ 0 for t KT and x M. Proof. By Proposition 3.1 and Lemma 4.1, u(mKT; ;un ; ;f) un ; ( ) 8m 1: (4.9) 39 Moreover, by the arguments of Lemma 4.1, u(mKT;x;un ; ;f) 0 8m 1; x n : It then su ces to prove that for any M 1, inf t2[KT;(K+1)T];x2H;x 2[ n ;M] u(t;x;un ; ;f) > 0: (4.10) Suppose that (4.10) does not hold. Then there are tn 2 [KT;(K + 1)T] and xn 2H with xn 2[ n ;M], kxnk!1 such that u(tn;xn;un ; ;f)!0 as n!1: (4.11) Without loss of generality, we may assume that tn!t as n!1for some t 2[KT;(K + 1)T]. We then have ku(tn; ;un ; ;f) u(t ; ;un ; ;f)k!0 as n!1 and hence u(tn;xn;un ; ;f) u(t ;xn;un ; ;f)!0 as n!1: Observe that u(t ;xn;un ; ;f) = u(t ;0;un ; ( +xn);f( ; +xn; )) 8n 1 and f(t;x+xn;u) f0(t;x+xn;u)!0 40 as n!1 uniformly in (t;x;u) on bounded sets. Observe also that there is n n such that un ; ( +xn) un ; ( ) 8n 1: By Propositions 3.1 and 3.3, we have u(t ;0;un ; ( +xn);f( ; +xn; )) u(t ;0;un ; ( );f( ; +xn; )) and u(t ;0;un ; ( );f( ; +xn; )) u(t ;0;un ; ( );f0( ; +xn; ))!0 (4.12) as n!1. Without loss of generality, we may also assume that there is ~x2H such that f0(t;x+xn;u)!f0(t;x+ ~x;u) as n!1 uniformly in (t;x;u) on bounded sets. Then by Proposition 3.3 again, u(t ;0;un ; ( );f0( ; +xn; ))!u(t ;0;un ; ( );f0( ; + ~x; )) as n!1: (4.13) By Proposition 3.1, u(t ;0;un ; ( );f0( ; + ~x; )) > 0: (4.14) It then follows from (4.12), (4.13), and (4.14) that lim infn!1 u(tn;xn;un ; ;f) > 0; which contradicts to (4.11). Therefore, (4.10) holds and the lemma thus follows. Observe that for any M M0, u(t;x) M is a supersolution of (1.3) on H. Hence u(T;x;M;f) M 41 and then by Proposition 3.1, u(nT;x;M;f) decreases at n increases. De ne u+(x) := limn!1u(nT;x;M;f): (4.15) Then u+(x) is a Lebesgue measurable and upper semi-continuous function. In the following, we x an M maxfM0; 0g. Lemma 4.3. There exists > 0 such that u+(x) for x2RN. Proof. Let 0( ) be as in (4.4) and u i(x) = 0( x ei); i = 1;2; ;N: By Proposition 3.1, u(t; ;M;f) u(t; ;u i;f) for t 0 and i = 1;2; ;N. By Lemma 4.2, there is and T > 0 such that u(t;x;u i;f) 8t T; x ei 0; i = 1;2; ;N: It then follows that u(t;x;M;f) 8t T; x2H: This implies that u(mT;x;M;f) 8m 1; x2H and then u+(x) 8x2H: The lemma thus follows. Now we prove the existence of time periodic positive solutions 42 Proof of Theorem 2.1(3). We rst claim that (u(nT; ; =2);u(nT; ;M))!0 (4.16) as n!1. Assume this is not true. Let n = (u(nT; ; =2);u(nT; ;M)) and 1 = limn!1 n (the existence of this limit follows from Proposition 3.4). Then 1> 0, e 0 1 e 0u(nT; ;M) u(nT; ; =2) e 0u(nT; ;M) e 0M for n = 0;1;2; and (u(nT; ; =2);u(nT; ;M)) 1 for n = 1;2; . By Proposition 3.4, there is > 0 such that (u(nT; ; =2);u(nT; ;M)) 0 n 8n = 1;2; : This implies that 1 = 1, a contradiction. Therefore, (4.16) holds. By (4.16), there is K1 1 such that =2 u(K1T; ; =2) and then =2 u(nK1T; ; =2) u(nK1T; ;M) M 8n = 1;2; : 43 It then follows that u(nK1T;x;M) u+(x) u(nK1T;x; =2) 8x2H; n = 1;2; : Therefore 0 u(nK1T;x;M) u+(x) u(nK1T;x;M) u(nK1T;x; =2) u(nK1T;x;M)(1 1e n ) M(1 1e n ): This implies that limn!1u(nK1T;x;M) = u+(x) uniformly in x2H and u+( )2X++. Moreover, by u(nK1T; ;M) u(kT; ;M) u((n+ 1)K1; ;M) 8nK1 k (n+ 1)K1; we have lim k!1 u(kT;x;M) = u+(x) uniformly in x2H and then u(T; ;u+) = u+( ): This implies that u (t;x) = u(t;x;u+) is a time periodic strictly positive solutions of (1.3). 4.3 Tail property In this section, we prove the tail property of time periodic strictly positive solutions of (1.3). Throughout this subsection, we assume the conditions in Theorem 2.1 (4). 44 Proof of Theorem 2.1 (4). Suppose that u (t;x) is a time periodic strictly positive solution of (1.3). Observe that u (t;x) = u(t;x;u+), where u+ is as in the proof of Theorem 2.1(3). We claim lim r!1 sup x2H;kxk r ju (t;x) u 0(t;x)j= 0: (4.17) To prove (4.17), we rst show that limr!1 sup x2H;kxk r ju+(x) u+0 (x)j= 0: (4.18) Recall u+(x) := limn!1u(nT;x;M;f) and u+0 (x) := limn!1u(nT;x;M;f0): Assume (4.18) is not true. Then there exists 0 > 0 andfxng2R withkxkk!1such that ju+(xk) u+0 (xk)j> 0 for k 1. Since both u(nT;x;M;f)!u+(x) and u(nT;x;M;f0)!u+0 (x) uniformly on x2H, there is N such that for n N, ju+(nT;xk;M;f) u+(nT;xk;M;f0)j> 0 8k 1: (4.19) 45 Note that there is ~x0 2H such that f0(t;x+xk;u)!f0(t;x+ ~x0;u) as k!1 uniformly in (t;x;u) on bounded sets. Note also that f(t;x+xk;u) f0(t;x+xk;u)!0 as k!1 uniformly in (t;x;u) on bounded sets. Hence f(t;x+xk;u)!f0(t;x+ ~x0;u) as k!1 uniformly in (t;x;u) on bounded sets. Then by Proposition 3.3, ju(NT;xk;M;f) u(NT;xk;M;f0)j =ju(NT;0;M;f( ; +xk; )) u(NT;0;M;f0( ; +xk; ))j ju(NT;0;M;f( ; +xk; )) u(NT;0;M;f0( ; + ~x0; ))j +ju(NT;0;M;f0( ; + ~x0; )) u(NT;0;M;f0( ; +xk; ))j !0 as k!1. This contradicts to (4.19). Therefore, (4.18) holds. Now we prove (4.17). Recall that u (t;x) = u(t;x;u+;f) and u 0(t;x) = u(t;x;u+0 ;f0): 46 Suppose that (4.17) does not hold for some t> 0. Then there are xk 2H with kxkk!1 and 0 > 0 such that ju(t;xk;u+;f) u(t;xk;u+0 ;f0)j 0 8k 1: Hence ju(t;xk;u+;f) u(t;x;u+0 ;f)j=ju(t;0;u+( +xk);f( +xk)) u(t;0;u+0 ( +xk);f0( +xk))j 0 for all k 1. By (H1), (4.18), Proposition 3.3, and the arguments in the proof of (4.18), lim k!1 [u(t;0;u+( +xk);f( +xk)) u(t;0;u+0 ( +xk);f0( +xk))] = 0: This is a contradicts again. Therefore, (4.17) holds. 47 Chapter 5 Spatial Spreading Speeds In this chapter, we investigate the spatial spreading speeds of (1.3) and prove Theorem 2.2. We rst prove a Lemma. Throughout this section, we assume the conditions in Theorem 2.2. Let u 0(t;x) be the unique time and space periodic positive solution of (1.1). Let 0 > 0 be such that 0 < 0 < inf (t;x)2R H u 0(t;x): Lemma 5.1. Let 2SN 1, c> 0 and u0 2X+ be given. If lim inf x ct;t!1 u(t;x;u0;f) > 0; then for any 0 0. Then there are and T > 0 such that u(t;x;u0;f) 8(t;x)2R+ H; x ct; t T : Assume that the conclusion of (1) is not true. Then there are 0 < c0 < c, 0 > 0, xn 2H, and tn2R+ with xn c0tn and tn!1 such that 48 ju(tn;xn;u0;f) u (tn;xn)j 0 8n 1: (5.1) Note that there are kn 2 Z+ and n 2 [0;T] such that tn = knT + n. Without loss of generality, we may assume that n! and xn!x as n !1 in the case that fkxnkg is bounded (this implies that f(t + tn;x + xn;u) ! f(t+ ;x+x ;u) uniformly in (t;x;u) in bounded sets) and f(t+tn;x+xn;u) f0(t+ ;x+xn;u)!0 as n!1 uniformly in (t;x;u) on bounded sets in the case that fkxnkg is unbounded. Let ~u0 2X+, ~u0(x) = 8x2H: Let M = sup x2H u0(x): By Theorem 2.1, there is ~T > 0 such that ju(t;x;M;f) u (t;x)j< 02 8t ~T; x2H; (5.2) ju( ~T;x; ~u0;f( + ; +x ; )) u ( ~T + ;x+x )j< 02 ; 8x2H; 2R (5.3) and ju( ~T;x; ~u0;f0( + ; ; )) u 0( ~T + ;x)j< 02 8x2H; 2R: (5.4) Without loss of generality, we may assume that tn ~T T for n 1. Let ~u0n2X+ be such that ~u0n(x) = for x c 0 +c 2 (tn ~T); 49 0 ~u0n(x) for c 0 +c 2 (tn ~T) x c(tn ~T); and ~u0n(x) = 0 for x c(tn ~T): Then u(tn ~T; ;u0;f) ~u0n( ) and hence u(tn;xn;u0;f) = u( ~T;xn;u(tn ~T; ;u0;f);f( +tn ~T; ; )) = u( ~T;0;u(tn ~T; +xn;u0;f);f( +tn ~T; +xn; )) u( ~T;0; ~u0n( +xn);f( +tn ~T; +xn; )): (5.5) Observe that ~u0n(x+xn)! ~u0(x) as n!1 uniformly in x on bounded sets. In the case that f(tn +t;x+xn;u) f0(t+ ;x+xn;u)!0 as n!1, by Proposition 3.3, u( ~T;0; ~u0n( +xn);f( +tn ~T; +xn; )) u( ~T;0; ~u0;f0( + ~T; +xn; ))!0 as n!1. Then by (5.4) and (5.5), u(tn;xn;u0;f) >u 0( ;xn) 0=2 for n 1: (5.6) By Theorem 2.1(4), 50 u 0( ;xn) >u ( ;xn) 0=2 for n 1: (5.7) By Proposition 3.1 and (5.2), u(tn;xn;u0;f) u(tn;xn;M;f) u (tn;xn) + 0 8n 1: (5.6), (5.7), and the continuity of u (t;x) imply, ju(tn;xn;u0;f) u (tn;xn)j< 0 for n 1: This contradicts to (5.1). In the case that xn!x , by Proposition 3.3 again, u( ~T;0; ~u0n( +xn);f( +tn ~T; +xn; ))!u( ~T;0; ~u0;f( + ~T; +x ; )) as n!1. By (5.3) and (5.5), u(tn;xn;u0;f) >u ( ;x ) 0=2 for n 1: (5.8) By the continuity of u ( ; ), u ( ;x ) >u ( n;xn) 0=2 for n 1: (5.9) By Proposition 3.1 and (5.2), u(tn;xn;u0;f) u(tn;xn;M;f) u (tn;xn) + 0 8n 1: This together with (5.8), and (5.9) implies that ju(tn;xn;u0;f) u (tn;xn)j< 0 for n 1: 51 This contradicts to (5.1) again. Hence lim sup x c0t;t!1 ju(t;x;u0;f) u (t;x)j= 0 for all 0 0 such that c 0( ) = ; ( )(f0( ; ;0)) ( ) and ; (f0( ; ;0)) >c 0( ) 8 0 < < ( ): We now prove Theorem 2.2. Proof of Theorem 2.2. We rst show that for any c>c 0( ), lim sup x ct;t!1 u(t;x;u0;f) = 0: (5.10) Let 0 < < ( ) be such that c = ; (f0( ; ;0)) : By Proposition 3.5, for any > 0, there are > 0 and a( ; )2Xp such that a(t;x) f0(t;x;0) + ; c 0( ) < ; (a) < ; (f0( ; ;0)) 0 such that f(t;x;u) f(t;x;0) a(t;x) 8t2R; x ~M; u 0: (5.12) Let M ku0k be such that f(t;x;uM(t;x)) a(t;x) 8t 0; x ~M: (5.13) It then follows from (5.11), (5.12), and (5.13) that uM(t;x) is a super-solution of (1.3). By Proposition 3.1, u(t;x;u0;f) uM(t;x) = Me (x ; (a) t) (t;x) 8t 0; x2H: This implies that (5.10) holds. Next, we prove that for any c 0 such that c< inf >0 ; (f0( ; ;0) ) 0 such that f(t;x;u) f0(t;x;u) 8t2R; x M; 0 u 1: (5.15) By Lemma 4.2, there are ~ > 0 and ~T > 0 such that u(t;x;u0;f) ~ 8t ~T; x M: (5.16) Consider equation ut = (Au)(t;x) + [f0(t;x;u) K u]u(x); x2H (5.17) By Proposition 3.8, (5.17) has a unique time and space periodic solution u 0;K (t;x;). Let K 1 be such that u0;K ( ~T;x) ~ : Let ~u0 2X+( ) be such that ~u0(x) minfu 0;K ( ~T;x);u( ~T;x;u0;f)g: By Proposition 3.1, u(t+ ~T;x;u0;f) u(t;x; ~u0; ~f0( + ~T; ; )) 8t 0; x M; where ~f0(t;x;u) = f0(t;x;u) K u. By Proposition 3.9, for any c0 ; (f0 ) , lim sup x c0t;t!1 ju(t;x; ~u0; ~f0( +~; ; )) u 0;K (t+ ~T;x)j= 0: (5.18) 54 By (5.16) and (5.18), lim inf x c0t;t!1 u(t;x;u0;f) > 0: This together with Lemma 5.1 implies that (5.14) holds. 55 Chapter 6 Another Method to Show the Time Independent Case In this chapter, we consider time independent monostable equations and provide some other method to prove Theorems 2.1 and 2.2. We also show Theorem 2.3. 6.1 Another method to prove Theorem 2.1 In this section, we investigate the existence of positive stationary solutions of (1.3) in the special case that f(t;x;u) f(x;u) and f0(x;u) f0(u) and give another proof of Theorem 2.1. Throughout this section, we assume f(t;x;u) f(x;u), f0(x;u) f0(u) and (H0), (H1)0 and (H2). We rst prove some lemmas. For convenience, we denote f(x;u) and f0(u) by fi(x;u) and f0i (u), respectively, in the case H=Hi and A=Ai for i = 1;2;3. We may write u(t;x;u0;fi) as ui(t;x;u0). Lemma 6.1. For any 1 i 3 and > 0, there are p = (p1;p2; ;pN) 2 NN and hi2Xi;p\CN(Hi;R) such that fi(x;0) hi(x) for x2Hi; ^^h i f0i (0) (hence (hi( )) f0i (0) ); and for the cases that i = 1 and 2, the partial derivatives of hi(x) up to order N 1 are zero at some x0 2Hi with hi(x0) = maxx2Hi(x), where ^^hi is the average of hi( ) (see (3.16) for the de nition). 56 Proof. Fix 1 i 3. By (H1)0, there are 0 < 0 1 and L0 > 0 such that fi(x;0) f0i (0) 0 for x2Hi; kxk L0: Let M0 = inf x2Hi;1 i 3 fi(x;0): Let h0 : R![0;1] be a smooth function such that h0(s) = 8 >>> >>>< >>> >>>: 1 for jsj 1 0 for jsj 2: For any p = (p1;p2; ;pN)2NN with pj > 4L0, let hi2Xi;p\CN(Hi;R) (i = 1;2;3) be such that hi(x) = f0i (0) 0 h0 kxk 2 L20 (f0 i (0) 0 M0) for x2 [ p12 ;p12 ] [ p22 ;p22 ] [ pN2 ;pN2 ] \Hi: Then fi(x;0) hi(x) 8x2Hi; 1 i 3: It is clear that for i = 1 or 2, the partial derivatives of hi(x) up to order N 1 are zero at some x0 2Hi with hi(x0) = maxx2Hihi(x)(= f0i (0) 0). For given > 0, choosing pj 1, we have ^^h i >f0i (0) : By Proposition 3.7, (hi( )) (^^hi) = ^^hi and hence (hi( )) f0i (0) : 57 The lemma is thus proved. Lemma 6.2. Suppose that ~u 2 : RN ![ 0;M0] is Lebesgue measurable, where 0 and M0 are two positive constants. If Z RN (y x)~u 2(y)dy ~u 2(x) + ~u 2(x) ~f2(x; ~u 2(x)) = 0 8x2RN; where ~f2(x;u) = f2(x;u) or f02 (u) for all x2RN and u2R, then ~u 2( )2X++2 . Proof. We prove the case that ~f2(x;u) = f2(x;u). The case that ~f2(x;u) = f02 (u) can be proved similarly. Let h (x) = Z RN (y x)~u 2(y)dy for x2RN: Then h ( ) is C1 and has bounded rst order partial derivatives. Let F(x; ) = h (x) + f2(x; ) 8x2RN; 2R: Then F : RN R!R is C1 and F(x; ~u 2(x)) = 0 for each x2RN. If > 0 is such that F(x; ) = 0, then 1 +f2(x; ) = h (x) < 0 and hence @ F(x; ) = 1 +f2(x; ) + @uf2(x; ) < 0: By Implicit Function Theorem, ~u 2(x) is C1 in x. Moreover, @~u 2(x) @xj = @h (x) @xj 1 +f(x; ~u 2(x)) +@uf2(x; ~u 2(x))~u 2(x) 8x2R N; 1 j N: Therefore, ~u 2 has bounded rst order partial derivatives. It then follows that ~u 2(x) is uni- formly continuous in x2RN and then ~u 2 2X++2 . 58 Lemma 6.3. Suppose that u i( )2X++i and u = u i( ) is a stationary solution of (1.3) with H=Hi, A=Ai, and f(t;x;u) = fi(x;u). Then u i(x)!u0i as kxk!1; where u0i is a positive constant such that f0i (u0i) = 0. Proof. We rst prove that u 1(x)!u01 as kxk!1: Assume that u 1(x)6!u01 as kxk!1. Then there are 0 > 0 and xn2RN such that kxnk!1 and ju 1(xn) u01j 0 for n = 1;2; : By the uniform continuity of u 1(x) in x2RN, without loss of generality, we may assume that there is a continuous function ~u 1 : RN ![ 0;M0] for some 0;M0 > 0 such that u1(x+xn)! ~u 1(x) as n!1 uniformly in x on bounded sets. Moreover, by a priori estimates for parabolic equations, ~u 1 is C2+ for some > 0 and we may also assume that u1(x+xn)! ~u 1(x) as n!1 uniformly in x on bounded sets. This together with f1(x + xn;u) !f01 (u) as n!1 uniformly in x on bounded sets and in u2R implies that ~u 1 + ~u 1f01 (~u 1) = 0; x2RN: 59 By Proposition 3.8, we must have ~u 1(x) u 1(x;f01 ( )) u01 and hence u 1(xn)!u01 as n!1: This is a contradiction. Therefore u 1(x)!u01 as kxk!1: Next, we prove that u 2(x)!u02 as kxk!1: Similarly, assume that u 2(x) 6!u02 as kxk!1. Then there are 0 > 0 and xn 2RN such that kxnk!1 and ju 2(xn) u02j 0 for n = 1;2; : By the uniform continuity of u 2(x) in x2RN, without loss of generality, we may assume that there is a continuous function ~u 2 : RN ![ 0;M0] for some 0;M0 > 0 such that u 2(x+xn)! ~u 2(x) as n ! 1 uniformly in x on bounded sets. By the Lebesgue Dominated Convergence Theorem, we have Z RN (y x)~u 2(y)dy ~u 2(x) + ~u 2(x)f02 (~u 2(x)) = 0 8x2RN: By Lemma 6.2, ~u 2 2X++2 . By Proposition 3.8 again, we have ~u 2(x) u02 and then u 2(xn)! u02 as n!1. This is a contradiction. Therefore u 2(x)!u02 as kxk!1. 60 Finally, it can be proved by the similar arguments as in the case i = 2 that u 3(j)!u03 as kjk!1: Lemma 6.4. There is u i 2 X++i such that for any > 0 su ciently small, u(t;x; u i ) is increasing in t > 0 and u ; ; 2X++i , where u ; ; (x) = limt!1u(t;x; u i ), and hence u = u ; ; ( ) is a stationary solution of (1.3) in X++i in the case H = H1, A = Ai, and f(t;x;u) = fi(x;u) (i = 1;2;3). Proof. Fix 1 i 3. Let M > 0 be such that fi(x;M ) < 0. Let > 0 be such that f0i (0) > 0: By Lemma 6.1, there are p2NN and hi( )2Xi;p\CN(Hi;R) such that fi(x;0) hi(x); and ^hi f0i (0) (> 0): Moreover, for i = 1 or 2, the partial derivatives of hi(x) up to order N 1 are zero at some x0 2Hi with hi(x0) = maxx2Hihi(x). Let u i be the positive principal eigenfunction of Ai +hi( )I withku ik= 1 (the existence of u i is well known in the case that i = 1 or 3 and follows from Proposition 3.5 in the case that i = 2). It is not di cult to verify that u = u i is a sub-solution of (1.3) for any > 0 su ciently small. It then follows that for any > 0 su ciently small, u i ( ) u(t1; ; u i ) ui(t2; ; u i ) 80 0 such that limt!1u(t;x; u i ) = u ; ; i (x) 8x2Hi: Moreover, by regularity and a priori estimates for parabolic equations, u ; ; 1 2X++1 . It is clear that u ; ; 3 2X++3 . By Lemma 6.2, u ; ; 2 2X++2 . Therefore for 1 i 3, u ; ; i 2X++i and u = u ; ; i ( ) is a stationary solution of (1.3) in X++i (i = 1;2;3). Lemma 6.5. Let M 1 be such that fi(x;M) < 0 for x 2 Hi (i = 1;2;3). Then limt!1u(t;x;u0) exists for every x2Hi, where u0(x) M. Moreover, u+; ;Mi ( ) 2X++i , where u+; ;Mi (x) := limt!1ui(t;x;u0), and hence u = u+; ;Mi ( ) is a stationary solution of (1.3) in X++i in the case H=Hi and A=Ai (i = 1;2;3). Proof. Fix 1 i 3. For any M > 1 with fi(x;M) < 0 for all x 2Hi, u = M is a super-solution of (1.3). Hence u(t2; ;M) u(t1; ;M) M 80 t1 1 such that i(u1; i ;u2; i ) = ln > 0: Note that 1 u 1; i u 2; i u1; i : By Lemma 6.3, lim kxk!1 u1; i (x) = u0i and lim kxk!1 u2; i (x) = u0i: This implies that there is > 0 such that 1 u 1; i (x) u 2; i (x) ( )u1; i (x) for kxk 1: By Proposition 3.1 and the arguments in Proposition 3.4, 1 u 1; i (x) 0 su ciently small and M > 0 su ciently large such that u i u0 M and u = u i is a sub-solution of (1.3) (u i is as in Lemma 6.4) and u = M is a super-solution of (1.3). Then u i ui(t; ; u i ) ui(t; ;u0) ui(t; ;M) M 8t 0: By (1), Lemmas 6.4 and 6.5, and Dini?s Theorem, ui(t;x; u i ) 0; x2Hi and limt!1ui(t;x; u i ) = limt!1ui(t;x;M) = u i(x) uniformly in x on bounded sets. It then follows that limt!1ui(t;x;u0) = u i(x) uniformly in x on bounded sets. We claim that kui(t; ;u0) u i( )k!0 as t!1. Assume the claim is not true. Then there are 0 > 0, tn!1, and xn with kxnk!1 such that jui(tn;xn;u0) u i(xn)j 0 8n2N: Then by Lemma 6.3, jui(tn;xn;u0) u0ij 02 8n 1: Let ~ > 0 and ~M > 0 be such that ~ ui(t; ;u0) ~M 8t 0: 64 For any > 0, let T > 0 be such that jui(T; ; ~ ;f0i ( )) u0ij< ; jui(T; ; ~M;f0i ( )) u0ij< : (6.1) Observe that ~ ui(tn T;xn +x;u0) ~M and ui(tn;xn + ;u0) = ui(T;xn + ;ui(tn T; ;u0)) = ui(T; ;ui(tn T; +xn;u0);fi( +xn; )) for n 1. Then ui(T; ; ~ ;fi( +xn)) ui(tn;xn + ;u0) ui(T; ; ~M;fi( +xn; )): (6.2) Observe also that fi(x+xn;u)!f0i (u) as n!1 uniformly in (x;u) on bounded sets. Then by Proposition 3.3, ui(T;x; ~ ;fi( +xn; ))!ui(T;x; ~ ;f0i ( )) and ui(T;x; ~M;fi( +xn; ))!ui(T;x; ~M;f0i ( )) as n!1 uniformly in x on bounded sets. This together with (6.1) implies that jui(T;0; ~ ;fi( +xn; )) u0ij< 2 ; jui(T;0; ~M;fi( +xn; )) u0ij< 2 for n 1 65 and then by (6.2), jui(tn;xn;u0) u0ij< 2 for n 1: Hence limn!1ui(tn;xn;u0) = u0i; which is a contradiction. Therefore kui(t; ;u0) u i( )k!0 as t!1. Finally, note that (4) follows from Lemma 6.3. 6.2 Spatial Spreading Speeds and Proofs of Theorems 2.2 and 2.3 In this section, we explore the spreading speeds of (1.3) in the special case thatf(t;x;u) = f(x;u) and f0(x;u) = f0(u). We provide another proof of Theorem 2.2 and give a proof of Theorem 2.3. Throughout this section, we assume f(t;x;u) f(x;u) and f0(x;u) f0(u), and assume (H0), (H1)0 and (H2). For convenience again, we denote f(x;u) and f0(u) by fi(x;u) and f0i (u), respectively, in the case H=Hi and A=Ai for i = 1;2;3. We may write u(t;x;u0;fi) as ui(t;x;u0) or ui(t;x;u0;fi), write c 0( ;f0i ) as c0i( ), and ; (f0i (0)) as i( ; ;f0i (0)). We rst prove two lemmas. Lemma 6.6. Let 2SN 1, c> 0, 1 i 3, and u0 2X+i be given. (1) If lim infx ct;t!1ui(t;x;u0) > 0, then for any 0 0, then for any 0 0, then for any 0 0. Then there are and T > 0 such that ui(t;x;u0) 8(t;x)2R+ Hi; x ct; t T: Assume that the conclusion of (1) is not true. Then there are 0 < c0 < c, 0 > 0, xn2Hi, and tn2R+ with xn c0tn and tn!1 such that jui(tn;xn;u0) u i(xn)j 0 8n 1: (6.3) Without loss of generality, we may assume that xn!x as n!1in the case thatfkxnkgis bounded (this implies that fi(x+xn;u)!fi(x+x ;u) uniformly in (x;u) in bounded sets) and fi(x+xn;u)!f0i (u) as n!1 uniformly in (x;u) on bounded sets in the case that fkxnkg is unbounded. Let ~u0 2X+i , ~u0(x) = 8x2Hi: 67 By Theorem 2.1, there is ~T > 0 such that ui( ~T;x;u0) u i(x) < 0 8x2Hi; (6.4) jui( ~T;x; ~u0;fi( +x ; )) u i(x+x )j< 02 ; (6.5) and jui( ~T;x; ~u0;f0i ) u0ij< 02 : (6.6) Without loss of generality, we may assume that tn ~T T for n 1. Let ~u0n2X+i be such that 8 >>> >>> >>>> >>> >< >>> >>>> >>> >>>> : ~u0n(x) = for x c0+c2 (tn ~T) 0 ~u0n(x) for c0+c2 (tn ~T) x c(tn ~T) ~u0n(x) = 0 for x c(tn ~T): Then ui(tn ~T; ;u0) ~u0n( ) and hence ui(tn;xn;u0) = ui( ~T;xn;ui(tn ~T; ;u0)) = ui( ~T;0;ui(tn ~T; +xn;u0);fi( +xn; )) ui( ~T;0; ~u0n( +xn);fi( +xn; )): (6.7) Observe that ~u0n(x+xn)! ~u0 68 as n!1 uniformly in x on bounded sets. In the case that fi(x+xn;u)!f0i (u); by Proposition 3.3, ui( ~T;0; ~u0n( +xn);fi( +xn; ))!ui( ~T;0; ~u0;f0i ( )) as n!1. By (6.6) and (6.7), ui(tn;xn;u0) >u0i 0=2 for n 1: (6.8) By Lemma 6.3, u0i >u i(xn) 0=2 for n 1: (6.9) By (6.4), (6.8), and (6.9), jui(tn;xn;u0) u i(xn)j< 0 for n 1: This contradicts to (6.3). In the case that xn!x ; by Proposition 3.3 again, ui( ~T;0; ~u0n( +xn);fi( +xn; ))!ui( ~T;0; ~u0;fi( +x ; )) as n!1. By (6.5) and (6.7), ui(tn;xn;u0) >u i(x ) 0=2 for n 1: (6.10) 69 By the continuity of u i( ), u i(x ) >u i(xn) 0=2 for n 1: (6.11) By (6.4), (6.10), and (6.11), jui(tn;xn;u0) u i(xn)j< 0 for n 1: This contradicts to (6.3) again. Hence lim x c0t;t!1 jui(t;x;u0) u i(x)j= 0 for all 0 0 be such that fi(x;u) < 0 for x2Hi, u2 [0;M], and i = 1;2;3. Then for any > 0, there are p 2 NN and gi : Hi R ! R such that for any u 2 R, gi( ;u)2Xi;p, gi( ; ) satis es (H0) and (H2), and fi(x;u) gi(x;u) 8x2Hi; u2[0;M]; ^gi(0) f0i (0) ; where ^^gi( ) is as in (2.20) (i = 1;2;3). Proof. By Lemma 6.1, for any > 0, there are p2NN and hi( ) 2Xi;p\CN(Hi;R) such that fi(x;0) hi(x) 8x2Hi 70 and ^^h i f0i (0) for i = 1;2;3. Fix 1 i 3 and choose Mi > 0 such that fi(x;u) gi(x;u) := hi(x) Miu for x2Hi; 0 u M: It is not di cult to see that gi( ; ) (1 i 3) satisfy the lemma. Recall that for given 2SN 1, c01( ) = inf >0 f 0 1 (0) + 2 = 2 q f01 (0); c02( ) = inf >0 R RN e z (z)dz 1 +f0 2 (0) ; and c03( ) = inf >0 P k2Kak(e k 1) +f0 3 (0) : Observe that i( ; ;f0i (0)) (i = 1;2;3) exist, 1( ; ;f01 (0)) = f01 (0) + 2; 2( ; ;f02 (0)) = Z RN e z (z)dz 1 +f02 (0); and 3( ; ;f03 (0)) = X k2K ak(e k 1) +f03 (0): If no confusion occurs, we may denote i( ; ;f0i (0)) by i( ; ) (i = 1;2;3). Observe also that v1(t;x) = e (x 1( ; ) t); v2(t;x) = e (x 2( ; ) t); 71 and v3(t;j) = e (j 3( ; ) t) are solutions of vt(t;x) = v(t;x) +f01 (0)v(t;x); x2RN; (6.12) vt(t;x) = Z RN (y x)v(t;y)dy v(t;x) +f02 (0)v(t;x); x2RN; (6.13) and vt(t;j) = X k2K ak(v(t;j +k) v(t;j)) +f03 (0)v(t;j); j2ZN; (6.14) respectively. We now give another proof of Theorem 2.2. Proof of Theorem 2.2. Fix 2SN 1 and 1 i 3. We rst prove that for any c0 > c0i( ) and u0 2X+i ( ), lim sup x c0t;t!1 ui(t;x;u0) = 0: (6.15) To this end, take a c such that c0 >c>c i( ). Note that there is i > 0 such that c0i( ) = i( ; i) i and there is 2(0; i) such that c = i( ; ) : Take d>M > 0 such that u0(x) M and u0(x) de x 8x2Hi; fi(x;M) < 0 8x2Hi; (6.16) 72 and fi(x;u) = f0i (u) for x 1 ln Md (> 0): (6.17) Observe that by (6.16) and (H2), for (t;x)2(0;1) Hi with de (x ct) M, i.e., x 1 ln Md +ct, fi(x;de (x ct)) < 0 0 x2Hi: (6.18) This implies that (6.15) holds. Next, we prove that for any c0 0 be such that u0(x) M and fi(x;M) < 0 for all x2Hi: 73 Then u M is a super-solution of (1.3) and ui(t;x;u0) M 8t 0; x2Hi: For any > 0, let gi( ; ) be as in Lemma 6.7. By Corollary 3.2, for > 0 su ciently small, c i( ;gi( ; )) c i( ;^^gi( )) >c: By Propositions 3.1 and 3.10, lim inf x ct;t!1 ui(t;x;u0) lim inf x ct;t!1 ui(t;x;u0;gi) > 0: (6.19) then follows from Lemma 6.6. By (6.15) and (6.19), c i( ) exists and c i( ) = c0i( ) for i = 1;2;3. Moreover, (2.26) holds Finally, prove Theorem 2.3. Proof of Theorem 2.3. (1) Fix 2SN 1 and 1 i 3. Let u0 2X+i satisfy that u0(x) = 0 for x2Hi with jx j 1. Then there are u+0 2X+i ( ) and u 0 2X+i ( ) such that u0(x) u 0 (x) 8x2Hi: By Proposition 3.1 and Theorem 2.2, lim sup x c0t;t!1 ui(t;x;u0) lim sup x c0t;t!1 ui(t;x;u+i ) = 0 8c0 >c i( ) and lim sup x ( ) c0t;t!1 ui(t;x;u0) lim sup x ( ) c0t;t!1 ui(t;x;u i ) = 0 8c0 >c i( ) 74 It then follows that lim sup jx j c0t;t!1 ui(t;x;u0) = 0 8c0 > maxfc i( );c i( )g: (2) Fix 2SN 1 and 1 i 3. For given 0 < c0 < minfc i( );c i( )g, take a c > 0 such that c0 0 be such that u0(x) M and fi(x;M) < 0 for all x2Hi: Then u M is a super-solution of (1.3) and ui(t;x;u0) M 8t 0; x2Hi: For any > 0, let gi( ; ) be as in Lemma 6.7. By Corollary 3.2, for > 0 su ciently small, c i( ;gi( ; )) c i( ;^gi( )) >c: By Propositions 3.1 and 3.10, lim inf jx j ct;t!1 ui(t;x;u0) lim inf jx j ct;t!1 ui(t;x;u0;gi) > 0: It then follows from Lemma 6.6 that lim sup jx j c0t;t!1 jui(t;x;u0) u i(x)j= 0: 75 (3) Fix 2SN 1 and 1 i 3. Let c> sup 2SN 1 c i( ): Let u0 2X+i be such that u0(x) = 0 for kxk 1: Note that for every given 2SN 1, there is ~u0( ; )2X+i ( ) such that u0( ) ~u0( ; ): By Proposition 3.1, 0 ui(t;x;u0) ui(t;x; ~u0( ; )) for t> 0 and x2Hi. It then follows from Theorem 2.2 that 0 lim sup x ct;t!1 ui(t;x;u0) lim sup x ct;t!1 ui(t;x; ~u0( ; )) = 0: Take any c0 >c. Consider all x2Hi withkxk= c0. By the compactness of @B(0;c0) = fx2Hijkxk = c0g, there are 1; 2; ; L2SN 1 such that for every x2@B(0;c0), there is l (1 l L) such that x l c: Hence for every x2Hi with kxk c0t, there is 1 l L such that x l = kxkc0 c0 kxkx l kxkc0 c ct: By the above arguments, 0 lim sup x l ct;t!1 ui(t;x;u0) lim sup x l ct;t!1 ui(t;x; ~u0( ; l)) = 0 76 for l = 1;2; L. This implies that lim sup kxk c0t;t!1 ui(t;x;u0) = 0: Since c0 >c and c> sup 2SN 1 c i( ) are arbitrary, we have that for c> sup 2SN 1 c i( ), lim sup kxk ct;t!1 ui(t;x;u0) = 0: (4) It can be proved by similar arguments as in (2). To be more precise, for given 0 0 such that c0 0 be such that u0(x) M and fi(x;M) < 0 for all x2Hi: Then u M is a super-solution of (1.3) and ui(t;x;u0) M 8t 0; x2Hi: For any > 0, let gi( ; ) be as in Lemma 6.7. By Corollary 3.2, for > 0 su ciently small, c i( ;gi( ; )) c i( ;^gi( )) >c: By Propositions 3.1 and 3.10, lim inf kxk ct;t!1 ui(t;x;u0) lim inf kxk ct;t!1 ui(t;x;u0;gi) > 0: 77 It then follows from Lemma 6.6 that lim sup kxk c0t;t!1 jui(t;x;u0) u i(x)j= 0: 78 Chapter 7 E ects of Temporal and Spatial Variations In this chapter, we prove Theorem 2.4 on the e ects of temporal and spatial variations on the spatial spreading dynamics of monostable equations. Proof of Theorem 2.4. (1) Suppose that ( ^f0( ;0)) > 0. By Proposition 3.6, (f0( ; ;0)) ( ^f0( ;0)) > 0: Then by Theorem 2.1, (1.3) has a time periodic strictly positive solution u (t;x). By The- orem 2.2, (1.3) has a spatial spreading speed c ( ) in the direction of for each 2SN 1, and c ( ) = c 0( ): Recall that c 0( ) = inf >0 ; (f0( ; ;0)) : By Proposition 3.6 again, ; (f0( ; ;0)) ; ( ^f0( ;0)): It then follows that c ( ) = c 0( ) inf >0 ; ( ^f0( ;0)) = ^c 0( ): (2) Suppose that ( ^^f0(0)) > 0. By Proposition 3.7, we have ( ^f0( ;0)) ( ^^f0(0)) > 0: 79 It then follows from (1) that (1.3) has a time periodic strictly positive solution u (t;x) and has a spatial spreading speed c ( ) in the direction of for each 2SN 1. Recall again that c 0( ) = inf >0 ; (f0( ; ;0)) : By Proposition 3.7 again, ; ( ^f0( ;0)) ; ( ^^f0(0)): This together with (1) implies that c ( ) = c 0( ) ^c 0( ) ^^c 0( ) for each 2SN 1. 80 Chapter 8 Concluding Remarks and Future Plan 8.1 Concluding Remarks In this dissertation, we studied the semilinear dispersal evolution equations of the form ut(t;x) = (Au)(t;x) +u(t;x)f(t;x;u(t;x)); x2H; where H = RN or ZN, A is a random dispersal operator or nonlocal dispersal operator in the case H= RN and is a discrete dispersal operator in the case H= ZN, and f is periodic in t, asymptotically periodic in x (i.e. f(t;x;u) f0(t;x;u) converges to 0 as kxk!1 for some time and space periodic function f0(t;x;u)), and is of KPP type in u. It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if u 0 is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction. Moreover, we developed multiple methods to achieve the two results, and considered the e ects of temporal and spatial variations of the media on the uniform persistence and spatial spreading speeds of monostable equations. It should be pointed out that the Liouville type property for (1.3) in the case that H = RN, A = u, and f(t;x;u) = f(x;u) has been proved in [8]. However, many tech- niques developed in [8] are di cult to apply to (1.3). Several important new techniques are developed in the current dissertation. Both the results and techniques established in the current paper can be extended to more general cases (say, cases that A is some linear 81 combination of random and nonlocal dispersal operators and/or f(t;x;u) is almost periodic in t and asymptotically periodic in x). It should also be pointed out that, if u = 0 is a linearly unstable solution of (1.1) with respect to periodic perturbations, then (1.1) has traveling wave solutions connecting 0 and u 0( ; ) and propagating in the direction of with speed c>c 0( ) for any 2SN 1. But (1.3) may have no traveling wave solutions connecting 0 and u ( ; ) (see [63]) (hence, localized spatial inhomogeneity may prevent the existence of traveling wave solutions). 8.2 Future plans 8.2.1 Single-species population model For the single-species model, we established the existence of strictly positive solution u and spreading speeds. Next topic I am interested in is to generalize traveling wave solution, which is a global-in-time solution, connecting the positive solution u and trivial solution u = 0, in the locally inhomogeneous media. Due to the spatial heterogeneity, the global-in- time wave-like solutions lose some classical features, compared to homogeneous equations, such as, constant in shape (up to a spatial shift) and the unchanged speed for each pro le. Recently, James Nolen together his collaborates considered (1.3) in temporal independent setting, and showed that a su ciently strong localized inhomogeneity may prevent existence of global-in-time wave-like solutions. They created a time-global bump-like solution, which blocks any wave-like solution. I am curious to know whether such interesting scenario happen in more general heterogeneous media (say, a spatially locally perturbed spatial periodic media). How about the existence of wave-like solutions in nonlocal and discrete dispersal framework? As a short term plan, the following problem is listed in my agenda. Explore the existence, uniqueness and stability of of global-in-time wave-like solutions in spatially locally perturbed media. 82 8.2.2 Multi-species population model Comparing with single species model, much less is known about the multi-species system, especially in the heterogeneous (say, spatially locally inhomogeneous) environments. For instance, for two species competitive system, main results established in the litera- ture are related to spatial-temporal independent nonlinearities, 8 >>>> >>< >>>> >>: @u @t = u+uf(u;v); x2R N; @v @t = v +vg(u;v); x2R N; where f(u;v) < 0 and g(u;v) < 0 for u;v 0 with u2 + v2 1, fu(u;v) < 0, fv(u;v) < 0, gu(u;v) < 0, and gv(u;v) < 0 for u;v 0 (see [28], [36], [38], [47], [48], [53], etc.) Here are three problems being considered. Explore the existence of traveling wave solutions to the systems in periodic media with nonlocal dispersal. 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