Traveling wave solutions in a plant population model with a seed bank

Transcription

1 J. Math. Biol. DOI /s x Mathematical Biology Traveling wave solutions in a plant population model with a seed bank Bingtuan Li Received: 13 December 2010 / Revised: 2 October 2011 Springer-Verlag 2011 Abstract We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions. Keywords Seed bank Integro-difference model Spreading speed Laplace transform Traveling wave solution Mathematics Subject Classification (2000) 92D40 92D25 1 Introduction Integro-difference models are of great interest in the studies of invasions of populations with discrete generations and separate growth and dispersal stages. They describe a process in which individuals first undergo reproduction and then redistribute their offspring before reproduction occurs once again. This is the case with annual plants This research was partially supported by the National Science Foundation under Grant DMS B. Li (B) Department of Mathematics, University of Louisville, Louisville, KY 40292, USA bing.li@louisville.edu

2 B. Li and many insect species. The classical integro-difference model takes the form of convolution integral with a density-dependent fecundity function and a flexible dispersal kernel. The shape of the dispersal kernel can be fitted to data from mark-recapture experiments and include rare, long-distance dispersal events observed with invasive spread (Silvermann 1986; Southwood 1978; Tarter and Lock 1993; Kot et al. 1996). The classical integro-difference model has been used to predict changes in gene frequency (Lui 1982a,b, 1983; Slatkin 1973; Weinberger 1978), and applied to ecological problems (Hardin et al. 1988a,b; Hastings and Higgins 1994; Kot 1992, 2001; Kot and Schaffer 1986; Neubert et al. 1995). It has been known for a long time (see Weinberger 1982) that if the fecundity function is a nondecreasing function, the classical integro-difference model predicts that there is a spreading speed and it can be characterized as the slowest speed of a class of traveling wave solutions. Under the additional assumption that the population dynamics exhibit no Allee effect, the spreading speed can be computed explicitly through linearization. The results have been recently extended to the case where the fecundity function is a nonmonotone function (Hsu and Zhao 2008; Li et al. 2009). Loosely speaking, the spreading speed of a species is the asymptotic rate at which the species with uniformly positive initial distribution over a large interval and zero distribution outside an interval expands its spatial range. A traveling wave solution describes the propagation of a species as a wave with a fixed shape and a fixed speed. The spreading speeds and traveling wave solutions provide important insight into the spatial patterns and rates of invading species in space. In plant population biology, the description of population processes is complicated by the fact that the seeds may persist for several years in the soil before germinating. Some seeds, the seed bank, may lie dormant in the soil and germinate at some later time, if they survive (Edelstein-Keshet 1988; Ritland 1983; Templeton and Levin 1979). Several non-spatial models have been developed for a population of species of annual plant, of which a proportion of the seeds remain dormant for at least one year (e.g., MacDonald and Watkinson 1981; Edelstein-Keshet 1988; Pacala 1986; Rees 1997). In the presence of a seed bank in a plant population, due to the fact that seeds in the seed bank do not disperse, the classical integro-difference equation is not suitable to describe the spatial spread of the population. Allen et al. (1996) proposed integrodifference models to study spatial dynamics of a single plant population and dynamics of two competing plant populations with a seed bank. The model that describes the dispersal of an annual plant population with a seed bank given in Allen et al. (1996) is a very fundamental one. It incorporates a piecewise linear (monotone) recruitment function, a term involving a convolution integral, and a term outside the integral to describe the effect of the seed bank. Allen et al. pointed out that the models do not satisfy the compactness assumption which is required in Weinberger (1982) for establishing the existence of traveling wave solutions, and provided numerical simulations for traveling wave solutions. In this paper we shall propose a single plant population model with a seed bank that includes a general fecundity function which is allowed to be nonmonotone, and show analytically the existence of traveling wave solutions in the model. This model may be viewed as an extension of the single annual plant population model given in Allen et al. (1996). The formulation of the new model is based on the non-spatial

3 Waves for a plant population bottleneck model given in MacDonald and Watkinson (1981). The results from Weinberger (1982), Hsu and Zhao (2008) and Li et al. (2009) assure the existence of a spreading speed in the model, even when the fecundity function is nonmonotone. However, since the recursion operator in the model is not compact, the existing results do not show the existence of traveling wave solutions in the model. We shall show that even though the recursion operator in the seed bank model is not compact, there are reasonable additional conditions on the recruitment function which do not necessarily imply monotonicity, and under which the spreading speed can be characterized as the lowest speed of a family of traveling waves. The proof of the existence of traveling wave solutions makes use of an asymptotic fixed point theorem. An important observation made in MacDonald and Watkinson (1981) is that a seed bank can stabilize a non-spatial nonmonotone plant system. We provide numerical simulations to demonstrate the stabilizing effect of a seed bank on the spatial patterns of traveling wave solutions in the spatial seed bank model. This paper is organized as follows. In the next section, we present an integrodifference equation model with a seed bank and a general fecundity function that may be nonmonotone, and show the existence of a spreading speed in the model. Our results on the existence of traveling wave solutions in the model are given in Sect. 3. Section 4 is about simulations of traveling wave solutions. Section 5 includes some concluding remarks and discussions. 2 The model and spreading speed MacDonald and Watkinson (1981) gave the the difference equation N n+1 = (1 γ)g(n n ) + γρn n (2.1) in the study of an annual plant population with a seed bank. Here N n represents the mature plant population at time n; g(n n ) describes the seed production, the fraction of seeds that survive the dormant period, the fraction of these surviving seeds that germinate, and the fraction of seedlings that survive to yield a mature plant; γ and ρ are respectively the fraction of seeds in the seed bank that do not germinate, and the fraction of non-germinating seeds that survive for a whole year. The term (1 γ)g(n n ) describes the growth from new seeds, while the term γρn n represents the growth due to old seeds in the seed bank. In model (2.1), no distinction is made between this year s quota of seeds and those from earlier years persisting in the seed bank. For a more detailed description of model (2.1), the reader is referred to MacDonald and Watkinson (1981). MacDonald and Watkinson showed that the seed bank can exert both a damping effect and an inertial effect in the difference equation model (2.1) when g is nonmonotone. One can further show that the seed bank can even change the state of chaos to limit cycle (see Sect. 4). We now incorporate the spatial aspect into model (2.1) to allow for spatial movement. Since new seeds disperse after seed production and old seeds in the seed bank do not move, the spatial model corresponding to the difference equation (2.1) takes the form

4 u n+1 (x) = (1 γ) + B. Li k(x y)g(u n (y))dy + γρu n (x) (2.2) where u n (x) represents the density of the mature plant population at time n, and k is the seed dispersal kernel. In (2.2)thetermγρu n (x) is outside the convolution integral, which makes the recursion operator of (2.2) not compact. This poses a challenge to establishing the existence of traveling wave solutions for (2.2). We shall make the following assumptions on the model (2.2). Hypotheses 2.1 i. 0 γ<1and 0 <ρ 1. ii. There is a positive constant β 0 such that a. g(u) is continuous for 0 u β 0, g(0) = 0, and 0 g(u) 1 γρ 1 γ β 0 for 0 u β 0 ; b. g(u) >0for0< u β 0 ; and c. g(u) is differentiable at u = 0 with g (0) >(1 γρ)/(1 γ), and g(u) g (0)u for 0 u β 0. iii. There exist positive constants d, σ, and d β 0 such that for 0 u d g(u) g (0)u du 1+σ. iv. k(x) is a continuous nonnegative function such that i. k(x)dx = 1; ii. and the integral K (μ) := is finite for all real values of μ. Define the function g + (u) := max 0 v u g(v) k(x)e μx dx for 0 u β 0 (This function is called G(u, 0) in Thieme 1979.) g + (u) is a continuous and nondecreasing function, and 0 g(u) g + (u) β 0 for 0 u β 0. It is easily seen that there is at least one positive root of the equilibrium equation (1 γ)g + (u) + γρu = u on the interval (0,β 0 ].Weuseβ + to denote the smallest of such roots. We have that (1 γ)g + (u) + γρu > u for 0 < u <β +.

5 Waves for a plant population We now define the function g (u) := min g(v) for 0 u u v β β+. + (This function is called G(u,α)in Thieme 1979.) Clearly, g (u) g(u). The definition of g shows that the equilibrium equation (1 γ)g (u) + γρu = u has at least one positive root on the interval (0,β + ].Weuseβ to denote the smallest of such roots. We observe that g ± are differentiable at 0, and g (0) = g + (0) = g (0). (2.3) In fact, since g(u) is differentiable at u = 0, for any ɛ>0, there is a positive constant d <β such that for 0 u d (1 ɛ)g (0)u g(u) (1 + ɛ)g (0)u. On the other hand, the definitions of g ± show that there exists a positive constant d 1 d such that for 0 u d 1, g + (u) = max 0 v u g(v) (1 + ɛ)g (0)u, and g (u) = { } min g(v) = min min g(v), u v β + u v d g (d 1 ) (1 ɛ)g (0)u. We therefore have that for 0 u d 1, (1 ɛ)g (0)u g (u) g(u) g + (u) (1 + ɛ)g (0)u. Since ɛ is arbitrary, the definition of derivative yields (2.3). The definitions of g ± and (2.3) show that for 0 u β +, g ± (u) g ± (0)u = g (0)u. Proposition 2.1 Suppose that the Hypotheses 2.1 are satisfied. Define the numbers c := inf (1/μ) ln (1 γ)g (0) μ>0 e μy k(y)dy + γρ (2.4)

6 B. Li and c ( 1) := inf (1/μ) ln (1 γ)g (0) μ>0 e μy k(y)dy + γρ. (2.5) Then c is the asymptotic rightward spreading speed of the recursion (2.2) in the following sense: If the continuous initial function u 0 (x) is zero for all sufficiently large x, u 0 0, and 0 u 0 (x) β +, then for any positive ɛ the solution of u n of the recursion (2.2) has the following properties i. u n (x) β + for all x and n. ii. [ ] lim n sup u n (x) x n(c +ɛ) = 0. iii. c + c ( 1) >0, and [ ] lim inf inf u n(x) n n(c ( 1) ɛ) x n(c ɛ) β. (2.6) The proof of this proposition is similar to that of Proposition 3.1 in Li et al. (2009) and is omitted. Remark 2.1 c ( 1) defined by (2.5) is the leftward asymptotic spreading speed of (2.2). It is possible that one of c and c ( 1) is negative when k(x) is asymmetric. 3 The existence of traveling wave solutions In this section we study the traveling wave solutions for model (2.2). Define Q[u](x) := (1 γ) + k(x y)g(u(y))dy + γρu(x) (3.1) which is on the right-hand side of (2.2). A traveling wave solution u n (x) = w(x nc) of (2.2) satisfies w(x) = Q c [w](x)

7 Waves for a plant population where Q c [u] =Q[u](x + c). (3.2) We have that w is a traveling wave solution of (2.2) if and only if it is a fixed point of Q c. We introduce the notation C r := {u : u(x) is continuous, and 0 u(x) r for all x}. We first consider the case that g is monotone. Proposition 3.1 Let the Hypotheses 2.1 be satisfied. Assume that g(u) is nondecreasing for 0 u β with 0 <β β 0 and β the smallest positive root of the equilibrium equation (1 γ)g(u)+γρu = u. Then for c c, the equation (2.2) has a continuous nonincreasing traveling wave w c (x nc) with w c () = β and w c (+ ) = 0. A continuous nonnegative traveling wave solution w c (x nc) in (2.2) with w c ( ) = 0 and lim inf x w c (x) >0 does not exist if c < c. Proof The proof of the second statement is similar to that of Theorem 4.1 in Li et al. (2009), and is omitted. We now prove the first statement of the Proposition. Given a function ψ C β and a bounded interval I =[a, b], we define a function ψ I (x) C(I, R) by ψ I (x) = ψ(x). Here C(I, R) is equipped with the maximum norm I. For any subset D C β,we define D I := {ψ I C[I, R] :ψ D}. We use α(d I ) to denote the Kuratowski measure of noncompactness of D I. We refer to Liang and Zhao (2010) and Martin (1976) for the precise definition of the Kuratowski measure of noncompactness and various properties of it. Define R[u](x) := (1 γ) for u C β. Clearly + k(x y)g(u(y))dy, and P[u](x) := γρu(x) (3.3) Q = R + P. P is a linear operator that has the property that for ψ I D I, P[ψ] I = γρ ψ I. Since R is compact (Li et al. 2009), for any D C β,wehaveα((r(d)) I ) = 0. It follows that α((qd) I ) α((rd) I ) + α((pd) I ) γρα(d I ).

8 B. Li Since γρ < 1, the first statement of the Proposition follows from Theorem 4.2 in Liang and Zhao (2010). The proof is complete. Let (μ) := (1/μ) ln (1 γ)g (0) e μy k(y)dy + γρ. Then c defined by (2.4)isgivenby c = inf (μ). (3.4) μ>0 The following lemma follows from Lemma 3.1 and its proof in Volkov and Lui (2007). Lemma 3.1 i. The infimum in (3.4) is attained at a finite real number μ >0. ii. (μ) < 0 for 0 <μ< μ. iii. ( μ) = 0 and ( μ) > 0. For c c,weuseμ c to denote the root of for 0 <μ μ. Note that μ c = μ. (μ) = c Proposition 3.2 Let the Hypotheses 2.1 be satisfied. Assume that g(u) is nondecreasing for 0 u β with 0 <β β 0 and β the smallest positive root of the equilibrium equation (1 γ)g(u)+γρu = u. Let w c (x nc) be a continuous nonincreasing traveling wave solution of (2.2) with w c () = β and w c (+ ) = 0 for c c. Then for c > c, lim x e μ cx w c (x) = AforsomeA> 0, and lim x e μx w c (x)/x = B for some B > 0. Proof Since the integral of k(x)dx is convergent, there is, for any prescribed ɛ>0, an L > 0 such that L L k(x)dx (1 ɛ). Since w c (x) decreases to 0 as x, we can choose L sufficiently large such that for x L g(w c (x)) (1 ɛ)g (0)w c (x).

9 Waves for a plant population For c c, w c (x c) L (1 γ) k(x y)g(w c (y))dy + γρw c (x) L (1 ɛ)(1 γ)g(w c (x + L)) + γρw c (x) (1 ɛ) 2 (1 γ)g (0)w c (x + L) + γρw c (x + L) for x L.Itfollowsthatforx L + c w c (x) [(1 ɛ) 2 (1 γ)g (0) + γρ]w c (x + L + c). (3.5) We choose ɛ so small that (1 ɛ) 2 (1 γ)g (0) + γρ > 1. Let δ := 1 L + c ln[(1 ɛ)2 (1 γ)g (0) + γρ] > 0. We choose L sufficiently large such that L + c > 0. It follows from (3.5) that for x L + c e δ(x+l+c) w c (x + L + c) e δx w c (x) so that e δx w c (x) is bounded for large x. It follows that the two-sided Laplace transform with positive argument converges for 0 <ν<δ.let W c (ν) := M[u](x) := (1 γ) + e νx w c (x)dx k(x y)g (0)u(y)dy + γρu(x). M is the linearization of Q near zero. We write the traveling wave equation in the form w c (x) M[w c ](x + c) = (1 γ) k(x + c y)[g(w c (y)) g (0)w c (y))]dy. (3.6)

10 B. Li Let F c (ν) := 1 [(1 γ)g (0)K c (ν) + γρe νc ]. (3.7) where K c (ν) = eνx k(x + c)dx. It is easily seen that F c (ν) = 1 [(1 γ) g (0)K (ν) + γρ]e νc = 1 e ν( (ν) c). We observe that e νx M[w c ](x + c)dx = (1 F c (ν))w c (ν) = e ν( (ν) c) W c (ν). Therefore the Laplace transform of (3.6) takes the form of F c (ν)w c (ν) = e νx k(x + c y)[g(w c (y)) g (0)w c (y))]dydx. (3.8) If we define G c (w c ; ν) := we see from (3.8) that e νx k(x + c y)[g(w c (y)) g (0)w c (y))]dydx, W c (ν) = G c (w c ; ν)/f c (ν). (3.9) Since g(u) g (0)u and g(u) <g (0)u for some u with 0 < u <β, the continuity of w c (y) in y shows that G c (w c ; ν) < 0 when G c (w c ; ν) is convergent. By Lemma 3.1, (μ) is decreasing for 0 <μ<μ c and (μ c ) = c. We therefore have that F c (ν) < 0 for 0 <ν<μ c, and that F c (μ c ) = 0forc c. Because of Hypotheses 2.1 ii.c and iii, G c (w c ; ν) converges for 0 <ν<(1 + σ)δ, and the formula (3.9) shows that W c (ν) is real analytic in 0 <ν<min{(1+σ)δ,μ c }. If (1+σ)δ < μ c, we repeat this argument q 1 times for a q such that (1+σ) q δ μ c. We conclude that W c (ν) is analytic for 0 <ν<μ c. This and Hypotheses 2.1 ii.c and iii show that G c (w c ; ν) is analytic for 0 <ν<(1 + σ)μ c. We note that W c (ν), G c (w c ; ν) and F c (ν) have natural extensions to the complex plane which are obtained by replacing ν by ν 1 + iν 2 in their integral definitions. We now show γρe μc < 1 (3.10) for 0 μ μ c. In the case of c 0, this clearly is true due to the fact that γρ < 1. On the other hand the definition of μ c shows that ln(γρ)/μ c < (μ c ) = c, which

11 Waves for a plant population leads to (3.10) with μ replaced μ c. When c < 0, the left-hand side of (3.10) isan increasing function of μ. It follows that (3.10) is still valid when c < 0. The definition of μ c shows that F c (μ c ) = 0, that is (1 γ)g (0) e μ cx k(x + c)dx + γρe μ cc = 1. (3.11) If F c (ν) = 0 has a complex zero in the from μ c + iν 2 = 0, then a direct calculation shows that (1 γ)g (0) e μ cx k(x + c) cos(ν 2 x)dx + γρe μ cc cos(ν 2 c) = 1. (3.12) Comparing (3.12) with (3.11), we easily see that ν 2 = 0. Therefore F(ν) = 0 has only a real zero μ c on the vertical line ν 1 = μ c. The Lemma of Riemann Lebesgue (Titchmarsh 1986) and (3.10) show that F c (ν) givenby(3.7) has no zero for ν = ν 1 + iν 2 with 0 ν 1 μ c and ν 2 sufficiently large. This and the analyticity of F c (ν) show that the zeros of the function F c (ν), which lies in a vertical strip S ɛ ={ν 1 + iν 2 : μ c ɛ < ν 1 μ c } where ɛ is a small positive number, are finite. By using this and the fact that F c (ν) has only the real zero μ c on the vertical line Reν = μ c, we obtain that for ɛ sufficiently small, F c (ν) has only one zero at ν = μ c in the strip S ɛ. On the other hand by Lemma 3.1, F c (ν) has a zero of order 1 at μ c for c > c and a zero of order 2 at μ c. Moreover, G c (w c ; μ c )<0forc c, F c (μ c) = μ c (μ c )>0when c > c, and F c (μ c ) = 0 and F c (μ c ) = μ c (μ c ) < 0. If follows from the Laurent expansion of G c (w c ; ν)/f c (ν) at ν = μ c in the complex plane and Lemma 6.1 in Diekmann and Kaper (1978) (a modified version of Ikehara theorem) that for c > c, lim x e μcx w c (x) = A for some A > 0. The argument given in line 21 line 34 on page 922 in Lui (1982a) (that involves an application of an extension of Ikehara theorem ) shows that lim x e μ c x w c (x)/x = B for some B > 0. The proof is complete. Remark 3.1 The uniqueness of traveling wave solutions of (2.2) for the case that g is monotone can be established under some appropriate assumptions on g. Assume that the conditions of Proposition 3.2 are satisfied. If in addition g(u) g(v) g (0) u v for all u,v [0,β], then there is one nonincreasing traveling wave solution (up to translation) in (2.2) with speed c > c connecting 0 with β; and if g(u)/u is nonincreasing for 0 < u β and g (u) >0for0 u <σwhere σ = sup{u : g(u) <β}, then there is one nonincreasing traveling wave solution (up to translation) in (2.2) with speed c connecting 0 with β. This result can be shown by essentially using Proposition 3.2, the proof of Theorem 6.4 in Diekmann and Kaper (1978), and the second part of the proof of Theorem 4 in Lui (1982a). Theorem 3.1 Assume that Hypotheses 2.1 are satisfied. Then for any c c,the equation (2.2) has a traveling wave solution w(x nc) with w(x) β + for all x,

12 B. Li w( ) = 0, and lim inf x w(x) β. A traveling wave solution w(x nc) with w( ) = 0 and lim inf x w(x) >0 does not exist if c < c. Proof The proof of the second statement is similar to that of Theorem 4.1 in Li et al. (2009), and is omitted. We prove the first statement of the Theorem. Let Q + c defined by (3.2) and (3.1) with g(u) replaced by g +. Choose 0 <β<β and define g β (u) := min{g (u), (1 γρ)β/(1 γ)}. Let Q c defined by (3.2) and (3.1) with g(u) replaced by g β (u). Clearly, g β (0) = 0, and gβ (u) is nonincreasing on [0,β], on which β is the only positive root of the equation (1 γ)gβ (u) + γρu = u. It is easily seen that for u C β + Q c [u] Q c[u] Q + c [u]. Propositions 3.1 and 3.2 show that for c c, there exist nonincreasing functions (x), such that w (±) c and Q + c [w+ c ](x) = w+ c (x), w+ c () = β+,w c + ( ) = 0, Q c [w c ](x) = w c (x), w c () = β, w c ( ) = 0, lim x eμcx w c ± (x) = A± > 0, for c > c lim x eμ c x w ± c (x)/x = B± > 0. By translating wc if necessary we can assume that A + > A and B + > B so that for c c there exists a real number l (1) c such that w c + (x) w c (x) for x l(1) c. On the other hand, since for c c, w c + () = β+ β >β= wc (), there exists a real number l (2) c such that w c + (x) w c (x) for x l(2) c. Since w c ± (x) are nonincreasing, for b c = max{l (1) c l (2) c, 0}, Define w c (x + b c) w + c (x). E c ={u(x) : u(x) is continuous,w c (x + b c) u(x) w + c (x)} for c c. E c is a convex and closed subset of the Banach space C(R, R) with the norm μc 2 given by u μc = sup 2 x R u(x) e ( μc 2 )x <. (3.13)

13 Waves for a plant population For u E c, w c (x + b c) = Q c [w c ](x + b c) = Q c [w c ( +b c)](x) Q c [u](x) Q c[u](x) and Q c [u](x) Q + c [u](x) Q+ c [w+ c ](x) = w+ c (x). It follows that Q c takes E c into itself. Q c is of the form R c + P c where R c [u] = R[u](x + c) and P c [u](x) = P[u](x + c) with R and P given by (3.3). R c is continuous and compact with respect to the norm (3.13) (for a proof see Page 784 in Hsu and Zhao 2008). P c is a strict contraction (i.e., P c u γρ u ). Consequently Q c is a γρ-set-contraction with γρ < 1. The asymptotic fixed point theorem given on page 495 in Nussbaum (1969) shows that Q c has a fixed point w c (x) E c, which represents a traveling wave solution of (2.2). Since w c (x) w c + (x), w c(x) β + for all x, w c ( ) = 0, and lim inf x w c (x) β. Choose a continuous nonincreasing function u 0 (x) such that u 0 (x) = 0 for all sufficiently large x, u 0 () >0, and w c (x) u 0 (x) for all x. Then induction shows that the solution u n (x) of (2.2) has the property that w c (x nc) u n (x) for all x. Property (2.6) shows that for 0 <ɛ<1/2(c + c ( 1)) [ ] lim inf inf w c(x nc) n n(c ( 1) ɛ) x n(c ɛ) [ ] lim inf n inf n(c ( 1) ɛ) x n(c ɛ) u n(x) β, which yields [ ] lim inf inf w c(x) n n(c ( 1)+c ɛ) x n(c c ɛ) β. (3.14) Note that c c ɛ<0 and that the intervals [ n(c ( 1) + c ɛ),n(c c ɛ)] overlap for large n due to the fact that c + c ( 1) >0 and the choice of ɛ.itfollows from this and inequality (3.14) that lim inf x w c (x) β. The proof is complete. 4 Numerical simulations In this section we present some approximations to traveling waves of model (2.2) with the Ricker recruitment function g(u) = ue r u (4.15) with r > 0. This function has been used to study plant populations (see for example Kuang and Chesson 2009; Norghauer and Newbery 2010). We first examine the dynamics of the difference equation (2.1) with the given g and γ = 0.2 and ρ = 0.7.

14 B. Li Fig. 1 The bifurcation diagram of (2.1) with g givenby(4.15) for 0 r 7 with γ = 0.2 and ρ = 0.7 Fig. 2 The bifurcation diagram of (2.1) with g givenby(4.15) for 0 r 7 with γ = 0 The bifurcation diagram (compressed vertically by a factor r) of the difference equation (2.1) with the above g is given in Fig. 1. It is interesting to compare it with Fig. 2, the bifurcation diagram (compressed vertically by a factor r) of the system with γ = 0 (i.e., there is no seed bank). We observe that the seed bank has the effect of stabilizing the system, and it can change the dynamics from the state of chaos to limit cycle. We next show simulations for traveling wave solutions for (2.2) with g given by (4.15), 0 <γ <1, 0 <ρ<1, and the Laplace kernel k( x y ) = 100e 200 x y. The positive solution of the equilibrium equation (1 γ)g(u) + γρu = u is given by r + ln 1 γ 1 γρ. Moreover

15 Waves for a plant population { g(u), for u 1 g + (u) = e r 1, for u 1, so that β + = r + ln 1 γ 1 γ, for 0 < r + ln 1 γρ 1 γρ 1 (1 γ)e r 1 1 γρ, for r + ln 1 γ 1 γρ 1, while { g (u) = min g(u), g ( (1 γ)e r 1 1 γρ )}, so that β = r + ln 1 γ 1 γρ, ( 1 γ 1 γρ g (1 γ)e r 1 1 γρ 1 γ for 0 < r + ln 1 γρ 1 ), for r + ln 1 γ 1 γρ 1. Numerical simulations for the model with the given g and k for the case of γ = 0 were provided in Li et al. (2009). We shall use the algorithm similar to what described in Sect. 5 in Li et al. (2009)to conduct simulations for traveling waves of the seed bank model (2.2). The algorithm uses the fact that a traveling wave of speed c > c of (2.2) behaves like multiple of e μx at infinity, where μ is the smaller root of the equation (μ) = c, i.e., (1/μ) ln{(1 γ)e r /[1 (μ/200) 2 ]+γρ}=c. A small positive number μ is chosen, and c is determined from this formula. The following figures show approximations to the graphs of w(x)/r where w(x nc) is a traveling wave of speed c of (2.2) with γ = 0.2, ρ = 0.7, for several choices of r and c. We first choose r = 0.9. It is easily seen that in this case the positive equilibrium of the model is β = so that β/r = , and g(u) is increasing for 0 u β. Figure 3 shows that the traveling wave appears to be nonincreasing. For r = 2.1, in the absence of the seed bank, the corresponding difference equation has a limit cycle with period 2; see Fig. 2. (It is easily seen that a period doubling bifurcation occurs at r = 2.) The simulation work in Li et al. (2009) shows that in this case the traveling wave solution for (2.2) with γ = 0 has an oscillating tail with period 2c for large negative x (see Figure 3 in Li et al. 2009). Figure 4 shows the graph

16 B. Li population density [ w(x)/r ] location (x) Fig. 3 A numerical approximation to the graph of w(x)/r with r = 0.9andc = population density [ w(x)/r ] location (x) Fig. 4 A numerical approximation to the graph of w(x)/r with r = 2.1andc = of w(x)/r for the seed bank model oscillates about the equilibrium.96556, and the oscillations are damped out for large negative x. We next choose r = 3. The corresponding difference equation in the absence of the seed bank exhibits chaos; see Fig. 2. It was shown numerically in Li et al. (2009) that in this case the traveling wave solution in (2.2) with γ = 0 has a chaotic tail (see Figure 4 in Li et al. 2009). Figure 5 shows that the graph of w(x)/r for the seed bank model seems to continue oscillating all the way to. It appears that the oscillations for large negative x are periodic of period 2c.

17 Waves for a plant population 3 population density [ w(x)/r ] location (x) Fig. 5 A numerical approximation to the graph of w(x)/r with r = 3andc = population density [ w(x)/r ] location (x) Fig. 6 A numerical approximation to the graph of w(x)/r with r = 5andc = As r becomes large, the periods in the tail of a traveling wave solution in (2.2) can be expected to become shorter, so the the tail will eventually look chaotic. This behavior is illustrated by Fig. 6. In summary, a seed bank can stabilize the patterns of traveling wave solutions in a plant population model, and it is possible for such a model to have a traveling wave solution with a chaotic tail.

18 B. Li 5 Concluding remarks An integro-difference equation model was presented to model the dispersal of a plant population with a seed bank. The recursion operator of the model is not compact, so that the existing theory on traveling waves cannot be applied to the model particularly when the fecundity function is nonmonotone. We showed analytically that under biologically reasonable assumptions, the model has a spreading speed that can be characterized as the slowest speed of a class of traveling wave solutions, and the spreading speed can be calculated based on the model parameters. We demonstrated numerically that a seed bank can stabilize spatial patterns of traveling wave solutions. The analytical techniques developed in this paper can be used to show the existence of traveling wave solutions in noncompact spatial models in other contexts. Mistro et al. (2005) proposed an integro-difference model to study the dispersal of the Africanized honey bees, which is slightly different from (2.2). In the model, the population reproduction function is given by the Ricker function. The existence of traveling wave solutions for the model can be established using the arguments analogous to those in Sect. 3 of the present paper for the case that the dispersal depends on the signed distance between the location of birth and the location of settlement. Our numerical simulations suggest that the tail of a traveling wave solution of (2.2) can oscillate periodically or chaotically. It would be interesting to analytically determine the conditions under which oscillations occur. We leave this problem for future investigation. The model formulation of the present investigation could be generalized in several ways. It is assumed in the model (2.2) that all seeds in the seed bank have the same survival rate and gemination rate. These rates, however, are likely depend on the age of seeds in the seed bank, and the model could be age-structured. While the single species model were considered here, more general models could include two species and competition interaction between them. Acknowledgments The author would like to thank Professor Hans F. Weinberger for fruitful discussions and his helpful suggestions. The author also wants to thank two anonymous referees for their constructive comments which have considerably improved the paper. References Allen EJ, Allen LJS, Gilliam X (1996) Dispersal and competition models for plants. J Math Biol 34: Diekmann O, Kaper H (1978) On the bounded solutions of a nonlinear convolution equation. Nonlinear Anal 2: Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New York Hardin DP, Takac P, Webb GF (1988a) Asymptotic properties of a continuous-space discrete-time population model in a random environment. Bull Math Biol 26: Hardin DP, Takac P, Webb GF (1988b) Dispersion population models discrete in time and space. J Math Biol 28:1 20 Hastings A, Higgins K (1994) Persistence of transients in spatially structured ecological models. Science 263: Hsu S-B, Zhao X-Q (2008) Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J Math Anal 40: Kot M (1992) Discrete-time traveling waves: ecological examples. J Math Biol 30: Kot M (2001) Elements of mathematical ecology. Cambridge University Press, Cambridge

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