Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase Online First, pp.1-20 https://doi.org/10.5206/mase/15145 LONG-TIME BEHAVIOR OF A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION ZHENZHEN LI AND BINXIANG DAI Abstract. This paper is devoted to a nonlocal dispersal logistic model with seasonal succession in one-dimensional bounded habitat, where the seasonal succession accounts for the effect of two different seasons. Firstly, we provide the persistence-extinction criterion for the species, which is different from that for local diffusion model. Then we show the asymptotic profile of the time-periodic positive solution as the species persists in long run. 1. Introduction The nonlocal diffusion as a long range process can well describe some natural phenomena in many situations (Andreu-Vaillo et al. [1], Fife [9]). Recently, nonlocal diffusion equations have attracted much attention and have been used to simulate different dispersal phenomena in material science (Bates [2]), neurology (Sun et at. [25]), population ecology (Hutson et al. [13], Kao et al.[15]). Especially, the spectral properties of nonlocal dispersal operators and the essential differences between them and local dispersal operators are studied in Coville [5], Coville et al. [6], Garćıa-Melián and Rossi [10], Shen and Zhang [22] and Sun, Yang and Li [25]. A widely used nonlocal diffusion operator has the form (J ∗u−u)(t,x) := ∫ R J(x−y)u(t,y)dy −u(t,x), which can capture the factors of ‘long-range dispersal’ as well as ‘short-range dispersal’. Time-varying environmental conditions are important for the growth and survival of species. Seasonal forces in nature are a common cause of environmental change, affecting not only the growth of species but also the composition of communities [7, 8]. The growth of species is actually driven by both external and internal dynamics. For instance, in temperate lakes, phytoplankton and zooplankton grow during the warmer months and may die or lie dormant during the winter. This phenomenon is termed as seasonal succession. In the present paper, we are concerned with the nonlocal dispersal logistic model with seasonal succession as follows:  ut = −δu, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ut = d( ∫ l2 l1 J(x−y)u(t,y)dy −u) + u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, u(0,x) = u0(x), x ∈ [l1, l2], (1.1) Received by the editors 27 October 2022; accepted 10 December 2022; published online 14 December 2022. 2020 Mathematics Subject Classification. Primary 35K57; 92D25; Secondary 35B40. Key words and phrases. Nonlocal dispersal; Seasonal succession; Persistence-extinction. Z. Li was supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts040) and B. Dai was supported by the National Natural Science Foundation of China (No. 11871475). 1 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/15145 2 Z. LI AND B. DAI where u(t,x) is the population density of a species at time t and location x in the one-dimensional bounded habitat [l1, l2] ⊂ R. All parameters δ,a,b and d are positive constants. The kernel function J : R → R is assumed to satisfy (J): J ∈ C(R) ∩L∞(R) is nonnegative, even, J(0) > 0 and ∫ R J(x)dx = 1. Here the parameter d stands for the diffusion rate of the species. Let J(x − y) be the probability distribution of the species jumping from location y to location x, then ∫ R J(x−y)u(t,y)dy represents the rate where individuals are arriving at location x from all other places and −u(t,x) = − ∫ R J(x − y)u(t,x)dy is the rate at which they are leaving location x to travel to all other sites. In such model, the integral operator ∫ l2 l1 J(x−y)u(t,y)dy−u can be viewed as the nonlocal dispersal counterpart of the elliptic operator uxx with homogeneous Dirichlet type boundary condition. The initial function u0(x) is nonnegative continuous function. Here and in what follows, unless specified otherwise, we always take i ∈ Z+ = {0, 1, 2, · · ·}. In (1.1), it is assumed that the species u undergoes two different seasons: the bad season and the good season. In the bad season: iω < t < (i + ρ)ω, for instance, from winter to spring, the species can not get enough food to feed themselves and its density are declining exponentially. During this season, the population has no ability to move in space. In the good season (for instance, from summer and autumn): (i + ρ)ω < t ≤ (i + 1)ω, we assume that the spatiotemporal distribution of the species u are governed by the classical nonlocal dispersal logistic equation. Parameters ω and 1 − ρ represent the period of seasonal succession and the duration of the good season, respectively. In fact, if we define the time-periodic finctions D(t) = { 0, t ∈ (iω, (i + ρ)ω], d, t ∈ ((i + ρ)ω, (i + 1)ω], ā(t) = { −δ, t ∈ (iω, (i + ρ)ω], a, t ∈ ((i + ρ)ω, (i + 1)ω], b̄(t) = { 0, t ∈ (iω, (i + ρ)ω], b, t ∈ ((i + ρ)ω, (i + 1)ω], (1.2) then model (1.1) can be rewritten as{ ut = D(t)(J ∗u−u)(t,x) + u(t,x)(ā(t) − b̄(t)u(t,x)), t > 0, l1 ≤ x ≤ l2, u(0,x) = u0(x), x ∈ [l1, l2], (1.3) which is a nonlocal dispersal piecewise smooth time-periodic system. The models with seasonal succession have been investigated by several authors. Ignoring the spatial evolution of the involved species, the effects of seasonal succession on the dynamics of population can be analysed by ODE models, see [14, 16] and references therein. There are also some investigations on it by the numerical method, see, e.g. [12, 20]. In [14], Hsu and Zhao first considered the single species model with seasonal succession:  zt = −δ, iω < t ≤ (i + ρ)ω, zt = z(a− bz), (i + ρ)ω < t ≤ (i + 1)ω, z(0) = z0 ∈ R+ := [0,∞), (1.4) where z(t) denotes the population density of a species at time t. They showed the threshold dynamics of model (1.4): when a(1 − ρ) − δρ ≤ 0, the unique solution of (1.4) converges to zero among all nonnegative initial value, while when a(1 −ρ) − δρ > 0, it converges to the unique positive ω-periodic solution of (1.4) for all positive initial value. A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 3 Taking spatial factor into account, Peng and Zhao [18] investigated the following local diffusion model with seasonal succession:  ut = −δu, iω < t ≤ (i + ρ)ω, x ∈ (l1, l2), ut −duxx = u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, x ∈ (l1, l2), u(t, l1) = u(t, l2) = 0, t ≥ 0, u(0,x) = u0(x) ≥ 0, x ∈ (l1, l2), (1.5) where the parameter d stands for the intensity of random diffusion. The positive constants ω,ρ,δ,a,b have the same biological interpretations as in (1.1), and the initial function u0 ∈ C2([l1, l2]). Denote by λl1 the principal eigenvalue of the eigenvalue problem  ϕt = −δϕ + λϕ, iω < t ≤ (i + ρ)ω, x ∈ (l1, l2), ϕt −dϕxx = aϕ + λϕ, (i + ρ)ω < t ≤ (i + 1)ω, x ∈ (l1, l2), ϕ > 0, (i + ρ)ω < t ≤ (i + 1)ω, x ∈ (l1, l2), ϕ(t, l1) = ϕ(t, l2) = 0, t ≥ 0, ϕ(t,x) = ϕ(t + ω,x), x ∈ (l1, l2). One can calculate exactly that λl1 = (1 − ρ)( π2d (l2−l1)2 − a) + ρδ. By the consequence of [27, Theorem 2.3.4], Peng and Zhao [18] has showed that, the solution of (1.5) converges to zero among all nonnegative initial value if λl1 ≥ 0, while when λl1 < 0, it converges to the unique positive ω-periodic solution of (1.5) for all nonnegative and not identically zero initial value. Specially, we can observe that (i) if (1 − ρ)a − ρδ > 0, then the solution of (1.5) converges to the unique positive ω-periodic solution of (1.5) for all nonnegative and not identically zero initial value; (ii) if (1−ρ)a−ρδ < 0, then there exists a critical value l̂ such that the solution of (1.5) converges to the unique positive ω-periodic solution of (1.5) for all nonnegative and not identically zero initial value if and only if l2 − l1 > l̂. The dynamics of the time-periodic nonlocal dispersal logistic equation have been studied by many authors (see [19, 24, 21, 23]). In [19], Rawal and Shen studied the eigenvalue problems of time-periodic nonlocal dispersal operator, and then showed that the existence of positive periodic solution relies on the sign of principal eigenvalue of a linearized eigenvalue problem. Sun et al. [24] considered a time- periodic nonlocal dispersal logistic equation in spatial degenerate environment. Shen and Vo [21] and Su et al. [23] have studied the asymptotic profiles of the generalised principal eigenvalue of time-periodic nonlocal dispersal operators under Dirichlet type boundary conditions and Neumann type boundary conditions, respectively. The models considered in the above mentioned work are all smooth periodic systems. The purpose of current paper is to study the dynamical properties of nonlocal dispersal model (1.1). Clearly, system (1.1) is in time-periodic environment and the dispersal term and reaction term are both discontinuous and periodic in t caused by the seasonal succession. Note that, by general semigroup theory (see [17]), (1.1) has a unique local solution u(t, ·; u0) with initial value u(0, ·; u0) = u0 ∈ C([l1, l2]), which is continuous in t. If u0 is nonnegative over [l1, l2], then by a comparison argument, u(t, ·; u0) exists and is nonnegative for all t > 0 (see Lemma 2.2). Next, we have the following theorem on the long time behavior of model (1.1). Theorem 1.1. Assume that (J) holds and −∞ < l1 < l2 < +∞. Let u(t, ·; u0) be the unique solution to (1.1) with the initial value u0(x) ∈ C([l1, l2]), where u0(x) is nonnegative and not identically zero. Then the following statements are true: 4 Z. LI AND B. DAI (1) If (1−ρ)a−ρδ > (1−ρ)d, then lim n→∞ u(t + nω,x; u0) = u ∗ (l1,l2) (t,x) in C([0,ω]× [l1, l2]), where u∗ (l1,l2) (t,x) is the unique ω-periodic positive solution of  ut = −δu, iω < t ≤ (i + ρ)ω, x ∈ [l1, l2], ut = d ∫ l2 l1 J(x−y)u(t,y)dy −du(t,x) + u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, x ∈ [l1, l2], u(t,x) = u(t + ω,x), t ≥ 0, x ∈ [l1, l2]; (1.6) (2) If 0 < (1−ρ)a−ρδ ≤ (1−ρ)d, then there exists a unique `∗ > 0 such that lim n→∞ u(t+nω,x; u0) = u∗ (l1,l2) (t,x) in C([0,ω] × [l1, l2]) if and only if l2 − l1 > `∗; (3) If (1−ρ)a−ρδ ≤ 0, then 0 is the unique nonnegative solution of (1.6), and lim t→∞ u(t,x; u0) = 0 uniformly for x ∈ [l1, l2]. Theorem 1.1 shows a complete classification on all possible long time behavior of system (1.1) with the assumption (J). The criteria governing persistence and extinction of the species show that: (i) When the duration of the bad season is too long (namely, ρ is close to 1), or the season is too bad (for example, bad weather and food shortages contributes to the large death rate δ) such that (1−ρ)a−ρδ ≤ 0, then the species will die out eventually regardless the initial population size; (ii) If the bad season is not long, or the food resource a is not small such that ρδ < (1−ρ)a ≤ (1−ρ)d + ρδ, then both persistence and extinction are determined by the range of the habitat of the species; (iii) When the good season is very long (i.e., ρ is close to 0), or the species has enough food such that (1 −ρ)(a−d) −ρδ > 0, then the species can persist for long time, which is different from that for the local diffusion model (1.5). The following conclusion concerns the asymptotic profile of the ω-periodic positive solution u∗ (l1,l2) of (1.6). Theorem 1.2. Assume that (J) holds. If (1 − ρ)a − ρδ > 0, then there exists ˆ̀ > 0 such that λ1(−L(l1,l2)) < 0 for every interval (l1, l2) with l2 − l1 > ˆ̀ and hence (1.6) admits a unique positive ω-periodic solution u∗ (l1,l2) (t,x). Moreover, lim −l1,l2→+∞ u∗(l1,l2)(t,x) = z ∗(t) in Cloc([0,ω] ×R), where z∗(t) is the unique ω-periodic positive solution of the following equation  zt = −δz, iω < t ≤ (i + ρ)ω, zt = z(a− bz), (i + ρ)ω < t ≤ (i + 1)ω, z(t + ω) = z(t), t ≥ 0. (1.7) The rest part of this paper is organized as follows. Sections 2 are devoted to the global existence and uniqueness of solution of (1.1). In Section 3, we then study the long-time dynamical behavior of system (1.1) based on the results for the time-periodic eigenvalue problem and time periodic upper- lower solutions. We also show some numerical simulations and discussion in Section 4 and the final section, respectively. 2. Well-posedness In this section, we show the existence and uniqueness of the global solution of (1.1). Before the statement of well-posedness of solution to (1.1), we provide a maximum principle. A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 5 Lemma 2.1 (Maximum principle). Let m be a positive integer. Assume that (J) holds and −∞ < l1 < l2 < +∞. Suppose that v,vt ∈ C([0,mω] × [l1, l2]),c ∈ L∞([0,mω] × [l1, l2]) and  vt ≥−δv, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, vt ≥ d ∫ l2 l1 J(x−y)v(t,y)dy −dv(t,x) + c(t,x)v, (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, v(0,x) ≥ 0, x ∈ [l1, l2], (2.1) where i = 0, 1, · · · ,m − 1. Then v(t,x) ≥ 0 for (t,x) ∈ [0,mω] × [l1, l2]. Moreover, if v(0,x) 6≡ 0 in [l1, l2], then v(t,x) > 0 for (t,x) ∈ (ρω,mω] × (l1, l2); if v(0,x) > 0 in (l1, l2), then v(t,x) > 0 for (t,x) ∈ (0,mω] × (l1, l2). Proof. Let V (t,x) = ektv(t,x). Then V (0, ·) ≥ 0 and V (t,x) satisfies  Vt ≥ p0V (t,x), iω < t ≤ (i + ρ)ω, x ∈ [l1, l2], Vt ≥ d ∫ l2 l1 J(x−y)V (t,y)dy + p1(t,x)V (t,x), (i + ρ)ω < t ≤ (i + 1)ω, x ∈ [l1, l2], (2.2) where p0 = k − δ, p1(t,x) = k + c(t,x) −d. Due to the boundedness of c, there exists k > 0 such that p0 > 0 and inf t∈[0,mω],x∈[l1,l2] p1(t,x) > 0. We now claim that V (t,x) ≥ 0 in [0,mω] × [l1, l2]. Let p1,0 = supt∈[0,mω],t∈[l1,l2] p1(t,x) and T0 = min { mω, 1 2(p0+d+p1,0) } . In the following, we will show that the claim holds for t ∈ (0,T0],x ∈ [l1, l2]. Assume to the contrary that Vinf := inft∈(0,T0),x∈[l1,l2] V (t,x) < 0. Then there exists (t0,x0) ∈ (0,T0] × [l1, l2] such that Vinf = V (t0,x0) < 0. Notice that there are tn ∈ (0, t0] and xn ∈ [l1, l2] such that V (tn,xn) → Vinf as n →∞. We only need to consider the following two cases. Case 1. t0 ∈ (i0ω, (i0 + ρ)ω] for some i0 ∈{0, 1, · · · ,m− 1}. In this case, tn ∈ (i0ω,t0] for large n. Then it follows from (2.2) that V (tn,xn) −V (0,xn) = i0−1∑ i=0 (∫ (i+ρ)ω iω Vtdt + ∫ (i+1)ω (i+ρ)ω Vtdt ) + ∫ tn i0ω Vtdt ≥ i0−1∑ i=0 ∫ (i+ρ)ω iω p0V (t,xn)dt + ∫ tn i0ω p0V (t,xn)dt + i0−1∑ i=0 ∫ (i+1)ω (i+ρ)ω [ d ∫ l2 l1 J(xn −y)V (t,y)dy + p1(t,xn)V (t,xn) ] dt ≥ ∫ tn 0 p0Vinf dt + d ∫ tn 0 ∫ l2 l1 J(xn −y)Vinf dydt + ∫ tn 0 p1,0Vinf dt ≥ tn(p0 + d + p1,0)Vinf ≥ t0(p0 + d + p1,0)Vinf for large n. Recall that V (0,xn) ≥ 0 for n = 0, 1, 2, · · · . Thus we have V (tn,xn) ≥ t0(p0 + d + p1,0)Vinf 6 Z. LI AND B. DAI for large n. Taking the limit as n →∞, it holds that Vinf ≥ t0(p0 + d + p1,0)Vinf ≥ 1 2 Vinf, which is a contradiction. Case 2. t0 ∈ ((i0 + ρ)ω, (i0 + 1)ω] for some i0 ∈{0, 1, · · · ,m− 1}. Similarly, we can also derive a contradiction since V (tn,xn) −V (0,xn) = i0−1∑ i=0 (∫ (i+ρ)ω iω Vtdt + ∫ (i+1)ω (i+ρ)ω Vtdt ) + ∫ (i0+ρ)ω i0ω Vtdt + ∫ tn (i0+ρ)ω Vtdt ≥ tn(p0 + d + p1,0)Vinf ≥ t0(p0 + d + p1,0)Vinf for large n. Therefore, V (t,x) ≥ 0 for (t,x) ∈ (0,T0] × [l1, l2] and then v(t, ·) ≥ 0 for t ∈ [0,T0]. If T0 = mω, then v(t,x) ≥ 0 in [0,mω]× [l1, l2] follows directly; while if T0 < mω, we can repeat the above process by replacing V (0, ·) and (0,T0] as V (T0, ·) and (T0,mω]. Obviously, this process can be repeated in finite many times, and consequently, v(t, ·) ≥ 0 for t ∈ [0,mω]. Now we assume that v(0,x) 6≡ 0 in [l1, l2]. To finish the proof, it suffices to prove that V > 0 in (ρω,ω] × (l1, l2). Suppose that there exists a point (t∗,x∗) ∈ (ρω,ω] × (l1, l2) such that V (t∗,x∗) = 0. First, we prove that V (t∗,x) = 0 for x ∈ (l1, l2). Otherwise, we can find x̃ ∈ [l1, l2] ∩∂{x ∈ (l1, l2) : V (t∗,x) > 0}. Then V (t∗, x̃) = 0 and it follows from (2.2) that 0 ≥ Vt(t∗, x̃) ≥ d ∫ l2 l1 J(x̃−y)V (t∗,y)dy > 0, by assumption (J). This is impossible, and hence V (t∗,x) = 0 for x ∈ (l1, l2). Thus, we can derive from (2.2) that for x ∈ [l1, l2] −V (0,x) = V (t∗,x) −V (0,x) = ∫ ρω 0 Vtdt + ∫ t∗ ρω Vtdt ≥ p0 ∫ ρω 0 V (t,x)dt + d ∫ t∗ ρω ∫ l2 l1 J(x−y)V (t,y)dydt + ∫ t∗ ρω p1(t,x)V (t,x)dt ≥ 0. This means that v(0,x) ≡ 0 in [l1, l2], which is a contradiction. � Lemma 2.2 (Existence and uniqueness). Assume that (J) holds and −∞ < l1 < l2 < +∞. Then for any nonnegative and bounded initial value u0(x) ∈ C([l1, l2]), problem (1.1) admits a unique global solution u ∈ C1,0((iω, (i + ρ)ω] × [l1, l2]) ∩ C1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) for i ∈ Z+. Moreover, u(t,x) > 0 for t > 0 and x ∈ (l1, l2), if u0(x) > 0 in (l1, l2). Proof. At first, we set û = e−δtu0(x) for t ∈ [0,ρω]. Then û ∈ C1,0((0,ρω] × [l1, l2]) satisfies{ ût = −δû, 0 < t ≤ ρω, l1 ≤ x ≤ l2, û(0,x) = u0(x), l1 ≤ x ≤ l2. A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 7 Consider the following problem  ut = d ∫ l2 l1 J(x−y)u(t,y)dy −du(t,x) + u(a− bu), ρω < t ≤ ω, x ∈ (l1, l2), u(ρω,x) = e−δρωu0(x), x ∈ [l1, l2]. (2.3) Then one can apply the Banach’s fixed theorem and comparison argument (see [1]) to conclude that (2.3) has a unique solution ū(t,x) ∈ C1,0((ρω,ω] × [l1, l2]). Moreover, by the Maximum principle and comparison argument, we have that 0 < ū(t,x) ≤ max { a b , max −h0≤x≤h0 u0(x) } for t ∈ (ρω,ω],x ∈ (l1, l2). Define u(t,x) = { û(t,x) in [0,ρω] × [l1, l2], ū(t,x) in [ρω,ω] × [l1, l2]. We have that u ∈ û ∈ C1,0((0,ρω] × [l1, l2]) ∩C1,0((ρω,ω] × [l1, l2]). Based on the above obtained function u, we let u1(t,x) = e −δ(t−ω)u(ω,x) for ω ≤ t ≤ (1 + ρ)ω. Then u1 ∈ C1,0((ω, (1 + ρ)ω] × [l1, l2]) satisfies{ u1,t = −δu1, ω < t ≤ (1 + ρ)ω, l1 ≤ x ≤ l2, u1(ω,x) = u(ω,x), l1 ≤ x ≤ l2. Likewise, the nonlocal dispersal problem  ut = d ∫ l2 l1 J(x−y)u(t,y)dy −du(t,x) + u(a− bu), (1 + ρ)ω < t ≤ 2ω, x ∈ (l1, l2), u((1 + ρ)ω,x) = e−δρωu(ω,x), x ∈ [l1, l2] has a unique solution ū1 ∈ C1,0(((1 + ρ)ω, 2ω] × [l1, l2]), in which 0 < ū1(t,x) ≤ max { a b , max −h0≤x≤h0 u0(x) } for t ∈ ((1 + ρ)ω, 2ω],x ∈ (l1, l2). Define u(t,x) =   u(t,x) in [0,ω] × [l1, l2], u1(t,x) in [ω, (1 + ρ)ω] × [l1, l2], ū1(t,x) in [(1 + ρ)ω, 2ω] × [l1, l2]. Then it holds that u ∈ C1,0((iω, (i + ρ)ω] × [l1, l2]) ∩C1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) for i = 0, 1. By repeating the above procedure, we therefore obtain the existence and uniqueness of the solution (u,g,h) of (1.1). � 3. Global dynamics In this subsection, we first establish the periodic upper-lower solutions method for model (1.1). Using this method, we can consider the long time behavior of model (1.1). 8 Z. LI AND B. DAI 3.1. The method of periodic upper-lower solutions. Following Hess [11], we can define the upper- lower solutions of (1.6) as follows. Definition 3.1. A bounded and continuous function ũ(t,x) is called an upper-solution of (1.6) if ũ(t,x) ∈ C1,0((iω, (i + ρ)ω] × [l1, l2]) ∩C1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) satisfies  ũt ≥−δũ, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ũt ≥ d ∫ l2 l1 J(x−y)ũ(t,y)dy −dũ(t,x) + ũ(a− bũ), (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, ũ(0,x) ≥ ũ(ω,x), x ∈ [l1, l2]. (3.1) for i ∈ Z+. Meanwhile, the function û(t,x) ∈ C1,0((iω, (i+ρ)ω]×[l1, l2])∩C1,0(((i+ρ)ω, (i+1)ω]×[l1, l2]) is called a lower-solution of (1.6) if the inequalities in (3.1) are reversed. Similarly, we can define the upper-solution (resp. lower-solution) of (1.1) by replacing the inequality ũ(0,x) ≥ ũ(ω,x) in (3.1) as ũ(0,x) ≥ ũ0(x) (resp. û(0,x) ≤ û0(x)). We say that a pair of upper-lower solution ũ and û are ordered if ũ(t,x) ≥ û(t,x) in [0, +∞) × [l1, l2]. Using the semigroup theory, we have the following result. Lemma 3.1. Let D(t), ā(t) and b̄(t) be defined as in (1.2). Assume that u(t,x) is bounded for (t,x) ∈ [0, +∞] × [l1, l2]. Then u(t,x) is a solution of (1.1) if and only if u(t,x) = u(0,x) + ∫ t 0 [ D(s) (∫ l2 l1 J(x−y)u(s,y)dy −u(s,x) ) + u(s,x)[ā(s) − b̄(s)u(s,x)] ] ds, t > 0, x ∈ [l1, l2]. (3.2) Proof. It follows from the semigroup method [17], we have that u(t,x) = e−tu(0,x) + ∫ t 0 e−(t−s) [ D(s) (∫ l2 l1 J(x−y)u(s,y)dy −u(s,x) ) + u(s,x) + u(s,x)[ā(s) − b̄(s)u(s,x)] ] ds, (3.3) which implies ∫ t 0 u(s,x)ds = (1 −e−t)u(0,x) + I1[u](t,x), (3.4) where I1[u](t, x) = ∫ t 0 ∫ s 0 e −(s−z) [ D(z) (∫ l2 l1 J(x − y)u(z, y)dy − u(z, x) ) + u(z, x)[1 + ā(z) − b̄(z)u(z, x)] ] dzds = ∫ t 0 ∫ t z e −(s−z) [ D(z) (∫ l2 l1 J(x − y)u(z, y)dy − u(z, x) ) + u(z, x)[1 + ā(z) − b̄(z)u(z, x)] ] dsdz + ∫ t 0 (1 − es−t) [ D(s) (∫ l2 l1 J(x − y)u(s, y)dy − u(s, x) ) + u(s, x)[1 + ā(s) − b̄(s)u(s, x)] ] ds. Therefore, (3.2) can be derived from (3.3) and (3.4). On the other hand, if u satisfies (3.2), then we can also show that (3.3) holds. � Similarly, we have the following result for (1.6). A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 9 Lemma 3.2. Assume that u(t,x) is bounded for (t,x) ∈ [0, +∞]× [l1, l2]. Then u(t,x) is a solution of (1.6) if and only if  u(t,x) = u(0,x) + ∫ t 0 [ D(s) (∫ l2 l1 J(x−y)u(s,y)dy −u(s,x) ) + u(s,x)[ā(s) − b̄(s)u(s,x)] ] ds, t > 0, x ∈ [l1, l2], u(t,x) = u(t + ω,x), t ≥ 0, x ∈ [l1, l2]. (3.5) Corollary 3.3. Assume that u(t,x) is bounded for (t,x) ∈ [0, +∞] × [l1, l2]. Then u(t,x) is a solution of (1.6) if and only if  u(t,x) = e−Ctu(0,x) + ∫ t 0 e−C(t−s) [ D(s) (∫ l2 l1 J(x−y)u(s,y)dy −u(s,x) ) + u(s,x)[C + ā(s) − b̄(s)u(s,x)] ] ds, t > 0, x ∈ [l1, l2], u(t,x) = u(t + ω,x), t ≥ 0, x ∈ [l1, l2], (3.6) where C is a constant. The following conclusion establishes a method of periodic upper-lower solutions. Theorem 3.4. Assume that (J) holds and u0(x) ∈ C([l1, l2]) is bounded. Let u(t,x) be the unique solution to (1.1) and ũ(t,x), û(t,x) be a pair of ordered and bounded upper-lower solutions to (1.6) satisfying û(0,x) ≤ u0(x) ≤ ũ(0,x) on [l1, l2]. Then the time periodic problem (1.6) admits a minimal solution u ∈ C1,0((iω, (i + ρ)ω] × [l1, l2]) ∩ C1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) and a maximal solution u ∈ C1,0((iω, (i + ρ)ω] × [l1, l2]) ∩C1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) (i ∈ Z+) satisfying û(t,x) ≤ u(t,x) ≤ lim n→∞ u(t + nω,x) ≤ lim n→∞ u(t + nω,x) ≤ u(t,x) ≤ ũ(t,x) for t ≥ 0,x ∈ [l1, l2]. Proof. For notational convenience, denote Q = (0, +∞) × [l1, l2] and J ∗u(t,x) = ∫ l2 l1 J(x−y)u(t,y)dy for u ∈ C(Q). Set I = [ −‖û‖L∞(Q) −‖ũ‖L∞(Q),‖û‖L∞(Q) + ‖ũ‖L∞(Q) ] . At first we take K > 1 such that for u ∈ I, u[ā(t)−b̄u] +Ku and ā(t)u−2(‖û‖L∞(Q) +‖ũ‖L∞(Q))b̄(t)u+ Ku are both increasing with respect to u. Define L[u](t,x) = u(t,x) −e−Ktu(0,x) − ∫ t 0 e−K(t−s)D(s) [J ∗u(s,x) −u(s,x)] ds and F(t,u) = u(t,x)[ā(t) − b̄(t)u(t,x)] + Ku(t,x). We construct two iterations sequences by the following linear nonlocal evolution equations{ L[un](t,x) = ∫ t 0 e−K(t−s)F(s,un−1(s,x))ds, (t,x) ∈ Q, un(0,x) = un−1(ω,x), x ∈ [l1, l2] (3.7) and { L[un](t,x) = ∫ t 0 e−K(t−s)F(s,un−1(s,x))ds, (t,x) ∈ Q, un(0,x) = un−1(ω,x), x ∈ [l1, l2], (3.8) 10 Z. LI AND B. DAI where n ≥ 1,u0(t,x) = ũ(t,x) and u0(t,x) = û(t,x). We can check that a sufficiently large constant is an upper-solution of (3.7) (resp. (3.8)) since K > 1. Then an application of Banach’s fixed point theorem and comparison principle yields that the linear initial value problem (3.7) (resp. (3.8)) has a unique bounded global solution un(t,x) (resp. un(t,x)) for any n ≥ 1. We complete the proof of this theorem by the following four steps. Step 1. The sequences {un}∞n=1 and {un}∞n=1 satisfy û(t,x) ≤ un(t,x) ≤ un+1(t,x) ≤ u(t + nω,x) ≤ un+1(t,x) ≤ un(t,x) ≤ ũ(t,x), (t,x) ∈ Q, (3.9) for n ≥ 1. Since ũ(t,x) is a bounded upper-solution of (1.6) and ũ(0,x) ≥ u0(x), we see that ũ(t,x) is also a bounded upper-solution of (1.1). Then by Lemma 2.1, we have ũ(t,x) ≥ u(t,x) in Q. It follows from (3.7) and Corollary 3.3 that u1(t,x) satisfies  u1t (t,x) = D(t) [ J ∗u1(t,x) −u1(t,x) ] + ũ(t,x)[ā(t) − b̄(t)ũ(t,x)] + K[ũ(t,x) −u1(t,x)], (t,x) ∈ Q, u1(0,x) = ũ(ω,x), x ∈ [l1, l2]. (3.10) Set w1(t,x) = u0(t,x) −u1(t,x) = ũ(t,x) −u1(t,x). Since{ ũt(t,x) ≥ D(t) [J ∗ ũ(t,x) − ũ(t,x)] + ũ(t,x)[ā(t) − b̄(t)ũ(t,x)], (t,x) ∈ Q, ũ(0,x) ≥ ũ(ω,x), x ∈ [l1, l2], there holds that { w1t (t,x) ≥ D(t) [ J ∗w1(t,x) −w1(t,x) ] −Kw1(t,x), (t,x) ∈ Q, w1(0,x) ≥ 0, x ∈ [l1, l2], which together with Lemma 2.1 implies that w1(t,x) ≥ 0 and so u1(t,x) ≤ u0(t,x) = ũ(t,x) for (t,x) ∈ Q. By a similar manner for lower-solution û(t,x), we have u(t,x) ≥ û(t,x) and u1(t,x) ≥ u0(t,x) = û(t,x) for (t,x) ∈ Q. Now we let w2(t,x) = u1(t,x) − u1(t,x). In view of û(t,x) ≤ u(t,x) ≤ ũ(t,x) in Q, by (3.7) and (3.8), we have w2(0,x) = ũ(ω,x) − û(ω,x) ≥ 0 and w2t (t,x) = D(t) [ J ∗w2(t,x) −w2(t,x) ] −Kw2(t,x) + ā(t)[ũ(t,x) − û(t,x)] − b̄(t)[ũ2(t,x) − û2(t,x)] + K[ũ(t,x) − û(t,x)] ≥ D(t) [ J ∗w2(t,x) −w2(t,x) ] −Kw2(t,x) in Q, where the conditions satisfied by K are used here. It follows from Lemma 2.1 that w2(t,x) ≥ 0 and hence u1(t,x) ≥ u1(t,x) in Q. Next, we show that u1(t,x) ≤ u(t + ω) ≤ u1(t,x) in Q. Let w3(t,x) = u1(t,x) −u(t + ω,x). Notice that u(t + ω,x) satisfies ut(t + ω,x) = D(t + ω) [J ∗ ũ(t + ω,x) − ũ(t + ω,x)] + ũ(t + ω,x)[ā(t + ω) − b̄(t + ω)ũ(t + ω,x)] = D(t) [J ∗ ũ(t + ω,x) − ũ(t + ω,x)] + ũ(t + ω,x)[ā(t) − b̄(t)ũ(t + ω,x)] in Q. (3.11) A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 11 Combining (3.10) and (3.11), there holds that w3(0,x) = u1(0,x) −u(ω,x) = ũ(ω,x) −u(ω,x) ≥ 0 for x ∈ [l1, l2] and w3t (t,x) = D(t) [ J ∗w3(t,x) −w3(t,x) ] + ũ(t,x) [ ā(t) − b̄(t)ũ(t,x) ] + K [ ũ(t,x) −u1(t,x) ] −u(t + ω) [ ā(t) − b̄(t)u(t + ω,x) ] = D(t) [ J ∗w3(t,x) −w3(t,x) ] + [ ā(t) − b̄(t) ( u1(t,x) + u(t + ω,x) )] w3(t,x) + ā(t) [ ũ(t,x) −u1(t,x) ] − b̄(t) [ ũ(t,x) + u1(t,x) ][ ũ(t,x) −u1(t,x) ] + K [ ũ(t,x) −u1(t,x) ] . Since ũ(t,x) ≥ u1(t,x) ≥ u1(t,x) ≥ û(t,x), by the condition satisfied by K, we see that ā(t) [ ũ(t,x) −u1(t,x) ] − b̄(t) [ ũ(t,x) + u1(t,x) ][ ũ(t,x) −u1(t,x) ] + K [ ũ(t,x) −u1(t,x) ] ≥ 0 in Q, which leads to that w3t (t,x) ≥ D(t) [ J ∗w3(t,x) −w3(t,x) ] + [ ā(t) − b̄(t) ( u1(t,x) + u(t + ω,x) )] w3(t,x) in Q. Due to the boundedness of u1(t,x) and u(t + ω,x), we can derive from Lemma 2.1 that w3(t,x) ≥ 0 and then u1(t,x) ≥ u(t + ω,x) in Q. Similarly, we also have u1(t,x) ≤ u(t + ω,x) in Q. Therefore, the following inequalities are true: û(t,x) ≤ u1(t,x) ≤ u(t + ω,x) ≤ u1(t,x) ≤ ũ(t,x), (t,x) ∈ Q. An induction argument implies the monotone property (3.9) immediately. Since {un} and {un} monotonically bounded sequences, there exist two bounded function u(t,x) and u(t,x) such that lim n→∞ un(t,x) = u(t,x) and lim n→∞ un(t,x) = u(t,x) and u(t,x) ≤ lim n→∞ u(t + nω,x) ≤ lim n→∞ u(t + nω,x) ≤ u(t,x) for each (t,x) ∈ Q. Thus from the dominated convergence theorem, we obtain that u(t,x) and u(t,x) are bounded solutions of the initial value problem{ ut(t,x) = D(t) [J ∗u(t,x) −u(t,x)] + u(t,x)[ā(t) − b̄(t)u(t,x)], (t,x) ∈ Q, u(0,x) = u(ω,x), x ∈ [l1, l2]. Step 2. We prove that u(t,x),u(t,x) ∈ C1,0((iω, (i+ρ)ω]×[l1, l2])∩C1,0(((i+ρ)ω, (i+ 1)ω]×[l1, l2]) for all i ∈ Z+. Since u(t,x) = u(0,x) + ∫ t 0 [ D(s) [ J ∗u(s,x) −u(s,x) ] + u(s,x) [ ā(s) − b̄(s)u(s,x) ]] ds, it holds that u(t + ε,x) −u(t,x) = ∫ t+ε t [ D(s) [ J ∗u(s,x) −u(s,x) ] + u(s,x) [ ā(s) − b̄(s)u(s,x) ]] ds for each fixed (t,x) ∈ Q, where |ε| > 0 is sufficiently small. Then, we have |u(t + ε,x) −u(t,x)| ≤ ∫ t+ε t ∣∣∣[D(s)[J ∗u(s,x) −u(s,x)] + u(s,x)[ā(s) − b̄(s)u(s,x)]]∣∣∣ ds ≤ C|ε|, where C > 0 is a constant independent of ε. This means that u(t,x) is continuous in t ∈ [0, +∞). The continuity of u(t,x) in x ∈ [l1, l2] follows from the argument in [1]. 12 Z. LI AND B. DAI For any t0 ∈ (0, +∞), there must exist a unique i0 ∈ Z+ such that either t0 ∈ (i0ω, (i0 + ρ)ω], or t0 ∈ ((i0 + ρ)ω, (i0 + 1)ω]. When t0, t0 + ε ∈ (i0ω, (i0 + ρ)ω], we see that lim ε→0 u(t + ε,x) −u(t,x) ε = lim ε→0 1 ε ∫ t+ε t [ − δu(s,x) ] ds = −δ lim ε→0 u(t + θε,x) (0 < θ < 1) = −δu(t,x). When t0, t0 + ε ∈ ((i0 + ρ)ω, (i0 + 1)ω], we have lim ε→0 u(t + ε,x) −u(t,x) ε = lim ε→0 1 ε ∫ t+ε t [ d [ J ∗u(s,x) −u(s,x) ] + u(s,x) [ a− bu(s,x) ]] ds = lim ε→0 d [ J ∗u(t + θε,x) −u(t + θε,x) ] + u(t + θε,x) [ a− bu(t + θε,x) ] (0 < θ < 1) = d [ J ∗u(t,x) −u(t,x) ] + u(t,x) [ a− bu(t,x) ] . Hence, u(t,x) ∈ C1,0((iω, (i + ρ)ω] × [l1, l2]) ∩C1,0(((i + ρ)ω, (i + 1)ω] × [l1, l2]) for all i ≥ 0 due to the arbitrariness of t0. The proof for u(t,x) is similar. Step 3. We prove that u(t,x) = u(t + ω,x) and u(t,x) = u(t + ω,x) for all t ≥ 0. Let v(t,x) = u(t + ω,x) −u(t,x). Note that D(t), ā(t) and b̄(t) are all ω−periodic in t. Then vt(t,x) = D(t) [∫ Ω J(x−y)v(t,y)dy −v(t,x) ] + ā(t)v(t,x) − b̄(t)[u(t + ω,x) + u(t,x)]v(t,x). (3.12) Since v(0,x) = u(ω,x) −u(0,x) = 0, the uniqueness of solution of initial value problem (3.12) implies that v(t,x) ≡ 0 in Q, equivalently, u(t,x) is ω-periodic in t. Similarly, we can also prove that u(t,x) = u(t + ω,x), and omit the details here. Step 4. We show the maximality of u(t,x) and minimality of u(t,x). Notice that every ω-periodic solution u∗(t,x) of (1.6) satisfies û(t,x) ≤ u∗(t,x) ≤ ũ(t,x). Meanwhile, u∗(t,x) is a lower-solution as well as a upper-solution of (1.6). By choosing ũ and u∗ as a pair of upper-lower solutions to (1.6), there holds that u∗(t,x) ≤ un(t,x) ≤ ũ(t,x) and hence u∗(t,x) ≤ u(t,x) ≤ ũ(t,x). On the other hand, if we take u∗ and û as a pair of upper-lower solutions to (1.6), then û(t,x) ≤ u(t,x) ≤ u∗(t,x). � 3.2. Proofs of Theorems 1.1 and 1.2. In this subsection, we complete the proof of Theorems 1.1 and 1.2. Linearizing model (1.1) at zero, we obtain the time-periodic eigenvalue problem{ vt + δv = λv, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, vt −d(J ∗v −v)(t,x) + av = λv, (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2. (3.13) It is well known (see, e.g., [3, 5, 6]) that the time independent eigenvalue equation d(J ∗φ−φ)(t,x) + aφ(x) = −σφ(x), x ∈ [l1, l2]. (3.14) admits a principal eigenvalue σ1, which satisfies σ1 < d−a. Moreover, from [4, Proposition 3.4], we see that Proposition 3.5. Assume that (J) holds and −∞ < l1 < l2 < +∞. Then the following hold true: (1) σ1 is strictly decreasing and continuous in ` := l2 − l1; (2) liml2−l1→+∞σ1 = −a; (3) liml2−l1→0+ σ1 = d−a. A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 13 Let φ1(x) be the positive eigenfunction of (3.14) associated with σ1. By defining σ(t) = { δ, t ∈ (iω, (i + ρ)ω], σ1, t ∈ ((i + ρ)ω, (i + 1)ω], we see that D(t) [J ∗φ1(x) −φ1(x)] + ā(t)φ1(x) = −σ(t)φ1(x), ∀t ∈ R,x ∈ Ω. Set ϕ(t,x) = exp [( (1 −ρ)σ1 + ρδ ) t− ∫ t 0 σ(s)ds ] φ1(x). (3.15) Then there holds that  ϕt + δϕ = [(1 −ρ)σ1 + ρδ]ϕ, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ϕt −d(J ∗ϕ−ϕ)(t,x) + aϕ = [(1 −ρ)σ1 + ρδ]ϕ, (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, ϕ(t,x) = ϕ(t + ω,x), t ≥ 0, l1 ≤ x ≤ l2. This means that (1 −ρ)σ1 + ρδ is a eigenvalue of (3.13) with the positive eigenfunction ϕ(t,x). In the proof of Theorems 1.1, we will show that (1−ρ)σ1 + ρδ serves as a threshold which determines whether the species can persist. Proof of Theorem 1.1. Let λ1 = (1 −ρ)σ1 + ρδ. we first consider two cases on the sign of λ1. Case 1. Suppose that λ1 < 0. One can easily check that a sufficiently large positive constant M is a upper-solution of (1.1) as well as the upper-solution of (1.6). Following the comparison argument in Theorem 3.4, we see that for any (t,x) ∈ [0,ω] × [l1, l2], u(t + nω,x; M) is non-increasing with respect to n. Then the function u+(t,x) := lim n→∞ u(t + nω,x; M), (t,x) ∈ [0,ω] × [l1, l2] is well-defined and upper semi-continuous. On the other hand, let ϕ(t,x) be defined as in (3.15). For any 0 < ε � 1, by λ1 < 0, we see that εϕ is a lower-solution of (1.1) as well as a lower-solution of (1.6). Again, by the comparison argument, u(t + nω,x; εϕ(0,x)) is non-decreasing as n increases for any (t,x) ∈ [0,ω] × [l1, l2]. Thus, the function u−(t,x) := lim n→∞ u(t + nω,x; εϕ), (t,x) ∈ [0,ω] × [l1, l2] is well-defined and lower semi-continuous. Obviously, u− ≤ u+. Next, we show u−(t,x) ≡ u+(t,x). For this purpose, we define γn := inf { ln α : 1 α u(t + nω,x; M) ≤ u(t + nω,x; εϕ(0,x)) ≤ αu(t + nω,x; M), (t,x) ∈ [0,ω] × [l1, l2] } . Since the sequence {u(· + nω, ·; εϕ(0, ·))}n is non-decreasing and {u(· + nω, ·; M)}n is non-increasing, u(· + nω, ·; εϕ(0, ·)) and u(· + nω, ·; M) will be closer to each other when n decreases. Consequently, {γn}n is a non-increasing sequence, and then the limit γ∗ := limn→∞γn exists. If γ∗ > 0, then by the comparison argument, we can construct some α∗ > 1 and 0 < σ � 1 such that 1α∗u(· + nω, ·; M) ≤ u(· + nω, ·; εϕ(0, ·)) ≤ α∗u(· + nω, ·; M) and ln α∗ < γn − σ for sufficiently large n. This causes a contradiction with the definition of γ∗. Hence, γ∗ = 0 and the equality u + = u− follows. Notice that u+ is upper semi-continuous and u− is lower semi-continuous. Then u∗ := u+ is con- tinuous and inf [0,ω]×[l1,l2] u∗ > 0. Using Dini’s Theorem, we have limn→∞u(· + nω, ·; M) = limn→∞u(· + nω, ·; εϕ(0, ·)) = u∗ uniformly for (t,x) ∈ [0,ω] × [l1, l2]. This also means that u(t + ω,x; u∗(0,x)) = lim n→∞ u(t + ω,x; u(nω, ·; M)) = lim n→∞ u(t + (n + 1)ω,x; M) = u∗(t,x). 14 Z. LI AND B. DAI This is, u(t,x; u∗(0,x)) is ω-periodic in t. The existence of time periodic positive solution of (1.6) is established. By the above contraction argument, we can obtain the existence of the solutions of (1.6). The uniqueness follows directly from Theorem 3.4 and the above argument. To emphasize the dependence of u∗(t,x) on l1, l2, denote by u ∗ (l1,l2) (t,x) the unique time periodic positive solution of (1.6). Since εϕ(t,x) ≤ u(t,x; u0) ≤ M, (t,x) ∈ [0,ω] × [l1, l2] for 0 < ε � 1 and M � 1, the above contraction argument also implies that lim n→∞ u(t + nω,x; u0) = u ∗ (l1,l2) (t,x) uniformly in C([0,ω] × [l1, l2]). The global stability of u∗(l1,l2) can also be inferred from Theorem 3.4 and the uniqueness of the solutions of (1.6). Case 2. Suppose that λ1 ≥ 0. At first, we show the nonexistence of positive solution of (1.6). By way of contradiction, suppose that v∗ is a positive solution of (1.6). Then we can choose � > 0 small enough such that �ϕ < v∗ in [0, +∞) × [l1, l2]. There holds that 0 ≤ λ1ϕ(t,x) = ∂tϕ(t,x) −D(t) [∫ l2 l1 J(x−y)ϕ(t,y)dy −ϕ(t,x) ] − ā(t)ϕ(t,x) ≤ ∂tϕ(t,x) −D(t) [∫ l2 l1 J(x−y)ϕ(t,y)dy −ϕ(t,x) ] − ā(t)ϕ(t,x) + b̄(t)ϕ2(t,x) in [0, +∞) × [l1, l2], which means ϕ is an upper-solution of (1.6). It follows from the comparison argument that v∗ ≤ ϕ in [0, +∞) × [l1, l2]. This is a contradiction. Hence the equation (1.6) admits no positive solution. Since λ1 ≥ 0, a simple calculation gives that for large e > 0, eϕ(t,x) and 0 are a pair of upper-lower solutions of (1.1) as well as a pair of upper-lower solutions of (1.6). It then follows from Theorem 3.4 that the time periodic problem (1.6) admits a minimal solution u and a maximal solution u satisfying 0 ≤ u(t,x) ≤ lim n→∞ u(t + nω,x; u0) ≤ lim n→∞ u(t + nω,x; u0) ≤ u(t,x) ≤ eϕ(t,x) in [0, +∞) × [l1, l2]. The nonexistence of positive solution to (1.6) implies that u = u = 0. Thus, the solution u(t,x; u0) of (1.1) converges to 0 point by point. Since u,u ∈ C([0, +∞) × [l1, l2]) and the sequences constructed in (3.9) are monotone, we have lim t→∞ u(t,x; u0) = 0 uniformly for x ∈ [l1, l2] by Dini’s Theorem. From the above two cases, one can obtain that the sign of (1 −ρ)σ1 + ρδ can completely determine the long time behavior of the species. Therefore, the conclusions of Theorem 1.1 follows from the above argument and Proposition 3.5. � We further discuss the behaviors of the positive ω-periodic solution to (1.6). Look at the ODE system (1.4). It is known from [14, Theorem 2.1] that (1.4) admits a unique positive ω-periodic solution z∗ satisfying the equation (1.7), if and only if (1 −ρ)a−ρδ > 0, where z∗ ∈ C1((iω, (i + ρ)ω]) ∩C1(((i + ρ)ω, (i + 1)ω]) is bounded. Moreover, if (1 −ρ)a−ρδ ≤ 0, then the solution z(t; z0) of (1.4) converges to 0 for all z0 ∈ [0, +∞) as t → +∞, while if (1−ρ)a−ρδ > 0, then limn→∞(z(t + nω; z0)−z∗(t)) = 0 in C([0,ω]) for all z0 ∈ (0, +∞). Using this fact, we can prove Theorem 1.2. Proof of Theorem 1.2. Since (1 −ρ)a−ρδ > 0, by Proposition 3.5, there holds that lim l2−l1→+∞ [ (1 −ρ)σ1 + ρδ ] = ρδ − (1 −ρ)a < 0, A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 15 and thus there exists a large ˆ̀> 0 such that (1 −ρ)σ1 + ρδ < 0 as l2 − l1 > ˆ̀. The existence and uniqueness of u∗ (l1,l2) (t,x) follow from Theorem 1.1. Note that z∗(t) satisfies  zt = −δz, 0 < t ≤ ρω, zt = z(a− bz), ρω < t ≤ ω, z(0) = z(ω). Then z∗(t) = e−δtz∗(0) for 0 ≤ t ≤ ρω. Meanwhile, u∗ (l1,l2) (t,x) = e−δtu∗ (l1,l2) (0,x) for 0 ≤ t ≤ ρω and l1 ≤ x ≤ l2. We make an assertion that for each 0 < � � 1, there exists `� ≥ ˆ̀ > 0 such that for each l1 ∈ (−∞,−`�) and l2 ∈ (`�, +∞), z∗(t) − � ≤ u∗(l1,l2)(t,x) ≤ z ∗(t) + �, (t,x) ∈ [ρω,ω] × [l1, l2]. (3.16) Only the proof for the lower bound will be given here since that for the upper bound is similar. Clearly, min[0,ω] z ∗(t) > 0. In fact, if there is t0 > 0 such that z ∗(t0) = 0, then z ∗(t) = 0 for all t ≥ t0, which is impossible as z∗ is periodic in t. Set 0 < � � 1. Then there exists η(�) > 0 such that ẑ(t) := (1 −η)z∗(t) ≥ z∗(t) − � > 0, t ∈ [ρω,ω]. Observe that ẑt(t) −d [∫ l2 l1 J(x−y)ẑ(t)dy − ẑ(t) ] − ẑ(t) [ a− bẑ(t) ] = (1 −η)z∗t (t) − (1 −η)z ∗(t)d [∫ l2 l1 J(x−y)dy − 1 ] − (1 −η)z∗(t) [ a− bz∗(t) ] − ẑ(t) [ a− bẑ(t) ] + (1 −η)z∗(t) [ a− bz∗(t) ] = −d(1 −η)z∗(t) [∫ l2 l1 J(x−y)dy − 1 ] − bη(1 −η)z∗2(t), ρω < t ≤ ω, l1 ≤ x ≤ l2. Denote El(t,x) := d(1 −η)z∗(t) [∫ l −l J(x−y)dy − 1 ] , (t,x) ∈ (ρω,ω] ×R. Since J ∈ C(R) ∩L∞(R) is nonnegative and lim l→+∞ ∫ l −l J(x)dx = 1, we know that El is non-decreasing with respect to l > 0 and continuous, bounded for all (l, t,x) ∈ (0, +∞) × (ρω,ω] × R. It then follows from Dini’s Theorem that El(t,x) converges to zero locally uniformly in (ρω,ω]×R as l → +∞. Hence, there exists `� ≥ ˆ̀> 0 such that for each l1 ∈ (−∞,−`�), l2 ∈ (`�, +∞), the following inequality holds ẑ(t) −d [∫ l2 l1 J(x−y)ẑ(t)dy − ẑ(t) ] − ẑ(t) [ a− bẑ(t) ] < 0, (t,x) ∈ (ρω,ω] × [l1, l2]. (3.17) It suffices to prove that for each l1 ∈ (−∞,−`�), l2 ∈ (`�, +∞), we have ẑ(t) ≤ u∗(l1,l2)(t,x) in [ρω,ω] × [l1, l2]. To this end, we fix any l1 ∈ (−∞,−`�), l2 ∈ (`�, +∞) and set β∗ = inf { β > 0 : ẑ(t) ≤ βu∗(l1,l2)(t,x) for all (t,x) ∈ (ρω,ω) × [l1, l2] } . We see that β∗ is well-defined and positive since min [ρω,ω]×[l1,l2] u∗ (l1,l2) (t,x) > 0 and ẑ(t) is bounded. It follows from the continuity of u∗ (l1,l2) and ẑ that ẑ(t) ≤ β∗u∗(l1,l2)(t,x) for all (t,x) ∈ (ρω,ω)× [l1, l2]. In particular, there must exist (t0,x0) ∈ (ρω,ω) × [l1, l2] such that ẑ(t0) = β∗u∗(l1,l2)(t0,x0). 16 Z. LI AND B. DAI When β∗ ≤ 1, the lower bound in (3.16) holds immediately. On the contrary, suppose that β∗ > 1. Let w(t,x) = ẑ(t) −β∗u∗(l1,l2)(t,x). Then by (3.17) and the equation satisfied by u ∗ (l1,l2) (t,x), a simple calculation yields that wt(t,x) < d [∫ l2 l1 J(x−y)w(t,y)dy −w(t,x) ] + ẑ(t) [ a− bẑ(t) ] −β∗u∗(l1,l2)(t,x)[a− bu ∗ (l1,l2) (t,x)] for (t,x) ∈ (ρω,ω)×[l1, l2]. However, by the definition of β∗, we have that wt(t0,x0) = 0. This together with ∫ l2 l1 J(x0 −y)w(t0,y)dy −w(t0,x0) ≤ 0 leads to that 0 = wt(t0,x0) ≤ ẑ(t0) [ a− bẑ(t0) ] −β∗u∗(l1,l2)(t0,x0)[a− bu ∗ (l1,l2) (t0,x0)] < ẑ(t0) [ a− bẑ(t0) ] −β∗u∗(l1,l2)(t0,x0)[a− bβ∗u ∗ (l1,l2) (t0,x0)] = 0, which is a contradiction. Consequently, β∗ ≤ 1 and so ẑ(t) ≤ u∗(l1,l2)(t,x) for all (t,x) ∈ (ρω,ω)×[l1, l2]. In fact, the domain (ρω,ω) × [l1, l2] can be extended to [ρω,ω] × [l1, l2] since ẑ(t) and u∗(l1,l2)(t,x) are both continuous and bounded. Hence, (3.16) holds true and so lim−l1,l2→+∞u ∗ (l1,l2) (t,x) = z∗(t) in Cloc([ρω,ω] ×R). On the other hand, when t ∈ [0,ρω], it holds that z∗(t) = e−δtz∗(0) = e−δtz∗(ω) and u∗ (l1,l2) (t,x) = e−δtu∗ (l1,l2) (0,x) = e−δtu∗ (l1,l2) (ω,x) for 0 ≤ t ≤ ρω and l1 ≤ x ≤ l2. This means that lim −l1,l2→+∞ u∗(l1,l2)(t,x) = z ∗(t) in Cloc([0,ρω] ×R). As a result, lim −l1,l2→+∞ u∗(l1,l2)(t,x) = z ∗(t) in Cloc([0,ω] ×R). The proof is completed. � 4. Simulations In this section, we present the simulations to illustrate some of our results. Referring to [26], we choose the form of J to be a simple Laplace kernel: J(x) = 1 2D e− |x| D with D = 20. Consider the following parameter sets: (P1) δ = 0.2,d = 0.6,a = 1.2,b = 0.6,ρ = 0.6,ω = 1; (P2) δ = 0.2,d = 1,a = 1.2,b = 0.6,ρ = 0.6,ω = 1; (P3) δ = 0.8,d = 0.6,a = 1.2,b = 0.6,ρ = 0.6,ω = 1; and the initial condition (IC) u0(x) = cos( πx l ),x ∈ (−l, l). Clearly, the parameter set (P1) satisfies the condition in Theorem 1.1 (1). Then Fig. 1 shows that when the domain length L := 2l = 0.4, the solution of (1.1) satisfying (P1) and (IC) converges to a spatially nonhomogeneous positive periodic solution. This is consistent with the conclusion of Theorem 1.1 (1). The parameter set (P2) satisfies the condition in Theorem 1.1 (2). Then Fig. 2 shows that when the domain length L = 2l = 0.4, the solution of (1.1) satisfying (P2) and (IC) converges to a spatially nonhomogeneous positive periodic solution, but when L = 2l = 8, the solution of (1.1) with the same parameters and initial condition converges to zero. This is consistent with the conclusion of Theorem 1.1 (2). A NONLOCAL DISPERSAL LOGISTIC MODEL WITH SEASONAL SUCCESSION 17 Figure 1. Numerical simulations of (1.1) with parameter set (P1) and initial condition (IC), where l = 0.2. Figure 2. Numerical simulations of (1.1) with parameter set (P2) and initial condition (IC). Left: l = 0.2; Right: l = 4. The parameter set (P3) satisfies the condition in Theorem 1.1 (3). Then Fig. 3 shows that when the domain length L = 2l = 8, the solution of (1.1) satisfying (P3) and (IC) converges to zero. This is consistent with the conclusion of Theorem 1.1 (3). 5. Discussion In this paper, we mainly examine a nonlocal dispersal logistic model with seasonal succession subject to Dirichlet type boundary condition. In Section 3, in order to study the long time behavior of the solutions to (1.1), we establish a method of time periodic upper-lower solutions, and show that the sign of the eigenvalue (1 − ρ)σ1 + ρδ of the linearized operator can completely determine the asymptotic behavior of the solutions to (1.1). Meanwhile, we see that the ω-periodic positive solution corresponding to the nonlocal dispersal model (1.1) behaves like the ω-periodic positive solution corresponding to the ODE model (1.4) when the range of the habitat tends to the entire space R. 18 Z. LI AND B. DAI Figure 3. Numerical simulations of (1.1) with parameter set (P3) and initial condition (IC), where l = 4. In the following, we give some remarks on a nonlocal dispersal logistic model under Neumann type boundary condition, which is associated with model (1.1), that is,  ut = −δu, iω < t ≤ (i + ρ)ω, l1 ≤ x ≤ l2, ut = d ∫ l2 l1 J(x−y) ( u(t,y) −u(t,x) ) dy + u(a− bu), (i + ρ)ω < t ≤ (i + 1)ω, l1 ≤ x ≤ l2, u(0,x) = u0(x) ≥ 0, x ∈ [l1, l2], (5.1) The kernel function J : R → R is assumed to satisfy (J). The integral operator ∫ l2 l1 J(x−y) ( u(t,y) − u(t,x) ) dy describes diffusion processes, where ∫ l2 l1 J(x−y)u(t,y)dy is the rate at which individuals are arriving at position x from all other places and ∫ l2 l1 J(x−y)u(t,x)dy is the rate at which they are leaving location x to travel to all other sites. Since diffusion takes places only in [l1, l2] and individuals may not enter or leave the domain [l1, l2], we call it Neumann type boundary condition. Linearizing system (5.1) at u = 0, we obtain the time-periodic operator: L̃(l1,l2)[v](t,x) =   −vt − δv, t ∈ (iω, (i + ρ)ω], x ∈ [l1, l2], −vt + d ∫ l2 l1 J(x−y) ( u(t,y) −u(t,x) ) dy + av, t ∈ ((i + ρ)ω, (i + 1)ω], x ∈ [l1, l2]. (5.2) A easy calculation yields that λ1 = δρ−a(1−ρ) is a eigenvalue of −L̃(l1,l2) with a positive eigenfunction. Moreover, one can also derive as in Theorem 1.1 that Theorem 5.1. Assume that (J) holds and −∞ < l1 < l2 < +∞. Let u(t, ·; u0) be the unique solution to (5.2) with the initial value u0(x) ∈ C([l1, l2]), where u0(x) is bounded, nonnegative and not identically zero. The following statements are true: (1) If δρ−a(1 −ρ) < 0, then lim n→∞ u(t + nω,x; u0) = z ∗(t) in C([0,ω] × [l1, l2]), where z∗(t) is the unique positive ω-solution to (1.7); (2) If δρ−a(1 −ρ) ≥ 0, then lim t→∞ u(t,x; u0) = 0 uniformly for x ∈ [l1, l2]. 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School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan, 410083, PR China Email address: zhzhli@csu.edu.cn Corresponding author, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan, 410083, PR China Email address: bxdai@csu.edu.cn 1. Introduction 2. Well-posedness 3. Global dynamics 3.1. The method of periodic upper-lower solutions 3.2. Proofs of Theorems 1.1 and 1.2 4. Simulations 5. Discussion References