Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

In this paper, we are concerned with a diffusive Leslie-Gower predator-prey model in heterogeneous environment. The global existence and boundedness of solutions are shown. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global stability of semi-trivial solutions. The existence of positive steady state solution bifurcating from semi-trivial solutions is obtained by using local bifurcation theory. The stability analysis of the positive steady state solution is investigated in detail. In addition, we explore the asymptotic profiles of the steady state solution for small and large diffusion rates.


1.
Introduction. In ecological systems, the interaction of predator population and prey population has abundant dynamical feature. Many biological models have been put forward since the pioneering works of Lotka [25] and Volterra [34]. Moreover, more realistic models are proposed according to laboratory experiments and observations. For example, the Leslie-Gower predator-prey model was proposed in [20,21,29], which takes the form where u(t) and v(t) represent the densities of prey and predators at time t, respectively; u/c represents the environmental carrying capacity of predators. Thus, the amount of predators is only affected by its favorite food u. When the quantity of the favorite food u is insufficient, the predator has to switch over to other population but its growth will be limited. To take into account the effect of this factor, Aziz-Alaoui and Okiya [3] have proposed the following Leslie-Gower predator-prey system with saturated functional responses where a 1 and a 2 stand for the growth rates per capita of prey u and predator v, respectively; b 1 measures the strength of intraspecific competition among individuals of species u, which is related to the carrying capacity of the prey; c 1 and c 2 represent the maximum value of the per capita reduction rate of u due to v and the maximum growth per capita of v due to predation of u, respectively; k 1 and k 2 measure the extent to which environment provides protection to prey u and predator v, respectively. In view of the inhibitory effect of high concentrations on microbial growth, Andrews [2] suggested a Monod-Haldane response function p(u) = mu k1+k2u+u 2 . In experiments on the uptake of phenol by pure culture of Pseudomonas later, Sokol and Howell [33] proposed a simplified Monod-Haldane type function of the form p(u) = mu k1+u 2 . In the interactions between predator and prey, we can obtain the following Leslie-Gower predator-prey model by using Monod-Haldane type functional response function Considering the effects of dispersal and environmental heterogeneity on the dynamic behavior of populations, our concern is the following Leslie-Gower prey-predator system in Ω × (0, ∞), in Ω × (0, ∞), where ∆ denote the Laplacian operator on R N , Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂Ω, n is the outward unit normal vector on ∂Ω. The homogeneous Neumann boundary condition means that the two species have zero flux across the boundary ∂Ω. Throughout this paper, we always assume that a(x) and b(x) ∈ C r (Ω) with r ∈ (0, 1) are nonconstant, and that a(x) > 0 and b(x) > 0 in Ω. For system (4) with Ω = (0, π), d 1 = 1, and constant-valued functions a(x) and b(x), Li et al. [24] investigated the Hopf bifurcation and steady-state bifurcation by taking d 2 as the bifurcation parameter and described both the global structure of the steady-state bifurcation from simple eigenvalues and the local structure of the steady-state bifurcation from double eigenvalues by using space decomposition and the implicit function theorem.
Many researchers have paid more attention to the effect of interspecies interaction on the dynamic behaviors of reaction-diffusion population models in which it is assumed that the environment is spatially homogeneous, that is, all the coefficients are constant [12,13,14,15,23,28,30,37,38]. It has been observed in many scientific experiments that spatial heterogeneity has a profound effect on ecosystems. Therefore, it is very important and significant to study the heterogeneous effects of the environments on the dynamics of biological population models. In recent two decades, more and more researchers have paid attention to the effects of the spatial heterogeneity on the dynamics of biological population models and have put forward many interesting mathematical problems; See, for example, [7,8,9,10,16,17,18,19,27,35,36]. In particular, some predator-prey models with degeneracy have been investigated in [7,10], in which some coefficients of the prey population vanish to zero in some areas of life. Du and Shi [9] proposed a predator-prey model with a protected area and assumed that only the bait population is free to enter or leave the protected area. In [16,17,18,19], the researchers considered the following diffusive Lotka-Volterra competition system with spatially heterogeneous resources In a weak competition case, Lam and Ni [19] showed that system (5) with m 1 (x) = m 2 (x) has an unique coexistence steady state solution which is globally asymptotically stable. In the case where b = c = 1 (i.e., the two species have the same competition) and Ω m 1 (x)dx = Ω m 2 (x)dx (i.e., the total resources are the same), He and Ni [16] analyzed the effect of the diffusion coefficients of system (5) on the stability of the steady state solution, and obtained the global asymptotic stability when the diffusion coefficients d 1 and d 2 are sufficiently large or sufficiently small, respectively. In particular, in the case where m 2 (x) is a constant, the species u with heterogeneous resource allocation has more obvious competitive advantage. He and Ni [17] analyzed the effect of diffusion coefficients on the stability of semi-trivial steady-states and coexistence steady state solutions of system (5). He and Ni [18] discussed the existence and uniqueness of coexistence steady state solutions and the global asymptotic stability of the semi-trivial steady state solution of system (5). Recently, Lou and Wang [27], Wang and Zhang [35] discussed the following reaction-diffusion predator-prey system with heterogeneous resource allocation More precisely, Lou and Wang [27] focused on the effect of the variation of diffusion coefficient on the stability of the semi-trivial steady-state solution of system (6), while Wang and Zhang [35] investigated the local bifurcation of semi-trivial steady state solutions of system (6). Note that the functional response function in system (5) is linear, and that only the prey population in system (6) has spatial resource heterogeneity. Our system (4) has not only a non-linear functional response function, but also has the spatial resource heterogeneity for both species. Because the functional response function is non-linear, there is some difficulty in the analysis of the existence and asymptotic behavior of positive steady-state solutions. Moreover, the structure of positive steady-state solutions will change essentially due to the heterogeneity of spatial resources. Therefore, it is very meaningful to investigate the influence of spatial resource heterogeneity on the dynamic behavior. Motivated by [16,19,35], in this paper we shall pay more attention to the existence, stability, and asymptotic behavior of positive steady-state solutions of system (4).
The organization of the remaining part of the paper is as follows. In section 2 some concepts are introduced for later use. In section 3 we obtain the global existence and boundedness of solutions to (4) by applying comparison methods. In section 4 we derive the stability of semi trivial solutions by analyzing the sign of the principal eigenvalue. Section 5 is devoted to the existence and stability of positive steady state solution which bifurcates from semi-trivial steady state of system (4). Finally, in section 6 we first employ the fixed point index theory in a cone to investigate the existence of positive steady-state solutions of system (4) and then investigate the limiting behaviours of positive steady state solution as the dispersal rates tend to 0 or ∞.
Throughout the paper, we denotě for any given continuous function f onΩ.

2.
Preliminaries. In this section, we will present several lemmas which shall be used in the subsequence analysis. First, we consider the following steady-state problem for the logistic equation with linear diffusion where m(x) ∈ C r (Ω) with r ∈ (0, 1) is nonconstant such that m > 0 in a set of positive measure in Ω. It is well known that (see, for example, [4]) system (7) has a unique positive solution, denoted by θ d,m , if and only if µ 1 (d, m) < 0, and that θ d,m ∈ C 2 (Ω), where µ 1 (d, m) < 0 is given later (see Definition 2.2). Dividing both sides of the first equation of (7) by θ d,m and integrating over Ω, we obtain that Next, we will present several results on properties of θ d,m , the unique positive solution of (7), which shall be used in the subsequent analysis.
To characterize the principal eigenvalue of system (7), we need to introduce the following eigenvalue problem with indefinite weight where h ≡ constant, could change sign in Ω. We say that λ 1 (h) is a principal eigenvalue if (8) with λ = λ 1 (h) has a positive solution (Notice that 0 is always a principal eigenvalue). Regarding the property of the principal eigenvalue λ 1 (h) of (8), detailed information can be found in [16]. Next, we collect some facts concerning the eigenvalue problem (8).
Definition 2.2. Given a positive constant d and a function h ∈ L ∞ (Ω), denote by µ k (d, h) the k-th eigenvalue (counting multiplicities) of the following eigenvalue problem: In particular, we call µ 1 (d, h) the principal eigenvalue of (8), which has the following variational characterization In the following, we give some important properties of µ 1 (d, h) in connection with λ 1 (h) and refer to [4] for a proof.
(ii): If Ω h < 0 and h changes sign in Ω, then is strictly increasing and concave in d > 0. Furthermore, whereh is the average of h.
in Ω. Assume in addition to that h is nonconstant, then

Global existence and boundedness.
In the section, we investigate the global existence and boundedness of the solution to (4). In what follows, we will state the relevant results.
Proof. First, the local existence of the solution to (4) follows from standard theory. Denote by T max the maximal existence time of solution. Since u(x, t) satisfies Then from comparison principle of the parabolic equations, it is easy to verify that where K =θ d1,a +φ. Again using the comparison principle of the parabolic equations, we obtain that Hence, it follows from [1] that T max = ∞. Moreover, we conclude that u(x, t) > 0 and v(x, t) > 0 due to the maximum principle and Hopf boundary lemma for parabolic equations. Therefore, the solution of system (4) exists globally and is bounded. The proof is completed.

Stability analysis of semi-trivial solutions.
It is easy to see that system (4) has a trivial steady state (0, 0) and two semi-trivial steady states (θ d1,a , 0) and . If a steady state (u, v) of (4) satisfying u ≥ 0 and v ≥ 0 is neither a trivial nor a semi-trivial steady state, then by the maximum principle, we have u > 0 and v > 0 onΩ. In this case, we call (u, v) a co-existing steady state. The purpose of this section is to analyze the local stability of semi-trivial steady states of (4) and to obtain the following results.
(ii): The semi-trivial steady state (0, θ d2,b ) of system (4) is locally asymptotically Proof. We first prove conclusion (i). From the linearization principle, the stability of (θ d1,a , 0) can be determined by studying the following eigenvalue problem x ∈ Ω, Let λ be an eigenvalue of (11) with an associated eigenfunction (Φ, Ψ). If Ψ = 0, then λ belongs to the spectrum of the self-adjoint operator −d 2 ∆ − b(x) (with zero Neumann boundary condition). Therefore, λ must be real and satisfy λ ≥ . Alternatively, if Ψ ≡ 0, then Φ = 0, and λ belongs to the spectrum of −d 1 ∆ − a(x) + 2θ d1,a (with zero Neumann boundary condition), which must again be real and satisfy λ ≥ Moreover, in view of proposition 1 (iv), we have then every eigenvalue of L is positive and hence L is invertible. Thus, min λ = µ 1 (d 2 , b(x)) < 0. This implies that the principal eigenvalue of (11) exists and is negative. It follows from [32] that the semi-trivial steady state (θ d1,a , 0) is unstable and hence the proof of conclusion (i) is completed. Next, we discuss the stability of (0, θ d2,b ), which is determined by the following eigenvalue problem Let λ be an eigenvalue of (12) with an associated eigenfunction (Φ, Ψ). From the first equation of system (12), it is easy to see that λ belongs to the spectrum of the self-adjoint operator −d 1 ∆ − a(x) + θ d2,b (with zero Neumann boundary condition). Hence, λ must be real and satisfy λ ≥ Similarly, it follows from the second equation of system (12) that λ must be real and satisfy Let ϕ be the first eigenfunction corresponding to is an eigenvalue of (12) with an associated eigenfunction (Φ, Ψ)=(ϕ, 0). Similarly, let ψ be the first eigenfunction associated with is an eigenvalue of (12) with an associated eigenfunction (Φ, Ψ)=(0, ψ). Hence, On the other hand, it follows from Proposition 1 (iv) that Based on the above discussion, we see that the principal eigenvalue of (12) exists and has the same sign as that of the first eigenvalue In this case, if a(x) − θ d2,b changes sign in Ω, then it follows from Proposition 1 that there exists d * This completes the proof of Theorem 4.1 (ii). Remark 1. Using a similar argument, we know that the linear stability of the trivial solution (0, 0) of system (4) is determined by In fact, the trivial solution (0, 0) is unstable for all d 1 > 0 and d 2 > 0.
More precisely, the steady state (0, θ d2,b ) is globally asymptotically stable if eitheř Proof. In virtue of Proposition 1 and (13), we have which together with Theorem 4.1 implies that (0, θ d2,b ) is locally asymptotically stable. Moreover, according to Proposition 1 (iv) and Lemma 2.1, we have In view of Proposition 1 (iii), we know that is strictly increasing with respect to d 1 and satisfies 1+â 2 changes sign in Ω, then it follows from Proposition 1 (ii) that there existsd 1 In what follows, we shall prove the global attractivity of (0, θ d2,b ). From the first equation of system (4) and v ≥ 0, we obtain that Then by the comparison principle of parabolic equations, it is easy to verify that lim t→∞ sup u(x, t) ≤ θ d1,a uniformly for x ∈Ω.
More precisely, for any given ε > 0, there exists t 1 > 0 such that Obviously, v(x, t) satisfies Similarly, we have Then there exists Therefore, in light of (14) and (16), there exists t ≥ t 3 = max{t 1 , t 2 } such that u(x, t) satisfies In view of (13), using the comparison principle, we have lim t→∞ u(x, t) = 0 uniformly for x ∈Ω.
Thus, there exists t > t 4 such that It then follows that v(x, t) satisfies By the standard comparison theorem again, we can get Hence, combining with (15) and the arbitrariness of ε, we obtain lim t→∞ v(x, t) = θ d2,b uniformly for x ∈Ω.

5.
Local bifurcation of steady states. In this section, we shall apply the local bifurcation theory [5,31] to analyze the bifurcation phenomena of semi-trivial steady states of (4) by regarding dispersal rates of the prey and predator as bifurcation parameters. Note that positive steady states of (4) satisfy the following equation: in Ω, To find solutions of (17), we define the operator F (d 1 , u, v): Note that for every fixed parameter value d 1 ∈ R + , we have F (d 1 , 0, θ d2,b ) = 0. By straightforward computations, it is easy to see that the derivatives D d1 F (d 1 , u, v), F (u,v) (d 1 , u, v) and D d1 F (u,v) (d 1 , u, v) exist and are continuous in a neighborhood of (d 1 , 0, θ d2,b ).
In what follows, we will study the linear stability of (u(s), v(s)) which bifurcates from semi-trivial steady state (0, θ d2,b ). Firstly, we need to prove the following Lemma.
Next, suppose that Φ ≡ 0 inΩ. Then Ψ ≡ 0 and In view of Proposition 1, we have

with homogeneous Neumann boundary condition.
Combining with the fact that (u(s), v(s)) → (0, θ d2,b ) as s → 0, we conclude that σ 1 < 0 for small s. Therefore, all eigenvalues of (22) have positive real parts, that is, (u(s), v(s)) is locally asymptotically stable. Hence, we complete the proof.
In view of the above lemmas, we have the following conclusion.
6. Asymptotic profiles of steady states. In this section, our interest is to investigate the asymptotic behavior of (17). To this end, we first obtain some a priori estimates for positive solutions and investigate the existence of positive steady state solutions of (17).
Define v(x 1 ) = minΩ v(x), then by using the maximum principle [11,26] for the second equation of system (17), we obtain Obviously, Therefore, the proof is completed.
Next, we will establish the existence result of positive solutions of (17) by the Leray-Schauder degree theory. Now, we recall the fixed point index theory in a cone [6,22]. Let E be a real Banach space and W ⊂ E be a closed convex set. For all β ≥ 0, if βW ⊂ W and W − W = E, then W is called a total wedge, moreover, when W ∩ (−W ) = 0, a wedge is said to be a cone. For any point y ∈ W , define W y = {x ∈ E : y + γx ∈ W for some γ > 0} and S y = {x ∈ W y : −x ∈ W y }. Let T be a compact linear operator on E which satisfies T (W y ) ⊂ W y , then we say that T has property α on W y if there exist t ∈ (0, 1) and w ∈ W y \ S y such that w − tT w ∈ S y . Let For a linear operator L, we denote the spectral radius of L by r(L). Then we give the relationship between µ 1 (d, h) and r(L) as follows (see [22]). For convenience, we introduce some notations: E = X × X, where X = {u ∈ C 1 (Ω) : ∂u ∂n = 0}; W = K × K, where K = {u ∈ X : u ≥ 0}; D = {(u, v) ∈ W : u < 1 +â, v < 1 + (1 +â 2 )b}; D = (int(D)) ∩ W . Therefore, it is easy to see that W (0,0) = K × K, S (0,0) = {(0, 0)}, W (θ d 1 ,a ,0) = X × K, S (θ d 1 ,a ,0) = X × {0},

Define an operator
where t ∈ [0, 1] and M is a sufficiently large number such that Then it follows from the standard elliptic regularity theory that A t is a completely continuous operator for all t ∈ [0, 1]. Therefore, it suffices to prove that A 1 has a nontrivial fixed point in D in order to show the existence of positive solutions of (17). For each t ∈ [0, 1], a fixed point of A t is a solution of the following problem in Ω, Furthermore, we have the following lemma.
In light of φ 1 ∈ K \ {0}, we have 1/t > 1 is an eigenvalue of the operator of B.
Based on the above lemma, we gives the following theorem for the existence of positive solutions of (17).   In what follows, we study the asymptotic behavior of positive solutions of (17) as d 1 → ∞. In virtue of the asymptotic analysis, we can obtain the following results.
Proof. We first prove part (i). Obviously, system (17) can be rewritten as It follows from Lemma 6.1 and a standard compactness argument that there exists a subsequence of d j (j = 1, 2), denoted by d jn satisfying d jn → ∞, such that a corresponding positive solution (u n , v n ) of (17) with d j = d jn satisfies u n → u ∞ and v n → v ∞ in C 1 (Ω) as n → ∞, where u ∞ ≥ 0 and v ∞ > 0 inΩ due to Lemma 6.1. Moreover, u ∞ and v ∞ satisfy the following equations Clearly, u ∞ and v ∞ are nonnegative constants. Definẽ u n = u n u n L ∞ , then (ũ n , v n ) satisfies x ∈ Ω, Hence, integrating the second equation of (24) yields Since v ∞ is a positive constant, letting n → ∞ in the above equation leads tō (iii): Suppose that a(x) > b(x). If d j → 0 (j = 1, 2), then Proof. We first prove (i). Lemma 6.1 and the usual compactness argument imply that there exist a subsequence of d 2 , still denote by d 2n satisfying d 2n → 0 as n → ∞, and a corresponding positive solution (u n , v n ) of (17) with d 2 = d 2n , such that u n → u * and v n → v * unfiormly onΩ, as n → ∞, where both u * and v * ∈ C 1 (Ω), u * ≥ 0 and v * > 0. Obviously, (u n , v n ) satisfies (17) with d 1 = d 1n , that is, Letting n → ∞ in the second equation of system (30), we have From the proof of Theorem 6.5, it is easy to see that u n → u * = u ∞ as d 1n → ∞ and u * is a nonnegative constant. Hence, (26) is satisfied. Letting n → ∞ in (26), we have Ω a(x) − u * − v * 1 + u 2 * dx = 0, which, together with (31), implies that u * =ā −b and v * = b(x)(1 + u 2 * ). This completes the proof of conclusion (i).