DYNAMICS OF A DIFFUSIVE PREY-PREDATOR SYSTEM WITH STRONG ALLEE EFFECT GROWTH RATE AND A PROTECTION ZONE FOR THE PREY

. In this paper, a diﬀusive prey-predator model with strong Allee eﬀect growth rate and a protection zone Ω 0 for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occur- rence and avoidance of overexploitation phenomenon are obtained. Further-more, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is eﬀective when Allee threshold is small and the protection zone is large.

1. Introduction. One important interaction between biological species is the preypredator interaction. Previously, the great mass of prey-predator models are expressed in terms of ordinary differential equations where spatial effects are ignored [9,13,17]. As we all know that the spatial diffusion of species can affect the complexity of ecosystems, such as ecological invasion and pattern formation, reaction-diffusion prey-predator models have emerged and been widely researched [14,15,19,21,25,29]. Considering the spatial distribution of the populations, a prototypical prey-predator system [8] is of the form where u and v represent the densities of prey and predator respectively. The parameters d 1 , d 2 , m and c are positive constants. Here f (u) and g(v) represent the growth of u and v respectively when the other species is absent. In most articles of prey-predator models, the prey is assumed to grow at a logistic form, i.e., f (u) = u(1 − u/K), where K is the carrying capacity. However, due to the factors of mate limitation, cooperative defense and environmental conditioning, it has been recently recognized that the prey species may have a growth rate with Allee effect [1,21,22]. Generally, the strong Allee growth rate is taken in the form f (u) = u(1 − u)(u − b), where 0 < b < 1 represents Allee threshold value. The function h(u) represents the functional response of the predator, which was first examined by Holling [11]. The Holling-type functional response is widely used by many researchers [16,30,31]. However, there are growing biological and physiological evidences [2,3,23,25] that a functional response depending on the ratio of prey to predator abundance is a suitable representation in some situations, for example, when predators have to search, share and compete for food. Essentially, the ratio-dependent theory asserts that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. In this case, it is necessary to change h(u) to h(u/v) in (1). Specially, we take h(u/v) in the form of u/(u + v).
For the sake of simplicity, we suppose that g(v) = −θv where θ > 0 is the death rate of the predator. Based on the above arguments, a diffusive prey-predator model [24] with strong Allee effect in prey can be written as follows: where Ω ⊂ R N is a bounded and smooth domain, u 0 (x), v 0 (x) ∈ C 2+α (Ω) and ∂u 0 (x)/∂n = ∂v 0 (x)/∂n = 0 on ∂Ω. The reference [24] shows that for any given u 0 (x), both of the prey and predator will be extinct if v 0 (x) is large enough. Such a feature is known as overexploitation phenomenon which is a distinctive character of dynamics of the prey-predator system with strong Allee effect in the prey growth. This result makes us think about how to save the endangered species. Then a idea comes out that setting up a protection zone for the prey. The prey species can enter and leave the protection zone freely but not the predator. Consequently, it is natural to have two questions: Does the protection zone protect the two species from the extinction caused by overexploitation? How do the Allee effect and protection zone affect the spatiotemporal dynamics of the species? In this paper, we will try to answer these two questions. Very recently, some efforts have been devoted to investigating the impact of protection zone on the prey-predator models [5,7,8]. They all showed the existence of a critical patch size of the protection zone and demonstrated that the ultimate fate of species changed with this critical patch size. The competition models with a protection zone also have been studied [6,27]. Compared with the prey-predator model, the protection zone had some essentially different effects on the fine dynamics of the competition model. Considering a protection zone in (2), we rewrite it as the following system where Ω * := Ω \ Ω 0 and Ω 0 is a smooth interior subdomain of Ω. The larger region Ω is the habitat of the prey with the protection zone Ω 0 . The predator species is initially driven out from Ω 0 and prohibited from entering Ω 0 again while the prey species can enter and leave Ω 0 freely. A no-flux boundary condition is assumed for both species on ∂Ω, so the prey and predator live in a closed ecosystem. Moreover, the no-flux boundary condition is imposed for the predator on ∂Ω 0 . We may think of a barrier along ∂Ω 0 that blocks v but not u. The initial data u 0 (x) is same as that in (2) while the initial data v 0 (x) ∈ C 2+α (Ω * ) is positive and satisfies ∂v 0 (x)/∂n = 0 on ∂Ω * . The function m(x) is zero for x ∈ Ω 0 , which implies that the prey species enjoys predation-free growth in Ω 0 . Note that though v is not defined for x ∈ Ω 0 , the interaction term in the first equation of (3) can still be regarded as properly Particularly, throughout this article, we assume that The meanings of other parameters are the same as those in (2). With a view to studying non-constant positive solutions of (3), we shall consider the stationary problem of (3), i.e., the following semi-linear elliptic problem: Recall the following results. For any given q ∈ L ∞ (Ω), we define λ D 1 (q, O) and λ N 1 (q, O) to be the principal eigenvalues of −∆ + q over the region O with Dirichlet and Neumann boundary conditions respectively. If the region O is omitted in the notation, then we understand that O = Ω. If the function q is omitted, then we understand that q = 0. It is well known that the properties of λ D 1 (q, O) and λ N 1 (q, O) [26,29]: The rest of the paper is arranged as follows. In Section 2, we present basic dynamic properties of (3), such as the global existence and long time behaviors of positive solutions, and the local stabilities of semi-trivial solutions. In Section 3, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. We demonstrate that the overexploitation phenomenon will probably disappear when Allee threshold b is small and the protection zone Ω 0 is large. If initial prey population is large enough, then the protection zone is effective. Unfortunately, we also prove that the prey and predator are destined to extinction when Allee threshold b is large or the protection zone Ω 0 is small. In Section 4, the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions are obtained.
2. Basic dynamic properties of (3). In this section, we shall investigate the global existence, uniqueness and long time behavior of the solutions to (3).
It is easy to see that, in the domain with u * = max x∈Ω u 0 (x) > 0. Letv(t) and v(t) be the unique positive solution of ,v(t)) and (u(x, t), v(t)) are the coupled ordered upper and lower solutions to (3) respectively. Actually, Moreover, by the strong maximum principle we also have u(x, t) > 0 for x ∈ Ω and t ≥ 0.
Before we study the long time behavior of solutions to (3), we first give a useful lemma in [29].
and the constant α > 0, then Thus, for any ε > 0, the and u * (t) is the unique solution of (6), it is deduced by the comparison principle that u(x, t) ≤ u * (t). Evidently, the first inequality of (7) holds. As a result, for any ε > 0, there exists T > 0 such that u(x, t) ≤ 1 + ε for all x ∈ Ω and t ≥ T . It If θ < c * , in view of Lemma 2.2 and the arbitrariness of ε, we have Similarly, Lemma 2.2 suggests that lim sup In either case, the second inequality of (7) holds.
Proof. The proof is divided into three cases.
is the unique solution of (7). It follows from u 0 (x) < b that lim t→∞ u(x, t) = 0 uniformly on Ω. Accordingly, for any given ε > 0, there exists T > 0 such that It follows from the strong maximum principle that As proved in Case 2, we get lim t→∞ u(x, t) = 0 uniformly on x ∈ Ω and lim t→∞ v(x, t) = 0 uniformly on x ∈ Ω * . Theorem 2.5. Let the parameters d 1 , d 2 , θ > 0 and 0 < b < 1. Suppose that m(x) and c(x) satisfy (4). If θ ≥ c * , then (u(x, t), v(x, t)) tends to (u a (x), 0) uniformly as t → ∞, where u a (x) is a non-negative solution of Proof. On account of the second inequality of (7), if θ ≥ c * , then lim t→∞ v(x, t) = 0 uniformly for x ∈ Ω * . Now the equation of u(x, t) is asymptotically autonomous [18], and its limit behavior is determined by the parabolic equation: It is well known that (9) is a gradient system and its every orbit converges to a steady state of corresponding elliptic problem for (9) [10]. Then from the theory of asymptotically autonomous dynamical system, the solution (u(x, t), v(x, t)) of (3) converges to (u a (x), 0) as t → ∞.
It is easy to check that the system (3) has semi-trivial solutions (1, 0), (b, 0) and (u s (x), 0), where u s (x) is a non-constant positive solution to (8) and the existence of u s (x) has been discussed in Subsection 3.2 of [28]. In the following, we will study the stabilities of the semi-trivial solutions to (3).
Proof. (i) The linearized eigenvalue problem of (5) at (b, 0) can be written as where µ is an eigenvalue and (h, k) is the corresponding eigenfunction of µ. It is obvious that µ = b(b − 1) > 0 is an eigenvalue with corresponding eigenfunction (φ 1 , 0), where φ 1 is the positive eigenfunction corresponding to the first eigenvalue of −∆ with Neumann boundary condition. Consequently, (b, 0) is always unstable.
(iii) According to the well-known results Theorem 2 in [4] , the non-constant solution u s (x) of (8) is unstable if Ω is convex. Hence, (u s (x), 0) is also an unstable solution of (5) when Ω is convex.
3. Overexploitation. In this section, we will investigate the occurrence and avoidance of the overexploitation phenomenon in (3). For this purpose, we first give two useful lemmas in [5].
the following statements hold true.
such that for 0 < d 1 < D 0 , the system (11) has at least two positive solutions. Moreover, the system (11) has a maximal solution U (x) such that for any solution u(x) of (11), U (x) > u(x) for x ∈ Ω 0 . Lemma 3.2. Suppose that β, A are positive constants, Ω is a bounded smooth domain in R N (N ≥ 1), and Ω 1 is a smooth interior subdomain of Ω. Then for any ε > 0, there existβ > 0 and K > 0 such that when β >β, the unique positive solution u(x) of the modified Helmholtz's equation where d(x, ∂Ω 1 ) is the distance from x to ∂Ω 1 . Proof. In order to get above result, it suffices to prove that there exists T * > 0 such that u(x, T * ) < b for x ∈ Ω in view of Theorem 2.4. In the following, we will find two time intervals I 1 and I 2 satisfying I 1 ⊃ I 2 . We will prove that The proof is divided into five steps.
Proof. Let w(x, t) be the solution of the Dirichlet boundary value problem Obviously, w(x, t) exists globally for all x ∈ Ω 0 and t > 0. It is derived from u 0 (x) ≥ U (x) and the comparison principle that w(x, t) ≥ U (x) for all x ∈ Ω 0 and t > 0. On the other hand, u(x, t) satisfies the first equation in (20) on Ω 0 , u(x, t) > 0 for x ∈ ∂Ω 0 , and u(x, 0) = u 0 (x). Thus u(x, t) is a upper solution of (20). Then u(x, t) ≥ w(x, t) ≥ U (x) for all x ∈ Ω 0 and t > 0. In conclusion, u(x, t) ≥ũ(x) for all x ∈ Ω and t > 0. (0, 1) and (0, b). In this section, we will investigate the bifurcation from semi-trivial solutions (1, 0) and (b, 0) using bifurcation theory. We fix d 1 , d 2 > 0 and 0 < b < 1, and take θ as the main bifurcation parameter. For p > 1, we define

Bifurcation from semi-trivial solutions
and For a given operator A, we denote the kernel and range of A with by N (A) and R(A) (sometimes, we simply write them as N A and RA), respectively. Before we study the bifurcation of (5), we first give a useful lemma.
Lemma 4.1. Let λ 1 (q) be the first eigenvalue of the operator −∆ + q in Ω with the homogeneous Neumann boundary condition, and φ 1 be the corresponding eigenfunction of λ 1 (q). Furthermore, if f ∈ L 2 (Ω) satisfying Ω f φ 1 dx = 0 and λ 1 (q) = 0, then the Neumann boundary value problem be the eigenvalues and corresponding eigenfunctions of −∆ + q in Ω with homogeneous Neumann boundary condition. We may assume that It follows from Ω f φ 1 dx = 0 that c 1 = 0. Due to the fact that the remainder of proof is similar to Theorem 2.7.2 in [26], we omit it.
Putting (θ n , u n , v n ) into the second equation of (5) and dividing it by v n p , it follows that for any n ≥ 1, which is equivalent to Since (−∆) −1 is compact operator, there exists a subsequence of v n / v n p , denoted by itself, and a function ϕ ∈ W 2,p (Ω * ), such that lim n→∞ v n v n p = ϕ > 0.
Take the limit on both sides of (23), we get Then we have Multiplying ϕ and η 1 to the second equation of (21) and the first equation (24), respectively, and integrating them over Ω * and then subtracting the results, we have Hence, θ 0 = θ * . The proof is completed.
By the perturbation theory of linear operators [12], we know that, when s are sufficiently small, L (s) has a unique eigenvalue µ(s) satisfying lim s→0 µ(s) = 0 and all the other eigenvalues of L (s) have positive real parts and are apart from 0. In order to simplify the notation, we denote L (s) = L and µ(s) = µ below. Now we determine the sign of Re µ as s > 0 is sufficiently small. Let ξ, η be the corresponding eigenfunction to µ such that (ξ, η) → (Φ * , Ψ * ). Multiplying the second equation of L (ξ, η) = µ(ξ, η) by v and integrating over Ω * , we get This fact combines with (25) to yield Noting that (u, v) = (1 + sΦ * + O(s 2 ), sΨ * + O(s 2 )) and ξ → Φ * , η → Ψ * , dividing (26) by s 2 and letting s → 0 + , it is deduced that lim − Ω * c(x)Φ * Ψ 3 * dx This implies that Reµ = 0 for s sufficiently small. Since all the other eigenvalues of L have positive real parts and are apart from 0, the local bifurcation coexistence state bifurcating from (θ * , 1, 0) is non-degenerate. In addition, the limits (27) suggests that the bifurcation coexistence state (u(s), v(s)) is linearly stable.