EFFECTS OF DISPERSAL FOR A PREDATOR-PREY MODEL IN A HETEROGENEOUS ENVIRONMENT

. In this paper, we study the stationary problem of a predator-prey cross-diﬀusion system with a protection zone for the prey. We ﬁrst apply the bifurcation theory to establish the existence of positive stationary solutions. Furthermore, as the cross-diﬀusion coeﬃcient goes to inﬁnity, the limiting be- havior of positive stationary solutions is discussed. These results implies that the large cross-diﬀusion has beneﬁcial eﬀects on the coexistence of two species. Finally, we analyze the limiting behavior of positive stationary solutions as the intrinsic growth rate of the predator species goes to inﬁnity.


1.
Introduction. For most predator-prey systems, the prey would become extinct due to the large growth rate of the predator or the high predation rate. Hence, in order to protect certain species, human interference is necessary. From this viewpoint, Du and Shi [3] studied the following predator-prey model with a protection zone: x ∈ Ω \ Ω 0 .
(1) From the results obtained in [3], we find that the protection zone has great influence on the dynamic behavior of the model (1). To be more precise, there is a critical size for the protection zone such that when the protection zone is below the critical size, the dynamical behavior is similar to the no-protection zone case: for small µ, the prey survives but the predator becomes extinct; for large µ, an opposite result occurs, that is, the predator survives but the prey is extinct; for intermediate µ, positive stationary solutions exist, and the system is permanent. However the dynamical behavior is changed once the protection zone is beyond the critical size: the two species can coexist even if µ is large, moreover, for sufficiently large µ, the two species stabilize at a unique positive stationary solution. Besides, the effect of a protection zone has been studied for predator-prey model with strong Allee effect [2], Leslie predator-prey model [6], ratio-dependent predator-prey model [23], Beddington-DeAngelis predator-prey model [7,21], and Lotka-Volterra competition model [5]. Finally, we point out that a related but different situation for the Holling type II predator-prey model, called a degeneracy, is studied in [4,[12][13][14] and references therein.
The effect of cross-diffusion is not considered in [3]. Therefore, in this paper, we introduce the cross-diffusion for the prey in (1) and study the effect of cross-diffusion on the dynamical behavior. That is, we will study the following predator-prey crossdiffusion system with a protection zone x ∈ Ω \ Ω 0 , t > 0, ∂ n u = 0, x ∈ ∂Ω, t > 0, ∂ n v = 0, x ∈ ∂Ω ∪ ∂Ω 0 , t > 0, u(x, 0) = u 0 (x) ≥ 0, x ∈ Ω, v(x, 0) = v 0 (x) ≥ 0, x ∈ Ω \ Ω 0 (2) in a bounded domain Ω ⊂ R N (N ≤ 3) with smooth boundary. Here k, λ are positive constants and µ is constant which may be negative values; Ω 0 is a subdomain of Ω with smooth boundary and is regarded as a protection zone for the prey where the predator cannot freely enter; in Ω 0 ; c(x) > 0 is a Hölder continuous function in Ω \ Ω 0 . One can refer to [3,17] for more detailed biological meaning of the parameters in (2). About the effect of cross-diffusion and protection zone, one can find more results for different reactiondiffusion models in [8-11, 17, 18, 20] and references therein.
Denote Ω 1 = Ω \ Ω 0 . The stationary solutions of (2) satisfy In this paper, we are mainly concerned with the stationary problem (3), that is, we mainly study the positive stationary solutions of (3). Since the set of positive stationary solutions may play an important role in the stationary patterns, it is extremely important to obtain more information on the set of positive stationary solutions. The main purpose of this present paper has three aspects: 1. Provide the sufficient condition for the existence of positive stationary solutions.
2. Discuss the effect of cross-diffusion on (3) and analyze the limiting behavior of positive stationary solutions as k → ∞.
3. Analyze the limiting behavior of positive stationary solutions as µ → ∞.
Focusing on these questions, we will state our main results (note that: some biological explanations related to these results will be given in last section). For this, we first introduce some notations. Let λ N 1 (q(x), O) and λ D 1 (q(x), O) be the principal eigenvalues of −∆ + q(x) over a region O, where O is any bounded domain in R N , with Neumann or Dirichlet boundary conditions respectively, in particular, one can think that q(x) = 0 in O when q(x) is omitted in these notations. For any By the results obtained in [3,17,18], we can summarize the following lemma.
By using the similar argument to Theorem 1.2, we can obtain the following result. Corollary 1. For any given Ω 0 = ∅ and k > 0.
passing to a subsequence.
(ii) For any fixed λ > λ * (µ) and µ < 0, passing to a subsequence, where (u, w) satisfies For the limiting system (4), we establish the bifurcation structure of positive solutions in the following theorem. It follows from Corollary 1 that when λ > λ * ∞ (k), (3) has a positive solution even for large µ. Therefore, our final theorem gives the limiting behavior of positive solutions of (3) as µ → ∞. Theorem 1.5. Let λ > λ * ∞ (k) and k > 0 be fixed. Suppose that (u µ , v µ ) is any positive solution of (3). Then, passing to a subsequence, where U λ,k is the unique positive solution of This paper is organized as follows. In Section 2, we prove some preliminary results which are used to show our main results. In Section 3, we complete the proof of our main results. In Section 4, we summary our results and discuss their biological implications.
2. Preliminary results. In this section, we show some preliminary results which will be used to prove our main results, including the non-existence of positive solutions, a priori estimates of any positive solution and the local bifurcation from semitrivial solutions. Denote To derive a priori estimates of any positive solution of (7), we cite Harnack inequality and maximum principle for weak solutions (see Lemma 2.2 and Lemma 2.3 in [15]).
where C depends on only ||m(x)|| q , q and O.
By means of Lemmas 2.1 and 2.2, we obtain a priori estimates of any positive solution of (7).
is any positive solution of (7). Then for each given (ii) Suppose that (U, v) is any positive solution of (7). Then for large k(> M ), there is a positive constant C = C (λ, |Ω|, |Ω 0 |) independent of µ and k such that Proof. (i) Suppose that (U, v) is any positive solution of (7). Then from the equation for v of (7), it follows that Thus, we apply Lemma 2.2 to get max{µ, We integrate the equation for U of (7) and apply the Schwarz inequality to obtain This implies that ||u|| 2,Ω ≤ λ|Ω| 1/2 , and thus As a result, we derive inf The equation for U of (7) is written as Note that Thus ||m(x)|| 2,Ω ≤ C 1 + ||u|| 2,Ω ≤ C 2 . Then we apply Lemma 2.1 to obtain where C is independent of µ.
(ii) By replacing the upper bound of |m(x)| by λ + u + 1/M , we can use the similar argument as above to derive the assertion (ii).
3. Proof of main results. In this section, by combining the results obtained in Section 2, we apply the global bifurcation theorem, elliptic regularity theory and various elliptic estimates to complete the proof of our main results.
3.1. Proof of Theorem 1.2. In this subsection, we apply the global bifurcation theorem (see Theorem 6.4.3 of López-Gómez [16] based on the global bifurcation theory of Rabinowitz [19]) to prove Theorem 1.2.
Proof of Theorem 1.2. For the case µ > 0. We define an operator: .
For each given λ > 0, it follows from the elliptic regularity theory that the second term of H is a compact operator. It is clear that (7) is equivalent to H(λ, U, v) = 0. By the standard argument, it is not hard to check that the conditions of Theorem 6.4.3 in [16] hold true (for details, one can see Theorem 3.2 in [10]). Therefore, by Proposition 2 and Theorem 6.4.3 in [16], the local bifurcation branch Γ local can be extended a maximal connected set Γ global ⊂ R × E, which satisfies Moreover, the maximal connected set Γ global satisfies one of the following: In the following, we will show only case (1) can occur. For this, we prove where P O = {w ∈ C 1 n (O) : w > 0 in O}. If this is not true, then there is a sequence Furthermore, by the maximum principle, (U ∞ , v ∞ ) satisfies one of the following: By the boundary condition of v i , we integrate the second equation of (7) with Suppose that (a) or (b) occurs. Then for sufficiently large i ∈ N , because of µ > 0. This shows that for sufficiently large i ∈ N , the left-hand side of (18) is positive, a contradiction. Suppose that (c) occurs. Then v ∞ satisfies Hence we must have v ∞ = µ in Ω 1 due to v ∞ > 0 in Ω 1 . By Proposition 2, we derive (λ ∞ , U ∞ , v ∞ ) = (λ * , 0, µ). This is impossible due to (14) and (17). Consequently, is true.
By a similar manner, we can show that (7) admit a positive solution if λ > λ * (µ) for the case µ < 0, and so we omit the details here.
As a last step, we discuss the case µ = 0. Since we have proved the existence of positive solutions for µ > 0 and µ < 0 respectively, we choose a sequence is a positive solution of (7) with µ = µ i . By Proposition 1 and the boundedness of {µ i } ∞ i=1 , we can further choose a subsequence such that where (U ∞ , v ∞ ) is a non-negative solution of (7) with µ = 0. Thus it follows from the maximum principle that (U ∞ , v ∞ ) satisfies either one of (a)-(c) or U ∞ > 0 and v ∞ > 0. From the first equation of (7) for any i ∈ N , thus (a) cannot occur because of λ > 0. We integrate the second equation of (7) for any i ∈ N , both (b) and (c) cannot occur due to lim i→∞ µ i = 0. Hence, the only possibility is that U ∞ > 0 in Ω and v ∞ > 0 in Ω 1 . This shows that for any fixed λ > 0, (7) with µ = 0 has a positive solution.

3.2.
Proof of Theorem 1.3. In this subsection, we will complete the proof of Theorem 1.3. For this, we establish the following lemmas.
Lemma 3.1. Suppose that (u ki , v ki ) is any positive solution of (3) with k = k i satisfying lim i→∞ k i = ∞. Then, passing to a subsequence, where U ki = (1 + k i ρ(x)v ki )u ki and U ∈ C 1 (Ω) is a non-negative function.
Proof. By Proposition 1(ii) and elliptic regularity theory, we may assume that

Lemma 3.2.
For any λ > λ * (µ) with µ ≤ 0. Suppose that (u ki , v ki ) is any positive solution of (3) with k = k i satisfying lim i→∞ k i = ∞, and suppose that passing to a subsequence. Here (u, w) is a positive solution of (4).
Proof. Denote w ki = k i v ki . Then (u ki , w ki ) is a positive solution of is bounded, then it is clear that v ki → 0 uniformly in Ω 1 . It follows from the elliptic regularity theory and Lemma 3.1 that Letting i → ∞ in (21), together with (22) and (23), we find that (u, w) satisfies (4). We next prove that u > 0 in Ω and w > 0 in Ω 1 . It follows from (4) and (23) that U is a non-negative solution of Thus either U > 0 or U ≡ 0 in Ω by the maximum principle. If U ≡ 0 in Ω, then, in view of (23), lim i→∞ u ki = 0 uniformly in Ω 1 . Due to λ > 0 and (22), we obtain This ia a contradiction. Hence we must have U > 0 in Ω, this implies that u > 0 in Ω. Similarly, it follows from the maximum principle that either w > 0 or w ≡ 0 in Ω 1 . If w ≡ 0 in Ω 1 , then The positivity of u implies that u ≡ λ in Ω. Then from the equation of v ki , it follows that , Ω 1 = 0.
Here we use the assumption λ > λ * (µ) and Lemma 1.1. This is a contradiction, which means that w must be positive in Ω 1 . Therefore, (u, w) is a positive solution of (4).
It remains to show µ < 0. We have shown that (u, w) is a positive solution of (4). Then Hence, it follows from the monotonicity of the principal eigenvalue that µ < 0.
is bounded. Thus, Lemma 3.2 implies µ < 0, which contradicts the assumption µ = 0. Note that w ki satisfies By Harnack inequality, there exists some positive constant C independent of i such that max Hence, it follows from (24) and ρ(x) = 1 in Ω 1 that For this part, it remains to prove that lim i→∞ u ki = λ in C 1 (Ω 0 ). By integrating the U -equation in (7) Here we use the fact that b(x) = 1 in Ω 1 and b(x) = 0 in Ω 0 . Furthermore, it follows from (25) that For U -equation in (7) with (U, v) = (U ki , v ki ), we divide it by U ki and integrate the obtained equation to derive After some arrangement we have By (24), the above inequality implies that It follows from (26) and (27) that Thus we must have u ≡ λ in Ω 0 . Therefore, we complete the proof of part (i).
(ii) According to the result of Lemma 3.2, we only need to verify is unbounded. Similar to Lemma 3.3, we can still apply Harnack inequality to prove is unbounded. Then we may assume that lim From Proposition 1 and (28), it follows that where max Ω1ṽ ki = 1. By the elliptic regularity theory, there exists some nonnegative functionṽ ∈ C 1 (Ω 1 ) with max Ω1ṽ = 1 such that lim i→∞ṽ ki =ṽ in C 1 (Ω 1 ).
By Harnack inequality, it is clear thatṽ > 0 in Ω 1 due to max Ω1ṽ = 1. This leads to µ = 0, a contradiction. Hence {max Ω1 k i v ki } ∞ i=1 is bounded. We complete the proof of part (ii).

3.3.
Proof of Theorem 1.4. Denote U := (1 + ρ(x)w)u. Then (4) can be written as where Let The following lemma gives the local bifurcation result for (32).
Proof. The proof is similar to that of Proposition 2, so we omit it.
The following lemma gives further information on the bifurcation curve Γ local obtained in Lemma 3.4.
Proof. By the global bifurcation theorem, we use the similar argument to Theorem 1.2 to claim that the local bifurcation curve Γ local can be extended into a global curve Γ global which is contained in the set of positive solutions of (32), moreover, Γ global is unbounded in R × E.
By Lemmas 3.4 and 3.5, we can immediately prove Theorem 1.4. Proof of Theorem 1.4. In order to complete the proof of Theorem 1.4, it suffices to obtain the convergence result of u µ by Lemma 3.5. Note that the convergence result of u µ can be proved by the similar argument to the first part (i) of Theorem 1.3. We complete the proof of Theorem 1.4.

3.4.
Proof of Theorem 1.5. In this subsection, we are devoted to the proof of Theorem 1.5. For this, we first show the existence and uniqueness of the single equation (5).
It follows from (37) and (38) that This shows that U * = U * in Ω 0 .
On the other hand, we directly integrate (5)  Since U * ≤ U * in Ω, we get U * = U * in Ω 1 .
By Proposition 1 and elliptic regularity theory, we may assume that lim i→∞ U µi = U ∞ in C 1 (Ω) as lim i→∞ µ i = ∞, where U ∞ ∈ C 1 (Ω) is a non-negative function.
It follows from (47) and (48) that After some arrangement we have Assume that U ∞ ≡ 0 in Ω. Then it follows from the assumption λ > λ * ∞ (k) and (43) that the left-hand side of (49) is positive for large i ∈ N . Clearly, this is impossible due to (49). This implies that U ∞ is a positive solution of (5). Therefore, the proof of Theorem 1.5 is complete.