Stability and bifurcation on predator-prey systems with nonlocal prey competition

In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.


(Communicated by the Chongchun Zeng)
Abstract. In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.
1. Introduction. Since diffusive predator-prey and competing systems have much significant roles in population dynamics, they have been investigated extensively in the literature. We refer to [5,6,7,8,10,13,14,30,31,33,34,39,41,42,43] on the aspect of existence and nonexistence of nonconstant steady state solutions, periodic solutions and traveling wave solutions. These results could be used to explain the complex pattern formation in ecology. For example, Yi et al. [42] considered the following diffusive Rosenzweig-MacArthur predator-prey model, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, (1.1) and investigated Hopf and steady state bifurcations to explain the complex spatiotemporal dynamics. Here u(x, t) and v(x, t) are the densities of the prey and predator at time t > 0 and a spatial position x ∈ Ω respectively; a > 0 and k > 0 represent the intrinsic growth rate and carrying capacity of the prey respectively; d > 0 represents the death rate of the predator; d 1 , d 2 > 0 are the diffusion coefficients of the predator and prey respectively; b, c > 0 measure the interaction strength between the predator and prey; and m > 0 represents the prey's ability to evade attack [14]. Through a priori estimates of positive steady states and the implicit function theorem, Peng and Shi [33] also proved the nonexistence of nonconstant positive steady states when the conversion rate c is sufficiently large, which implies that the global bifurcating branches of steady states obtained in [42] are bounded loops. We refer to [44] for the dynamics of system (1.1) with the homogeneous Dirichlet boundary conditions and [25] for a modification of (1.1) with a prey refuge. When the self-growth of the predator is logistic type, Du and Lou [14] analyzed the following model x ∈ ∂Ω, t > 0, (1.2) and investigated the positive steady states for large m. Here all the parameters are positive constants except d, which may change sign. They also analyzed the positive steady states with the homogeneous Dirichlet boundary conditions [12]. We remark that, for m = 0, system (1.2) was investigated in [26] and it was shown that every positive solution of system (1.2) converges to a constant steady state solution when time goes to infinity, which implies that system (1.2) has no nonconstant positive steady states. Moreover, Peng and Shi [33] proved the nonexistence of nonconstant positive steady states for m = 0. During the past thirty years, many researchers have focused on the nonlocal interspecific competition of a species for resources [4,18,20,21,23]. In [4], Britton firstly proposed the following single population model with nonlocal competition effect ∂u(x, t) ∂t = d∆u + λu 1 + αu − β they studied the the steady state bifurcation of Eq. (1.3) with the homogeneous Neumann boundary condition. Sun, Shi and Wang [38] also chose this spatial homogenous kernel and investigated the steady states of Eq. (1. 3) under the homogeneous Dirichlet boundary conditions. Then, the existence and bifurcations of steady states of diffusive logistic population models were investigated in [1,9,40] for more general kernel functions. We remark that for one-dimension domain Ω = (0, l), the following kernel function e −an 2 π 2 /l 2 cos nπx l cos nπy l (1.5) was used to model the growth of Nicholson's blowflies with age structure [37], which was also used in [20] for a food-limited population model. Moreover, researchers have also concentrated on the nonlocal intraspecific competition of prey for predator-prey systems [19,32]. For example, Merchant and Nagata [32] introduced the nonlocal competition of prey into the diffusive Rosenzweig-MacArthur predator-prey model and proposed the following model (1.6) When Ω = (−∞, ∞), they found that the nonlocal competition can destabilize the spatially homogeneous steady state and induce complex spatiotemporal patterns.
In this paper, we mainly consider model (1.6) when domain Ω is bounded. Firstly, we consider the case that m = 0, which means that the prey's ability to evade attack is weak, and then the interaction between the prey and predator is classical Lotka-Volterra type. For m = 0, model (1.6) has the following form Here Ω is a bounded domain in R N (N ≤ 3) with a smooth boundary ∂Ω, parameters d 1 , d 2 , a, b, c, and d are all positive constants, which have the same meanings as in Eq. (1.1), and the kernel function satisfies the following assumption: (K) K(x, y) is nonnegative and belongs to C 1 (Ω × Ω). Then, we also consider the following model (1.8) Here the predator can also survive without the specific prey, and all the parameters are positive constants except d, which may change sign. Actually, this model can also be obtained by introducing the nonlocal intraspecific competition of prey into model (1.2) for m = 0. Our main results for models (1.7) and (1.8) are as follows.
(I) There exists c 0 > 0 such that system (1.7) has a unique positive steady state for c > c 0 .

SHANSHAN CHEN AND JIANSHE YU
(II) If a > bd, then there exists c 0 > 0 such that system (1.8) has a unique positive steady state for c > c 0 , and if a < bd, then there exists c 0 > 0 such that system (1.8) has no positive steady states for c > c 0 . We remark that the method used here is motivated by [33,36]. However it cannot be used to model (1.6) for m = 0. The main reason is the lack of "order preserving property" of the nonlocal equation [11], and hence we can not estimate the upper bound for prey u. Finally, for model (1.6) with m = 0, we give the bifurcation analysis to show the existence of complex spatiotemporal patterns for a special kernel function, and find that the nonlocal competition of prey can induce some new dynamical behaviors.
The rest of the paper is organized as follows. In Section 2, we prove the existence and uniqueness of positive steady states for model (1.7) and (1.8), respectively. In section 3, we investigate the Hopf bifurcation for (1.6) with m = 0 and the spatially homogeneous kernel function.
2. Uniqueness of steady states. In this section, we will show the uniqueness of the steady states of models (1.7) and (1.8) for large conversion rate c. Therefore, complex pattern formation is impossible for large conversion rate.
2.1. Preliminaries. We first cite several results for later application. The first is the following result [27,33].
then, for any q ∈ [1, N N −2 ), there is a positive constant C, which is determined only by q, d, and Ω, such that The second is Harnack inequality [27,28,33,35].
Lemma 2.2. Suppose that Ω is a bounded Lipschitz domain in R N , and c(x) ∈ L q (Ω) for some q > N/2. If z ∈ W 1,2 (Ω) is a non-negative weak solution of the following problem ∆z + c(x)z = 0, x ∈ Ω, then there is a positive constant C, which is determined only by c(x) q , q, and Ω, such that sup Finally, we cite the following maximum principles [27,29,35].
Suppose that Ω is a bounded smooth domain in R N , and g ∈ C(Ω×R).
If z ∈ C 2 (Ω) ∩ C 1 (Ω) satisfies the inequalities and there is a constant K satisfying g(x, z) < 0 for z > K, then max x∈Ω z ≤ K.
Lemma 2.4. Suppose that Ω is a bounded smooth domain in R N , and g ∈ C(Ω×R).
If z ∈ C 2 (Ω) ∩ C 1 (Ω) satisfies the inequalities and there is a constant K satisfying g(x, z) > 0 for z < K, then min x∈Ω z ≥ K.

2.2.
Analysis of model (1.7). In this subsection, we will analyze the positive steady states of system (1.7), which satisfy x ∈ ∂Ω. (2.1) Let w = cu, z = bv and ρ = 1 c . Then w and z satisfy x ∈ ∂Ω. (II) There exists M 2 > 0 such that Proof. We first derive the existence of the upper bound. It follows from Eq.
It follows from Lemma 2.1 that, for q > N/2, there exists C 1 > 0 such that and consequently, Then, due to Lemma 2.2, there exists a constant C 4 > 0, depending on M 1 , such that sup This, combined with Eq. (2.4), implies that there exists a constant C 5 > 0, depending on M 1 , such that sup Next, we prove part (II). We first claim that there exists M 2 > 0 such that By way of contradiction, there exists a subsequence {ρ j } ∞ j=1 such that lim j→∞ ρ j = 0 and lim j→∞ inf x∈Ω w ρj = 0.
Then, it follows from Eq. (2.6) that w ρj → 0 uniformly on Ω as j → ∞. Hence, for sufficiently large j, which is a contradiction. Therefore, the claim is proved and Eq. (2.10) holds. Finally, we claim that there existsM 2 > 0 such that Similarly, we argue indirectly and assume that there exists a subsequence {ρ k } ∞ k=1 such that lim k→∞ ρ k = 0 and It follows from Eq. (2.8) that lim k→∞ sup x∈Ω z ρ k = 0. Since lim k→∞ ρ k = 0 and Ω w ρ k dx ≤ d|Ω|, we see that, for sufficiently large k, which is a contradiction. Therefore, Eq. (2.11) holds. Letting This completes the proof.
Then, we can obtain the following results on positive solutions of system (2.2) through the implicit function theorem.
Theorem 2.6. Assume that d 1 , d 2 , a, d, and ρ are all positive constants, Ω is a bounded domain in R N (N ≤ 3) with a smooth boundary ∂Ω, and K(x, y) satisfies the assumption (K). Then there exists ρ 0 > 0 such that system (2.2) has a unique positive solution for ρ < ρ 0 .
Proof. As in [36], we denote where p > N, and define an operator It follows from the embedding theorems that A is a continuously differentiable from [0, where M 2 is defined as in Lemma 2.5. Then (w ρ , z ρ ) is a solution of system (2.2) if and only if A(ρ, w ρ , z ρ ) = (0, 0) T . According to Lemma 2.4 [33], we see that A(0, w * , z * ) = 0, where (w * , z * ) = (d, a), and (w * , z * ) is non-degenerate in the sense that zero is not the eigenvalue of the linearized problem with respect to (w * , z * ). The Fréchet derivative of A with respect to (w, z) at (0, w * , z * ) is given by is an isomorphism. Then, it follows from the implicit function theorem that there exists ρ 0 > 0 such that, for 0 < ρ < ρ 0 , system (2.2) has a positive solution (w ρ , z ρ ) ∈ W 2,p ν (Ω) × W 2,p ν (Ω), which belongs to C 2 (Ω) × C 2 (Ω) from the regularity theory and embedding theorems. Then it remains to prove the uniqueness. From the implicit function theorem, we only need to verify that if (w ρ , z ρ ) is a positive solution of system (2.2), then From Lemma 2.5, we find that

SHANSHAN CHEN AND JIANSHE YU
where M 2 is defined as in Lemma 2.5 and It follows from Eq. (2.12) and the L p theory that Due to the embedding theorems, we find that Since and lim k→∞ ρ i k Ω K(x, y)w ρi k dy = 0 in C 1 (Ω), we see that, for some α ∈ (0, 1), and (w * , z * ) is a positive solution of Eq. (2.2) for ρ = 0. It follows from Lemma 2.4 [33] that (w * , z * ) = (w * , z * ). This completes the proof.
2.3. Analysis of model (1.8). In this subsection, we will analyze the positive steady states of system (1.8), which satisfy x ∈ ∂Ω. (2.13) Let w = cu, z = bv and ρ = 1 c . Then w and z satisfy (2.14) Similarly, the existence/nonexistence of positive solutions of system (2.13) for large c is also equivalent to that of system (2.14) for small ρ. As in Lemma 2.4 of [33], we first consider the positive solutions of system (2.14) when ρ = 0.
Lemma 2.7. Assume that ρ = 0, constants d 1 , d 2 , a, and b are all positive, and Ω is a bounded domain in R N with a smooth boundary ∂Ω. If a > bd, then system (2.14) has a unique positive solution Proof. We construct the following functional: An easy calculation implies that Therefore, if (w(x), z(x)) is a positive solution of system (2.14), then V (w, z) = 0 which leads to (w(x), z(x)) = (w * , z * ).
Then, from Lemmas 2.2-2.4, we obtain a priori estimates for positive solutions of system (2.14). Proof. We first prove part (I). From Eq. (2.14), we see that which imply that z ρ 2 L 2 ≤ (a + bd) z ρ L 2 |Ω| 1/2 − abd|Ω|. Therefore, there exists a constant C 1 > 0 such that Ω w ρ dx, z ρ L 2 ≤ C 1 for all ρ ≥ 0, which leads to Then, it follows from Lemma 2.2 that there exists a constant C 2 > 0, depending on M 1 , such that and consequently, Now, we prove part (II). We first claim that there exists M 2 > 0 such that We argue indirectly and assume that there exists a subsequence {ρ j } ∞ j=1 such that lim j→∞ ρ j = 0 and lim j→∞ inf x∈Ω w ρj = 0.
Then, it follows from Eq. (2.17) that w ρj → 0 uniformly on Ω as j → ∞. If d < 0, then, for sufficiently large j, which is a contradiction. If d ≥ 0, then, for sufficiently small ∈ 0, a − bd b , there exists j 0 > 0 such that max x∈Ω |w ρj | < and for j > j 0 . As a consequence, we have z ρj ≤ b(d + ) and It follows from Lemma 2.2 that lim k→∞ sup x∈Ω z ρ k = 0. Since lim k→∞ ρ k = 0 and Ω w ρ k dx ≤ C 1 , we see that, for sufficiently large k, which is a contradiction. Therefore, Eq. (2.21) holds. This completes the proof.
In what follows, we give our main results on solutions of system (2.14) for small ρ. Theorem 2.9. Assume that d 1 , d 2 , a, b, and ρ are all positive constants, Ω is a bounded domain in R N (N ≤ 3) with a smooth boundary ∂Ω, and K(x, y) satisfies assumption (K). Then the following two statements are true.
(I) If a > bd, then there exists ρ 0 > 0 such that system (2.14) has a unique positive solution, which is locally asymptotically stable for ρ < ρ 0 . (II) If a < bd, then there exists ρ 1 > 0 such that system (2.14) has no positive solutions for ρ < ρ 1 .
As in Theorem 2.6, we can prove that, for any sequence where (w * , z * ) is a positive solution of Eq. (2.14) for ρ = 0. It follows from Lemma 2.7 that (w * , z * ) = (w * , z * ). This completes the proof of part (I).
In the following, we consider the case that a < bd. We argue indirectly and assume that there exists a sequence {ρ i } ∞ i=1 such that lim i→∞ ρ i = 0 and system (2.14) has a positive solution (w ρi , z ρi ) for ρ = ρ i . It follows from Lemma 2.4 and the second equation of system (2.14) that z ρi ≥ bd, and consequently, Since lim i→∞ ρ i = 0, we see that, for sufficiently large i, As a consequence, w ρi ≤ 0 for sufficiently large i. This is a contradiction.
3. Hopf bifurcation for spatially homogeneous kernel. To show the existence of complex spatiotemporal patterns, in this section, we consider the Hopf bifurcation for model (1.6) with the homogeneous Neumann boundary conditions. Here we do not assume that m = 0, and kernel function K(x, y) is chosen as in Eq. (1.4) for simplicity. Moreover, to compare with the work of [42] and show the effect of nonlocal competition, we also choose Ω = (0, π) as in [42]. Then Eq. (1.6) has the following form An easy calculation implies that (u * , v * ) is locally asymptotically stable for m = 0. This, combined with Theorem 2.6, also implies that model (3.1) has a unique positive steady state, which is constant for large conversion rate c. Now, we consider the case that m = 0. Recall from [42] that the nondimensionalized form of (1.1) is Similarly, the nondimensionalized form of (3.1) is (3.4) where γ is equivalent to conversion rate c. Model (3.4) has a constant positive steady state (λ, v λ ), where if and only if γ > θ(1 + k) k (or equivalently, 0 < λ < k). Therefore, throughout this section, we always assume λ ∈ (0, k), and use λ (equivalent to conversion rate c) as a bifurcation parameter to study the Hopf bifurcation. Moreover, we will compare the bifurcation points between models (3.3) and (3.4) and investigate whether nonlocal competition can bring new dynamical behaviors. We denote N = {1, 2, 3, · · · } and N 0 = {0, 1, 2, 3, · · · }. Let and define the complexification of X to be X C : Then we obtain the sequence of the characteristic equations with respect to (λ, v λ ) as follows: where and for n ∈ N, (3.10) Therefore, (λ, v λ ) is locally asymptotically stable, if T n (λ) < 0 and D n (λ) > 0 for each n ∈ N 0 . Moreover, it follows from [24,42] that Hopf bifurcation value λ 0 satisfies the following condition: (H 1 ): There exists n ∈ N 0 such that T n (λ 0 ) = 0, D n (λ 0 ) > 0, and T j (λ 0 ) = 0, D j (λ 0 ) = 0 for j = n, (3.11) and the unique pair of complex eigenvalues α(λ) ± iω(λ) near λ 0 satisfy α (λ 0 ) = 0. (3.12) Denote and we can easily derive the following results for later application.
Lemma 3.1. Assume that k > 0. Then the following two statements are true.
Then we consider the occurrence of Hopf bifurcation for k ≤ 1.

14)
and define where C i (λ), λ i (i = 1, 2) are defined as in Eq. (3.13) and Lemma 3.1 respectively. Then the following two statements are true.
2. For ∈ ( 1 , ∞), when λ passes through λ H 1,− ( ) (or λ H 1,+ ( )) from left to right, (λ, v λ ) will change its stability from stability to instability (or instability to stability) through Hopf bifurcation. Moreover, the bifurcating periodic solutions are spatially nonhomogeneous. This is also a new dynamical behavior induced by the nonlocal effect. For model (3.3), (λ, v λ ) can also change its stability from stability to instability when λ passes through some Hopf bifurcation point, but the bifurcating periodic solutions near such bifurcation point are always spatially homogeneous. 3. As in [42], we can also prove that for ∈ ( n , n+1 ) with n ∈ N, there exist 2n Hopf bifurcation points λ H j,± ( ) (1 ≤ j ≤ n) satisfying (3.17) However, (λ, v λ ) will not change its stability or instability when λ passes through these points λ H j,± ( ) for 2 ≤ j ≤ n.
The case that k > 1 is more complex. In this case T 0 ((k − 1)/2) = 0, and consequently, (k − 1)/2 is also a possible Hopf bifurcation point. As in Theorem 3.2, we only concern with some special Hopf bifurcation points. When λ passes through such points, (λ, v λ ) will change its stability or instability.
Then, we have the following results for k ≥ 3, and omit the proof here.