SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS∗

A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.


Introduction
Predator-prey interaction systems with ratio-dependent functional response have been paid great attention by both applied mathematicians and ecologists, e.g. [2,4,13,16,18,19,29,32,33,35,39]. Ratio-dependent functional response shows that the growth rate of capital predator should be a function of the ratio of prey to predator abundance [1,3,30]. Since diffusive predator-prey systems play an important role in population dynamics, the existence and non-existence of non-constant positive solutions, periodic solutions and traveling wave solutions have been investigated extensively [7,8].
Most analyses of diffusive predator-prey systems are restricted to Neumann boundary conditions [37,40], which indicates no flux on the boundary. One interesting question is that the exterior environment is friendly and the species can move across the boundary of environment, with the number of species being constant on the boundary [10,17,27,28,31]. For example, in fish migratory behaviour [28], fishes were found to be dispersed between areas and [34] suggests an extensive drift of cod larvae from the North Sea into coastal Skagerrak. Moreover, nonhomogeneous Dirichlet boundary value problem has attracted much attention in recent years [5,14]. In this paper, we consider the following diffusive predator-prey model with non-homogenous Dirichlet boundary conditions and ratio-dependent functional response: where the habitat of both species Ω is a bounded domain in R N (N ≥ 1) with a smooth boundary ∂Ω; u(x, t) and v(x, t) represent the densities of prey and predator at the location x and time t, respectively, and d 1 , d 2 are the rescaled diffusion coefficient for the prey and the predator, respectively, the parameter σ is the intrinsic growth rate of predator, typically it admits that the carrying capacity of predator is proportional to the densities of prey. Moreover, p, g are assumed to satisfy the following hypotheses: The condition (a1) on the functional response p(u) includes the classical Leslie-Gower type, Holling-Tanner type, Sigmodial type and Ivlev type. Here ug(u) is the net growth rate of the prey, the prey u has a growth which is reflected from the assumption (a2). Note that (1.1) has a unique constant positive equilibrium e * = (u * , v * ) under assumptions (a1) and (a2), where u * satisfies g(u * ) = p(u * ), and non-homogeneous Dirichlet boundary conditions u(x, t) = v(x, t) = u * indicate that species are free to penetrate and cross borders. In some cases, the number of predators is the same as the number of prey on the boundary. When ug(u) is of a logistic growth on the prey, the kinetics of system (1.1) with Leslie-Gower type functional response has a globally asymptotically stable equilibrium [22]. For the kinetics of system (1.1) with logistic growth on the prey and Holling-Tanner type functional response, a complete dynamical analysis can be obtained in [6,11,20,24,25,39]. With homogeneous Neumann boundary conditions and under suitable conditions, there is a positive steady state solution which indicates that the predator and prey species coexist see [9,12,26] and the references therein. In these work, specific forms of function g(u) were used. For system (1.1) with homogeneous Neumann boundary conditions, Ko [21] investigated the existence and non-existence of non-constant positive solutions under some conditions. In [38], Wang studied a diffusive plant-herbivore system with homogeneous and non-homogeneous Dirichlet boundary conditions, which shows more richer dynamical behaviors.
Our purpose here is to investigate the spatiotemporal dynamics of system (1.1). We identify an explicit difference of the stability of equilibrium e * between nonhomogeneous Dirichlet boundary conditions and homogeneous Neumann boundary conditions for p(u * ) in some ranges. The paper is organized as follows. In Section 2, we study the linear stability and instability of the positive spatially homogeneous steady state of system (1.1). The difference between the former boundary conditions and the latter boundary conditions on the stability of system (1.1) is discussed. We investigate the occurrence of steady state bifurcation and Hopf bifurcation in Section 3. Some numerical results are given in Section 4.

The stability of the unique constant steady state solution
For such diffusive predator-prey model (1.1), it is normal to obtain the existence and boundedness of the unique solution by using the maximum principle and the comparison principle [23,41]. It is clear that e * = (u * , v * ) is the only constant positive equilibrium of system (1.1). In this section, we discuss the stability of the unique constant steady state. Linearizing the reaction-diffusion system (1.1) about e * = (u * , v * ) gives Let µ i be the sequence of eigenvalues of −∆ with Dirichlet boundary conditions such that 0 < µ 1 ≤ µ 2 ≤ . . . and lim t→∞ µ i = ∞ and φ i be the normalized eigenfunctions corresponding to µ i . After Fourier series expansions, the eigenvalues of L are determined by the characteristic equation: Therefore, all the eigenvalues of the operator L can be the union of the eigenvalues of J i for i ≥ 1. Denote The linear stability of the steady state e * = (u * , v * ) of system (1.1) is determined by the eigenvalues of the characteristic equation (2.3) and (2.4), if T i < 0 and D i > 0 for all i = 1, 2, 3, · · · , then e * = (u * , v * ) is locally asymptotically stable. Otherwise, e * = (u * , v * ) is unstable [36,42].
Remark 2.1. In fact, compared with homogenous Neumann boundary conditions, the stability of e * = (u * , v * ) does not change for M ≤ 0, see [21], but it is very different for M > 0.
Case II: M > 0 Now we study the stability change of e * = (u * , v * ) when M > 0.
Based on Lemma 2.3, we further justify the sign of D i for all i = 1, 2, 3, · · ·. To and Ω be a bounded domain with smooth boundary. Suppose (A1) holds. If p(u * ) > M , then the constant positive equilibrium (u * , v * ) of system (1.1) is locally asymptotically stable.
and p(u * ) > M , the coexistent equilibrium of system (1.1) with non-homogeneous Dirichlet boundary conditions is locally asymptotically stable for i = 1, 2, 3, · · ·. The stability of the coexistence equilibrium to the diffusive predator-prey system (1.1) with Neumann boundary conditions can be founded in [21]. Proof. In order to determine the sign of D i in (2.4), we need to consider the term Hence the constant positive equilibrium (u * , v * ) of system (1.1) is locally asymptotically stable. At here, we investigate the unstability for the positive equilibrium e * = (u * , v * ) with hypothesis ζ(D i ) > 0. Solving D i = 0 for σ gives the critical point of neutral stability: where p(u * ) > M , σ(µ i ) is increasing with respect to µ i for 0 < µ i < µ i and . At this critical wave number µ i , σ(µ i ) = max{σ(µ i ), i = 1, 2, · · · n}. 1 If σ > σ(µ i ), then the constant equilibrium (u * , v * ) is locally asymptotically stable for system (1.1).

Steady state bifurcation and Hopf bifurcation
In this section we consider non-constant steady state solutions of (1.1) bifurcating from the positive constant equilibrium (u * , v * ), using the predator intrinsic growth coefficient σ as the main bifurcation parameter while d 1 > 0, d 2 > 0, σ > 0 and Ω are fixed. It is clear that the positive constant coexistence steady state (u * , v * ) exists and the precise stability information of (u * , v * ) is determined by the trace and determinant of J i (i ≥ 0), which are defined in ( Then the sets ℵ = {(σ, q) ∈ R + : T (σ, q) = 0} and = {(σ, q) ∈ R + : D(σ, q) = 0} are potential Hopf bifurcation and steady state bifurcation curves sets. The studies in [15,36,42] show that the geometric properties of ℵ and play an important role in the bifurcation analysis of system (1.1).

Steady state bifurcation
In this subsection, we explore the occurrence of steady state bifurcation at the steady state (u * , v * ). Applying the abstract bifurcation theorem in [42], we know that a steady state bifurcation occurs if there exists a critical value σ S for some integer i ≥ 1, at which (S1) D i (σ S ) = 0, T i (σ S ) = 0, and D j (σ S ) = 0, for j = i; is a steady state bifurcation value. From D i (σ) = 0, we can obtain that σ = σ S i = then there exists a branch of non-constant positive solutions of system (1.1) bifurcating from (u * , v * ) when σ = σ S i , where µ i satisfies that µ 1 < µ 2 < · · · < M/d 1 to make sure that σ i > 0.

Hopf bifurcation
In this subsection, we analyze the properties of Hopf bifurcations for (1.1). To identify Hopf bifurcation values σ H , we recall the following sufficient condition from [42]: (T i (σ) and D i (σ) are defined in (3.1) and (3.2)). Firstly is a Hopf bifurcation point since T i (σ H i ) = 0, and T j (σ H i ) = 0 for any j = i. Next we verify that D i (σ H i ) > 0. In fact, from d 1 > 0, d 2 > 0 and M (p(u * ) − M ) > 0, we denote µ ± be the two roots of −d 1 µ 2 i − [2d 1 M + (d 1 + d 2 )p(u * )]µ i + M (p(u * ) − M ) = 0, then µ − < 0 < µ + . Therefore, for any σ ∈ (0, σ 1 (µ i )), if µ i satisfy then which also implies that a Hopf bifurcation point and a steady state bifurcation point do not overlap. Summarizing our analysis above and applying Theorem 2.1 in [42], we obtain the following results on the Hopf bifurcations:    (6x)). When only the boundary conditions are changed and the same parameters and initial value are kept the same as in Figure 1, the solution does not converge to (u * , v * ) = (0.0185, 0.0185).

Conclusions and numerical simulations
In this paper, we discuss a diffusive predator-prey model with ratio-dependent functional response subject to non-homogeneous Dirichlet boundary conditions. We consider the case of linear functional response g(u) = p − bu and Holling-Tanner type p(u) = u u+a , where g(u) and p(u) satisfy (a1)-(a2). From Theorems 2.1, 2.4, 2.6, 2.8, 3.1 and 3.2, we obtain a complete picture of the dynamics of system (1.1). Furthermore, we use some numerical simulations to illustrate our analytical results. With parameter set (i), (A1) and M < p(u * ) < M hold, (u * , v * ) of system (1.1) with non-homogeneous Dirichlet boundary conditions is locally asymptotically stable, as is shown in Figure 1, which confirms the analysis of Theorem 2.6. When  σ > M −µ 1 (d 1 +d 2 ) and M < p(u * ) < M , (u * , v * ) of system (1.1) with homogeneous Neumann boundary conditions becomes unstable, see Figure 2. As shown in Figure  1 and Figure 2, under the same parameters and initial value, compared the former conditions with the latter conditions, the change of boundary conditions alters the stability of (u * , v * ).
Remark 4.1. Since T 0 and D 0 are strictly greater or less than zero, there is no spatially homogeneous Hopf/steady state bifurcation considered in the paper. But we obtain spatially non-homogeneous Hopf bifurcation, also as shown in Figure 5, Figure 6.