POSITIVE STEADY STATES OF A DENSITY-DEPENDENT PREDATOR-PREY MODEL WITH DIFFUSION

. In this paper, we investigate the rich dynamics of a diﬀusive Holling type-II predator-prey model with density-dependent death rate for the preda- tor under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant posi- tive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we ﬁnd that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diﬀusion.

1. Introduction. In this paper, we study the positive steady states of the following reaction-diffusion predator-prey model with prey-dependent Holling type-II functional response and density-dependent death rate for the predator: x ∈ Ω, t > 0, where N (x, t), P (x, t) represent population densities of prey and predator at time t and location x ∈ Ω ⊂ R 2 , respectively. Ω is a bounded domain with smooth boundary ∂Ω and ν is the outward unit normal vector of the boundary ∂Ω. And cN a + N describes functional response of the predator, which is Holling type-II functional response. All parameters r, c, a, q, s, K, δ are positive. r is the intrinsic growth rate or biotic potential of the prey N , c the rate of capture, a half saturation constant, q the death rate of the predator, s the feed concentration, δ proportional to the density-dependent death rate and K the carrying capacity. The positive constants d 1 and d 2 are the diffusion coefficients of N (x, t) and P (x, t), respectively.
is the Laplacian in two dimensional space, which describes the random moving.
In mathematical ecology, the classical predator-prey system, due independently to Lotka and Volterra in the 1920s, reflects only population changes due to predation in a situation where predator and prey densities are not spatially dependent, and the corresponding model is systems of ordinary differential equations (ODE). Interaction between predator and prey has been a central research theme in ecology over many decades [1,2,4,12,15,28]. A wide variety of temporal predator-prey models have been investigated to help us understand the steady-state or oscillatory coexistence of both the species as well as the factors responsible for the system collapse through the extinction of one or both the species. Prey-dependant functional responses play an important role in dynamics of predator-prey models [1,2,17]. Gause type predator-prey models have been studied by many researchers [3,13,32,18]. The Gause type predator-prey models with predator's density-dependant functional response exhibit very rich dynamic behavior [17,13]. But the ODE model does not take into account either the fact that population is usually not homogeneously distributed, or the fact that predators and preys naturally develop strategies for survival. Both of these considerations involve diffusion processes which can be quite intricate as different concentration levels of predators and preys cause different population movements [16,37,38,11,36,31].
In [5], the authors established the following model with Allee effect (measured by the term m N + b ): x ∈ Ω, t > 0. (2) and investigated the asymptotical stability of the positive equilibrium, and gave the conditions of the existence of the Hopf bifurcation. Furthermore, in [6], based on the results in [5], the authors investigated the effect of density-dependent death rate

POSITIVE STEADY STATES OF A PREDATOR-PREY MODEL WITH DIFFUSION 3089
on the dynamics of the following model: and found that the density-dependent death rate δ in predator can induce Turing instability, and model (3) exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. Obviously, model (1) is the special case of m = 0 of model (3) in [6].
And there naturally comes a question: does model (3) without Allee effect, i.e., model (1), exhibit Turing pattern replication? This is one of our main goals in this paper.
The other goal of this paper is to investigate the existence, nonexistence and structure of nonconstant positive steady-state solutions to problem (1), specifically, we will concentrate on the following steady state system The rest of this article is organized as follows: in Section 2, we discuss the stability of constant steady state solution and give the conditions of Turing instability. In Section 3, we investigate the nonexistence/existence of nonconstant positive steady states. In Section 4, we analyze the local and global structure of nonconstant positive solutions, and give the direction of the bifurcation.
2. Constant steady state and Turing instability. In this section, we mainly discuss the stability of constant steady state solution. For convenience, we denote Then (1) can be written as Whereas the corresponding spatially homogeneous counterpart (i.e., d 1 = d 2 = 0) of problem (5) is as follows: The Turing instability refers to "diffusion driven instability", i.e., the stability of the positive constant steady state E * changing from stable for the ordinary differential equations (ODE) dynamics (6), to unstable, for the partial differential equations (PDE) dynamics (1) or (5).
Obviously, the ODE model (6) has the same constant steady states as the PDE model (5). And model (6) has a trivial steady state E 0 = (0, 0), a semitrivial steady 3090 K. HUANG, Y. CAI, F. RAO, S. FU AND W. WANG state E 1 = (K, 0). In additional, model (6) has at least one positive steady state , and N * is the positive roots of polynomial equation Lemma 2.1. (Shengjins discriminant [10]) Let equation  a i X i be a polynomial of degree n with real coefficients that has exactly p positive real roots, counted with multiplicities. Let v = var(a 0 , . . . , a n ) be the number of sign variations in its coefficient sequence. Then v ≥ p and v ≡ p(mod 2). If all roots of A(X) are real, then v = p.
Proof. (i) If ∆ > 0, by Lemma 2.1, (7) has a unique positive roots. Notice that the constant term D of the left hand of equation (7) is negative, hence equation (7) has a unique positive solution.
(ii) If ∆ ≤ 0, applying Lemma 2.1 again, then (7) has three real roots, (ii-1) if a > K/2, the signs of the coefficient 1, B, C, D of (7) may be + + +− or + + −−, the signs change only one time in these two cases above. By Lemma 2.2, we can claim that (7) has a unique positive root. It's easy to verify that the conclusion is correct in the case of a = K/2.
Based on the results above, we can obtain that (6) or (5) has a unique positive steady state E * = (N * , P * ).
Next, for simplicity, we will focus on the stability of the unique positive steady state E * for ODE model (6) and PDE model (5), respectively. First, we consider the stability of E * for ODE model (6). By simple calculation, we can obtain the Jacobian matrix of (6) evaluated at E * is given by where The characteristic equation of J(E * ) is where Q = a 11 + a 22 , P = a 11 a 22 − a 12 a 21 .
(11) It is obvious that E * is locally asymptotically stable if Q < 0 and P > 0. Thus, we can obtain the following theorem.
Theorem 2.4. The constant steady state solution E * of ODE model (6) is locally asymptotically stable, if one of the following conditions holds: (i) a 11 < 0; Next, we consider the stability of E * for PDE model (5). Consider the eigenvalue problem Let 0 = λ 0 < λ 1 < . . . be the sequence of eigenvalues for the elliptic operator − subject to the Neumann boundary condition on Ω, where λ i (i ≥ 1) has multiplicity m i ≥ 1, whose corresponding normalized eigenfunctions are given by φ ij , where j = 1, 2, . . . , m i . This set of eigenfunctions form an orthogonal basis in L 2 (Ω).
If a 11 > 0 and then we define i 0 to be the largest positive integer such that In this case, we let where d i 2 (E * ) is given by Therefore we can obtain the stability of E * of PDE model (5) as follows: 11 and 0 < d 2 <d 2 hold, E * is locally asymptotically stable.
Proof. Consider the linearization operator evaluated at E * : Suppose Φ = (ϕ, ψ) ∈ L is an eigenfunction of L corresponding to an eigenvalue η, and we can obtain Easy to know that η is the eigenvalue of L if and only if det B i = 0, which leads to ) .
(i) If a 11 < 0, then Q i > 0 and P i > 0, which implies that Re{η i } < 0 for all eigenvalues η. Therefore, the constant solution E * is locally asymptotically stable.
> 0 holds, then we have Q i > 0 and d 1 a 22 λ i − a 11 a 22 + a 12 a 21 < 0, and therefore, (ii-1) If a 11 > 0, d 1 λ 1 < a 11 and 0 < d 2 <d 2 , then d 1 λ i < a 11 and d 2 < d i 2 for all i ∈ [1, i 0 ]. Thus, One the other hand, if i > i 0 , then d 1 λ i > a 11 . So, we have P i > 0. The analysis yields the local asymptotical stability of E * .
(ii-2) If a 11 > 0, d 1 λ 1 < a 11 and d 2 >d 2 , then we may assume the minimum in (15) Hence, E * is unstable in this case. The proof is complete.
Remark 1. From Theorem 2.4 and 2.5, we can know that if a 11 > 0, the stability of the constant equilibrium E * may change from stable, for the ODE dynamics (6), to unstable, for the PDE dynamics (5), whereas those of other constant equilibria are invariant.
Remark 2. Compare these results above with [5] and [6], we can conclude that Turing instability is not induced by Allee effect, but density-dependent death rate δ.
Example 1. As an example, motivated by [6], we take the parameters in model (5) and (6) as: And easy to know that there is a unique constant positive steady state E * = (N * , P * ) = (1.2369, 2.3984). For the ODE model (6), from Theorem 2.4, easy to verify that E * is stable. For the PDE model (5) on one-dimensional space domain (0, π), after fixing d 1 = 0.015, from Theorem 2.5, we can know that if d 2 >d 2 = 0.4928, E * is Turing unstable, and model (5) exhibits Turing pattern. In Fig. 1, we show the numerical results of model (5) with different values of d 2 . Fig. 1(a) shows the numerical simulations of Turing instability in model (5) with d 2 = 0.6 >d 2 . And Fig. 1 3. Nonconstant positive solutions. In this section, we study the steady state problem (4), and we establish the existence and nonexistence results of positive nonconstant solutions, and in these results, the diffusion coefficients d 1 and d 2 play an important role. The mathematical techniques to be employed are energy method and implicit function theorem, respectively. From now on, let N (λ i ) ⊂ H 1 (Ω) be the eigenspace. Unless otherwise specified, in the following section, we always require that a 11 > 0 and

3.1.
A priori estimate for positive solutions. In this subsection, we derive some priori estimates of upper and lower bounds for positive solutions to (4), and these estimates will become fundamental in yielding the existence and nonexistence results of positive nonconstant solutions to (4) in the forthcoming subsections.
Lemma 3.2. (Harnack inequality [21]) Assume that c ∈ C(Ω) and let w ∈ C 2 (Ω) ∩ Then there exist a positive constant C * ( c ∞ ) such that Similar to Theorem 3.4 in [6], we can obtain the upper and lower bounds of the solutions to model (4).   (4) satisfies 3.2. Nonexistence of nonconstant positive steady state solutions. In this subsection, we will present several nonexistence results of nonconstant positive solutions to (4). First, similar to Theorem 4.4 in [6], we apply energy method to establish results of the nonexistence of nonconstant positive solutions of (4). For convenience, we denote Λ = Λ(r, K, s, c, a, q, δ).

POSITIVE STEADY STATES OF A PREDATOR-PREY MODEL WITH DIFFUSION 3095
Second, we apply the implicit function theorem method to establish results of the nonexistence of nonconstant positive solutions of (4). First of all, we show the following Lemma, which can be found in [8].
Let Ψ be the Fréchet derivative of F at (0, N * , 0, P * ) with respect to (h, z, P ), a direct computation yields where a ij is given in (10).
To complete the proof of this Theorem, we note that, by an implicit function theorem, there is a constant σ such that, for all 0 < ρ < σ, in a small neighborhood of (N * , 0, P * ), the equation F (ρ, h, z, P ) has a unique solution, which must be (N * , 0, P * ). Correspondingly, when d 1 is large, in a small neighborhood of (N * , P * ), the problem (4) has only the constant solution (N * , P * ). This fact, combined with Lemma 3.5, concludes the proof.
Remark 3. The results in this subsection demonstrate such a phenomenon: when all diffusion coefficients are large, no patterns exist (c.f., Theorem 3.4); or even if only one diffusion coefficient is large, the patterns do not exist (c.f., Theorem 3.6), which implies that the diffusion is helpful to create nonconstant positive steadystates solutions to the predator-prey model (1). That is, these results show more delicate dependence on the diffusion coefficients for the predator-prey system.
, where f and g are given in Section 2. Then the stationary problem of (4) can be written as In this subsection, we study the linearization of (17) at E * and then proceed to calculate the fixed point index of E * when it is an isolated solution. Further, we note that where γ is the sum of the algebraic multiplicities of all the negative eigenvalues of D E F (E * ).
Since the eigenvalues of D E F (E * ) and their algebraic multiplicities are the same regardless of whether we consider an operator in X or in Y , a straightforward calculation shows that, for each integer i ≥ 0 and each integer 0 ≤ j ≤ dimN (λ i ), X ij is invariant under D E F (E * ). Moreover, λ is an eigenvalue of D E F (E * ) if and only if, for some i ≥ 0, it is an eigenvalue of the matrix Thus, D E F (E * ) is invertible if and only if the matrix B i is nonsingular for all i ≥ 0.

POSITIVE STEADY STATES OF A PREDATOR-PREY MODEL WITH DIFFUSION 3097
Let λ be an eigenvalue of D E F (E * ). We now calculate its algebraic multiplicity, which we denote by σ(λ). Write and we can see that As a consequence, we have the following Lemma.
To facilitate our computation of index(F (·), E * ), we need to determine the sign of H(λ i ). In particular, we will concentrate on the dependence of H(λ i ) on d 2 . At this point, we note that A straightforward computation gives Since we always assume a 11 > 0 and ac 2 (a + N * ) 3 and (20) holds. Ifλ ∈ (λ n , λ n+1 ) for some n ≥ 1 and n i=1 dimN (λ i ) is odd, then there exists a positive constantd 2 such that, for d 2 ≥d 2 , model (4) has at least one constant positive solution.
Proof. The proof, which is by contradiction, is based on the homotopy invariance of the topological degree. Suppose on the contrary that the assertion is not true for some In what follows we fix d 2 = d * 2 . From Theorem 3.4, we know that there existsd 1 ,d 2 such that model (4) with Then E is a positive nonconstant solution of (4) if and only if it is such a solution of (21) for t = 1. It is obvious that E * is the unique constant positive solution of (21) for any 0 ≤ t ≤ 1. For any 0 ≤ t ≤ 1, E is a positive solution of (21) if and only if . From (18) and (19), we get where, zero is not an eigenvalue of the matrix Now, by Theorem 3.3, there exists a positive constant C such that, for all 0 ≤ t ≤ 1, the positive solution of (4) satisfies 1/C < N, P < C. Therefore, F (t; E) = 0 on ∂B(C) for all 0 ≤ t ≤ 1. By the homotopy invariance of the topological degree, deg(F (1; ·), 0, B(C)) = deg(F (0; ·), 0, B(C)).
On the other hand, by our supposition, both equations F (1; E) = 0 and F (0; E) = 0 have only the positive solution E * in B(C), and hence, by (22) and (23) This contradicts (24), and the proof is complete. In this subsection, we study the local structure of nonconstant positive solutions for model (4). In brief, by regarding d 2 as the bifurcation parameter, we verify the existence of positive solutions bifurcating from (d 2 , 0). The Crandall-Rabinowitz bifurcation theorem in [7] will be applied to obtain bifurcations from simple eigenvalues. Define the map F : (0, ∞) × X → Y by where f, g are given in Section 2. Then the solutions of boundary value problem (4) are exactly zero. With E * = (N * , P * ), we have If there is a number τ > 0 such that every neighborhood of (τ, E * ) contains zero of F in (0, ∞) × X not lying on the curve (d 2 , E * ), then we say that (τ, E * ) is a bifurcation point of the equation F = 0 with respect to this curve.
, and so condition (c) is satisfied. The proof is completed.
4.1.1. Direction of the bifurcation solutions. In this subsection, we investigate the direction of the bifurcation solutions of model (4) in the one-dimensional space domain. In the 1D interval Ω = (0, π), it is well known that the operator − with no-flux boundary conditions has eigenvalues and eigenfunctions as follows: for j = 1, 2, 3, . . .. We translate (N * , P * ) to the origin by the translation (N ,P ) = (N − N * , P − P * ). For convenience, we will denoteN ,P by N, P , respectively. Then we can obtain the following system ), x ∈ (0, π).

4.2.
Global structure of nonconstant positive solutions. Theorem 4.1 provides no information of the bifurcating curve Γ j far from the equilibrium. A further study is therefore necessary in order to understand its global bifurcation. we will prove that Γ j is unbounded, using the global bifurcation theory of Rabinowitz and the Leray-Schauder degree for compact operates.
Note that K(d 2 ) is a compact liner operator on H 1 for any given d 2 > 0, H(Ẽ) = o(|Ẽ|) forẼ near zero uniformly on closed d 2 sub-intervals of (0, ∞), and is a compact operator on H 1 as well. In order to apply Rabinowitz's global bifurcation theorem, we first verify that 1 is an eigenvalue of K(d j 2 ) of algebraic multiplicity one. From the argument in the proof of Theorem 4.1 it is seen that ker(K(d j 2 ) − I) = ker L 1 = span{Φ 1 }, so 1 is indeed an eigenvalue of K = K(d j 2 ), and dim ker(K − I) = 1. As the algebraic multiplicity of the eigenvalue 1 is the dimension of the generalized null space ∪ ∞ i=1 ker(K − I) i , we need to verify that ker(K − I) = ker(K − I) 2 , or ker(K − I) ∩ R(K − I) = 0.
And µ 2 (d 2 ) is an increasing function of d 2 , there is a small > 0 such that µ 1 (d j 2 + ) > 1, µ 1 (d j 2 − ) < 1. Consequently, K(d j 2 + ) has exactly one more eigenvalues that are larger than 1 than K(d j 2 − ) does, and by a similar argument above we can show this eigenvalue has algebraic multiplicity one. This verifies (32). And the proof is complete.