Qualitative analysis on the diffusive Holling–Tanner predator–prey model

. We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steady-state. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel’kov model to obtain the non-existence of positive non-constant steady-states if 0 < p ≤ 1


Introduction
In this paper, we consider the diffusive Holling-Tanner predator-prey model: (1.1) Here u and v are the density of prey and predator, respectively, Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, ν is the outward unit normal vector on ∂Ω, and the parameters d 1 , d 2 , a, b, m, γ are positive constants.The initial data u 0 and v 0 are C 1 (Ω) functions satisfying ∂ ν u 0 = ∂ ν v 0 = 0 on ∂Ω.The model describes real ecological interactions of various populations such as lynx and hare, sparrow and sparrow hawk (cf.[7,13,15]), and the Neumann boundary condition means that no species can pass across the boundary ∂Ω.We note that problem (1.1) has a unique positive global solution, see the Appendix for the proof.
Note that for fixed a, b and m, every global result in Theorems 1.1 and 1.2 excludes the case where γ is large.In this paper, we prove the following result that covers the case.
Then the positive equilibrium E * is globally asymptotically stable.
Then, as a consequence of Theorem 1.3, we obtain immediately Then there exists a positive constant γ 0 depending only on b, a, m such that E * is globally asymptotically stable for any γ ≥ γ 0 .
The steady-states of system (1.1) satisfy (1.3) Theorems 1.1-1.3obviously imply some conditions for the non-existence of positive nonconstant solutions of system (1.3), which are independent of the coefficients d 1 and d 2 .In [9], Peng and Wang gave some conditions for the non-existence of positive non-constant solutions of system (1.3), which depend on d 1 and d 2 , see [9, Theorems 3.1 and 3.5].For example, they proved that system (1.3) has no positive non-constant solution if d 1 and d 2 are sufficiently large, see [9,Theorems 3.1].By using a different approach from those in literature (see e.g.[8,10]), we prove the following result on the non-existence of positive non-constant solutions.
We point out that the approach used to show Theorem 1.6 can be applied to some interesting models to discuss non-existence of positive non-constant solutions, for instance, the steady-state Sel'kov model (see [12]): where θ, λ, p are positive constants, which had been studied in [6,8,14].For the case when 0 < p ≤ 1, Peng [8] proved the non-existence of positive non-constant solutions of system (1.3) if θ is sufficiently large.In the present paper, we remove the restriction on θ and obtain Theorem 1.7.Suppose θ, λ, p are positive constants.If 0 < p ≤ 1, then system (1.4) has no positive non-constant solution.
The rest of this paper is organized as follows.In Section 2, we will prove Theorems 1.2 and 1.3 by using Lyapunov function method.In Section 3, we will prove Theorems 1.6 and 1.7 by a novel approach.Finally, our conclusions are given in Section 4.

Proofs of Theorems 1.and 1.3
We begin with the following lemma.Proof.As for the conclusion (a), it it clear to see that the case where m ≥ bγ is trivial.We now suppose m < bγ.For the case, if m(m and then taking the square on the two sides of (2.1) yields m > abγ a+bγ .Note that the above reasoning process is also inverse since m > abγ a+bγ implies Thus the conclusion (a) is valid.
As for the conclusion (b), a simple calculation gives Solving the latter gives m > M 2 .This completes the proof of the lemma.
Proof of Theorem 1.2.Let (u, v) be a positive solution of system (1.1).Adapting the Lyapunov function in [2,3], we define (2.2) and It follows that We now assume that m > max{M 1 , M 2 }.Then, 1 − bγu * m(m+u * ) > 0 by Lemma 2.1(a), and am a+2m < u * by Lemma 2.1(b), so there exists a constant ε > 0 such that On the other hand, from (1.1), we have therefore, W ′ (t) ≤ 0 for all t ≥ T, and equality holds if and only if (u, v) = E * , so E * is globally attractive.Since m > M 1 (i.e., m(a + bγ) > abγ), E * is locally asymptotically stable according to Theorem 1.1(a), so is globally asymptotically stable.The proof of the theorem is complete.
We now are ready to show Theorem 1.3, whose proof is based on the following lemma.Proof.Like in [11], we set φ = v u .Then a simple calculation gives therefore, from the comparison principle, for any 0 < ε ≪ 1 there exists some constant By a similar argument to (2.5), there exists some constant Combining (2.6), (2.7) and (1.1) 1 , we obtain . Then, letting ε → 0 gives the desired result.
Proof of Theorem 1.3.We adapt the same Lyapunov function as that in (2.2).From Lemma 2.2 and (1.2), there exist some constants 0 < ε ≪ 1 and T ≫ 1 such that and (2.9) That is, Combining this and (2.3) with d 1 = d 2 yields W ′ (t) ≤ 0 for all t ≥ T, and equality holds if and only if (u, v) = E * , so E * is globally attractive.Since m > M 1 , E * is locally asymptotically stable according to Theorem 1.1(a), so is globally asymptotically stable.The proof is complete.
3 Proofs of Theorems 1.6 and 1.7 We first show Theorem 1.6.
Proof of Theorem 1.6.Assume that (u, v) is a positive solution of system (1.3).Multiplying (1.1) 1 by [(a − u)(m + u) − v] and integrating by parts over Ω, we have Multiplying (1.1) 2 by (u − v bγ ) and integrating over Ω, we obtain We first multiply (3.2) by d 1 /d 2 , and then add the resulting equation and (3.1) to get Since m ≥ a, the first term on the left hand side of (3.3) is non-positive and hence u and v must be constants.The proof is complete.
Proof of Theorem 1.7.Assume that (u, v) is a positive solution of system (1.4).Multiplying (1.4) by ( 1 u − v p ) and (uv p−1 − 1), respectively, and integrating by parts over Ω, we have and We first multiply (3.5) by pθ, and then add the resulting equation and (3.4) to obtain Consequently, u and v must be constants if p ∈ (0, 1].The proof is complete. Remark 3.1.In [14, Remark 2.1], the authors pointed out that it is difficult to expect the bifurcation of (1.4) near (u, v) = (1, 1) if 0 < p ≤ 1 since the constant positive solution (u, v) = (1, 1) is uniformly asymptotically stable for the corresponding reaction-diffusion system to (1.4) for the case.Our Theorem 1.7 shows that no bifurcation will happen for system (1.4) provided that 0 < p ≤ 1.

Conclusions
In this paper, we prove some new global stability results.In particular, the works by Chen and Shi [1] and Duan, Niu and Wei [2], mentioned above, have been improved.In addition, we derive a non-existence result of the positive non-constant steady-states for system (1.1) by using a different approach from those in literature.By virtue of the approach, we also obtain a complete understanding of the steady-state Sel'kov model for the case when 0 < p ≤ 1.
In summary, we have, for all ε ∈ (0, 1), Then, by a standard compactness argument, one can obtain a positive global solution of system (1.1).This proof is complete.