Global stability of a predator – prey model with Beddington – DeAngelis and Tanner functional response

In this paper, we study the global stability of a predator–prey system with Beddington–DeAngelis and Tanner functional response. By using the iteration method and comparison principle, we prove the global asymptotic stability of the unique positive equilibrium solution.


Introduction
The purpose of this paper is to consider the following predator-prey system with Beddington-DeAngelis and Tanner functional response where u(x, t) and v(x, t) are the densities of prey and predator, respectively, Ω is a bounded domain with smooth boundary ∂Ω, a, δ and β are positive constants.In this paper we assume that the two diffusion coefficients d 1 and d 2 are the diffusion coefficients corresponding to u and v, respectively, and are positive and equal, but not necessary constants.We use d to represent the common value.The admissible initial data u 0 (x) and v 0 (x) are continuous functions on Ω.
The functional response uv a+u+v was introduced by Beddington [1] and DeAngelis [3].They proposed the following predator-prey model with Beddington-DeAngelis functional response x = x(r − θx) − Exy a+bx+cy , y = −dy + βxy a+bx+cy . (1.2) Huang et al. [9,10] proposed a class of virus dynamics model with Beddington-DeAngelis functional response.Liu and Kong [11] studied the dynamics of a predator-prey system with Beddington-DeAngelis functional response and delays.
Besides the Beddington-DeAngelis functional responses mentioned above, there are several other well-known functional responses, such as Holling type (I, II, III, IV), Monod-Haldane type and Hassel-Verley type functional responses etc.Some authors studied and raised some open questions for structured predator-prey models with different types of functional responses.Especially, in [15], Peng and Wang considered the steady states of a diffusive Holling-Tanner prey-predador model They discussed the existence and non-existence of positive non-constant steady solutions for (1.3), and proved that (1.3) has no positive non-constant steady solution under a certain condition.In the another paper [16], by the construction of a Lyapunov function and a standard linearization procedure, they studied the stability of diffusive predator-prey system of Holling-Tanner type (1.3).Chen and Shi [2] concentrated on the steady states of (1.3).They used the comparison principle and defined iteration sequences to prove the global stability for the constant positive equilibrium.Their result improves the earlier one given in [16] which was established by Lyapunov method.We also note here that the (non-spatial) kinetic equation of system (1.3) was first proposed by Tanner [20] and May [14], see also [12,13].
Recently, Qi and Zhu [17] studied the global stability of diffusive predator-prey system (1.3).Indeed, in [17], they established improved global asymptotic stability of the unique positive equilibrium solution.For more detailed biological implications of the model, besides the references mentioned above, one can see [4-8, 18, 19].
Motivated by the previous works [17], in this paper by incorporating the diffusion and ratio-dependent Beddington-DeAngelis functional response into system (1.3), we study the stability of the positive equilibrium solution of (1.1).
A direct computation gives that (1.1) has a unique positive equilibrium (u * , v * ), where

Proof of the main result
Let w = v u , then we have Therefore the equation satisfied by w is Proposition 2.2.Suppose δ < 1 and ε 1 > 0 small.There exists a sufficiently large constant T > 0 such that the solution u of (1.1) for x ∈ Ω and t ≥ T, where and u 1 ≡ 1.
Proof.Since v > 0, a direct computation gives By a simple comparison argument and the well established fact that any positive solution of converges to the asymptotic stable equilibrium 1 as t → ∞, we get that for all ε 1 > 0, there exists a constant t 1 > 0, such that It is clear that the following equation about W(t) has three solutions: It is clear that W 1 (t) the unique asymptotically stable positive equilibrium point of (2.3), and W 0 (t) = 0 is unstable.Thus, all positive solutions W(t) of (2.3) converge to the unique positive asymptotically stable equilibrium point W 1 (t), since the trajectories of (2.3) cannot cross the x-axis.By a simple comparison argument, we get that there exists a positive constant for all x ∈ Ω and t ≥ t 2 .Consequently, v ≤ w 1 (ε 1 )u, and ) for all x ∈ Ω and t ≥ t 2 .The equation which is a stable equilibrium point of the ODE Thus, all positive solution of (2.6) converge to û, which implies that there exists for all x ∈ Ω and t ≥ t 3 .On the other hand, by using the second equation of (1.1), we get for all x ∈ Ω and t ≥ t 3 .Thus, there exists a constant t 4 > t 3 such that for all x ∈ Ω and t ≥ t 4 .Substituting v ≥ v 1 (ε 1 ) into the first equation of (1.1), we get (2.9) By comparison principle then yields there exists t 5 > t 4 such that if t ≥ t 5 , (2.10) Simple computation using (2.2), (2.4), (2.5) and (2.7)-(2.10)shows the expression of u 2 (ε 1 ) and that of u 1 (ε 1 ) and w 1 (ε 1 ) are valid.This completes the proof.
By repeating the above procedure, for any positive integer n, there exists T sufficiently large such that when t ≥ T, uniformly in Ω, where ) .
When ε 1 = 0, we have Then, we can obtain that {u n } is a decreasing sequence by induction.Similarly, since and where δ < 1, we obtain that {w n } is a decreasing sequence and {u n } is an increasing sequence.Thus, under the assumption of Theorem 2.1, we have (2.12) and the same to v n (ε 1 ), v n , v n (ε 1 ), v n and v * .Furthermore, there exists t M 1 such that when t ≥ t M , u N (ε 1 ) ≤ u(x, t) ≤ u N (ε 1 ) in Ω.