Global Asymptotic Stability of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

We study the predator-prey model proposed by Aziz-Alaoui and Okiye Appl. Math. Lett. 16 2003 1069–1075 First, the structure of equilibria and their linearized stability is investigated. Then, we provide two sufficient conditions on the global asymptotic stability of a positive equilibrium by employing the Fluctuation Lemma and Lyapunov direct method, respectively. The obtained results not only improve but also supplement existing ones.


Introduction
One of the important interactions among species is the predator-prey relationship and it has been extensively studied because of its universal existence.There are many factors affecting the dynamics of predator-prey models.One of the familiar factors is the functional response, referring to the change in the density of prey attached per unit time per predator as the prey density changes.In the classical Lotka-Volterra model, the functional response is linear, which is valid first-order approximations of more general interaction.To build more realistic models, Holling 1 suggested three different kinds of functional responses, and Leslie and Gower 2 introduced the so-called Leslie-Gower functional response.
Recently, Aziz-Alaoui and Daher Okiye 3 proposed and studied the following predator-prey model with modified Leslie-Gower and Holling-type II schemes,

1.1
Here, all the parameters are positive, and we refer to Aziz-Alaoui and Daher Okiye 3 for their biological meanings.System 1.1 can be considered as a representation of an insect pest-spider food chain, nature abounds in systems which exemplify this model; see 3 .Since then, system 1.1 and its nonautonomous versions have been studied by incorporating delay, impulses, harvesting, and so on see, e.g., 4-11 .In spite of this extensive study, the dynamics of 1.1 is not fully understood and some existing results are not true.For example, the main result Theorem 6 on global stability of a positive equilibrium of Aziz-Alaoui and Daher Okiye 3 is not true as the condition i and condition iii cannot hold simultaneously.In fact, it follows from condition i , 1/4a 2 b 1 a 2 r 1 r 1 4 which is impossible.One purpose of this paper is to establish several sufficient conditions on the global asymptotic stability of a positive equilibrium.
Let Ω 0 { x, y : x ≥ 0, y ≥ 0}.As a result of biological meaning, we only consider solutions x t , y t of 1.1 with x 0 , y 0 ∈ Ω 0 .Moreover, solutions x t , y t of 1.1 with x 0 , y 0 ∈ Ω 0 are called positive solutions.An equilibrium E * x * , y * of 1.1 is called globally asymptotically stable if x t → x * and y t → y * as t → ∞ for any positive solution x t , y t of 1.1 .System 1.1 is permanent if there exists 0 < α < β such that, for any positive solution x t , y t of 1.1 , The remaining part of this paper is organized as follows.In Section 2, we discuss the structure of nonnegative equilibria to 1.1 and their linearized stability.This has not been done yet, and the results will motivate us to study global asymptotic stability of 1.1 in Section 3. The obtained results not only improve but also supplement existing ones.

Nonnegative Equilibria and Their Linearized Stability
The Jacobian matrix of 1.1 is An equilibrium E of 1.1 is linearly stable if the real parts of both eigenvalues of J E are negative and therefore a sufficient condition for stability is Obviously, 1.1 has three boundary equilibria, E 0 0, 0 , E 1 r 1 /b 1 , 0 , and E 2 0, r 2 k 2 /a 2 , whose Jacobian matrices are respectively.As a direct consequence of 2.2 , we have the following result.
Besides the three boundary equilibria, 1.1 may have componentwise positive equilibria.Suppose that E x, y is such an equilibrium.Then,

2.4
One can easily see that x satisfies 5 can have at most two positive solutions, and hence 1.1 can have at most two positive equilibria.Precisely, we have the following three cases.
Case 1. Suppose one of the following conditions holds.
ii a 1 r 2 k 2 a 2 r 1 k 1 and B < 0.
Then, 1.1 has a unique positive equilibrium E 3,1 For a positive equilibrium E x, y , J E can be simplified to by using 2.4 .By simple computation, tr Then, one can easily see that det J E 3,1 > 0 for Case 1 i -ii , det J E 3,1 0 for Case 1 iii , det J E 3, > 0, and det J E 3,− < 0. Therefore, we obtain the following.

Proposition 2.2. i The positive equilibrium
ii The positive equilibrium E 3,− is unstable, while the positive equilibrium Remark 2.3.In 3, 7, 8 , only existence of the positive equilibrium of 1.1 for Case 1 i was considered, which is stable if either a r 1 ≤ r 2 and for a positive solution x t , y t of 1.1 .Therefore, system 1.1 is permanent if H1 holds.
With the help of these bounds, it was shown that E 2 is globally asymptotically stable if r 1 k 1 K < a 1 N holds see 5 .
In the coming section, we present two results on the global asymptotic stability of a positive equilibrium, which not only supplement Theorem 7 of Nindjin et al. 5 but also improve it by including more situations.

Global Asymptotic Stability of a Positive Equilibrium
The first result is established by employing the Fluctuation Lemma, and we refer to 12-16 for details.Theorem 3.1.In addition to H1 , further suppose that where M is defined in 2.8 .Then, system 1.1 has a unique positive equilibrium which is globally asymptotically stable.
Proof.Obviously, H1 implies a 1 r 2 k 2 < a 2 r 1 k 1 , that is, condition i of Case 1 holds.Thus, 1.1 has a unique positive equilibrium.Let x t , y t be any positive solution of 1.1 .By the results at the end of Section 2, We claim x x. Otherwise, x > x.According to the Fluctuation lemma, there exist sequences Letting n → ∞, we obtain that 0 ≤ r 2 − a 2 y/ x k 2 y and 0 ≥ r 2 − a 2 y/ x k 2 y.Hence, Similar arguments as above also produce Second, from the first equation of 1.1 , 3.5 Similarly, one can show that

3.6
Multiplying 3.5 by −1 and adding it to 3.6 , we have Due to x > x, one gets a 2 b 1 x x a 2 b 1 k 1 −a 2 r 1 −a 1 r 2 ≤ 0 which contradicts H2 .Therefore, x x, and the claim is proved.
The claim implies that lim t → ∞ x t exists and we denote it by x * .Then, it follows from 3.2 and 3.3 that lim t → ∞ y t exists and lim t → ∞ y t y 0. Then, one can see that x * , y * satisfies 2.4 , that is, x * , y * is a positive equilibrium of 1.1 .This completes the proof as the positive equilibrium is unique.Theorem 3.2.Suppose that 1.1 has a unique positive equilibrium E * x * , y * .Further assume that where L is defined in 2.7 .Then, E * is globally asymptotically stable.
Proof.Let x t , y t be any positive solution of 1.1 .From H3 , we can choose an ε > 0 such that Moreover, it follows from 2.7 that there exists T > 0 such that 0 < y t ≤ L ε for t ≥ T.

3.9
According to the proof of Theorem 6 in 3 , let Then, by the positivity of x, 3.8 , and 3.9 , Therefore, E * x * , y * is globally asymptotically stable, and this completes the proof.
V x, y x * k 1 x − x * − x * ln x x * a 1 x * k 2 a 2 y − y * − y * ln y y * .3.10