Dynamics of a modified Leslie–Gower predator–prey model with Holling-type II schemes and a prey refuge

We propose a modified Leslie–Gower predator–prey model with Holling-type II schemes and a prey refuge. The structure of equilibria and their linearized stability is investigated. By using the iterative technique and further precise analysis, sufficient conditions on the global attractivity of a positive equilibrium are obtained. Our results not only supplement but also improve some existing ones. Numerical simulations show the feasibility of our results.

As was pointed out by Aziz-Alaoui and Daher (2003), in the case of severe scarcity, y can switch over to other populations but its growth will be limited by the fact that its most favorite food x is not available in abundance. In order to solve such deficiency in system (1), Aziz-Alaoui and Daher (2003) proposed and studied the following predatorprey model with modified Leslie-Gower and Holling-type II schemes: where r 1 , b 1 , r 2 , a 2 have the samemeaning as in system (1). a 1 is the maximum value which per capita reduction rate of x can attain; k 1 and k 2 measure the extent to which environment provides protection to prey x and to predator y respectively. They obtained the boundedness and global stability of positive equilibrium of system (1). Since then, many scholars considered system (2) and its non-autonomous versions by incorporating delay, impulses, harvesting, stochastic perturbation and so on (see, for example, Yu 2012; Nindjin et al. 2006;Yafia et al. 2007Yafia et al. , 2008Nindjin and Aziz-Alaoui 2008;Gakkhar and Singh 2006;Guo and Song 2008;Song and Li 2008;Zhu and Wang 2011;Liu and Wang 2013;Kar and Ghorai 2011;Huo et al. 2011;Li et al. 2012;Gupta and Chandra 2013;Ji et al. 2009Ji et al. , 2011Yu 2014;Yu and Chen 2014;Yue 2015). In particular, Yu (2012) studied the structure, linearized stability and the global asymptotic stability of equilibria of (2) and obtained the following result (see Theorem 3.1 in Yu 2012): Theorem 1 Assume that hold, where M = r 1 k 1 −a 1 L b 1 k 1 and L = r 1 r 2 +b 1 r 2 k 2 a 2 b 1 , then system (2) has a unique positive equilibrium which is globally attractive.
As was pointed out by Kar (2005), mite predator-prey interactions often exhibit spatial refugia which afford the prey some degree of protection from predation and reduce the chance of extinction due to predation. In Kar (2005), Tapan Kumar Kar had considered a predator-prey model with Holling type II response function and a prey refuge. The author obtained conditions on persistent criteria and stability of the equilibria and limit cycle for the system. For more works on this direction, one could refer to Kar (2005), Srinivasu and Gayatri (2005), Ko and Ryu (2006), Huang et al. (2006), Kar (2006), González-Olivares and Ramos-Jiliberto (2003), Ma et al. (2009), Chen et al. (2009, 2012, Ji and Wu (2010), Tao et al. (2011) and the references cited therein.
Theorem 2 shows that lim t→∞ x(t) = x * , lim t→∞ y(t) = y * . Notice that x * and y * are only dependent with the coefficients of system (3), and independent of the solution of system (3). Thus we can get the following result: Corollary 1 Suppose that C 3 holds, then system (2) is permanent.
When m = 0 that is there is no prey refuge, (3) becomes to (2) and C 3 becomes to C 1 , so as a direct corollary of Theorem 2, we have: Corollary 2 Suppose that C 1 holds, then system (2) has a unique positive equilibrium which is globally attractive.
Comparing with Theorem 1, it follows from Corollary 2 that C 2 is superfluous, so our results improve the main results in Yu (2012). Moreover, when consider the case of no alternate prey, so k 2 = 0 (this is often called the Holling-Tanner model), by the similar proof of Theorem 2, we can obtain: Corollary 3 Suppose that holds, then system (3) with k 2 = 0 has a unique positive equilibrium (x * , y * ) which is globally attractive. (3) The remaining part of this paper is organized as follows. In section "Nonnegative equilibria and their linearized stability", we discuss the structure of nonnegative equilibria to (3) and their linearized stability. We prove the main result (i.e. Theorem 2) of this paper in section "Global attractivity of a positive equilibrium". Then, in section "Examples and numeric simulations", a suitable example together with its numeric simulations is given to illustrate the feasibility of the main results. We end this paper by a briefly discussion.

Nonnegative equilibria and their linearized stability
As for the existence of positive equilibria and linearized stability of equilibria, similar to the discussion in Yu (2012), we have the following results: Case 1. Suppose one of the following conditions holds.

Proposition 1 (i) Both
When m = 0 that is there is no prey refuge, Proposition 1 becomes to Propositions 2.1 and 2.2 in Yu (2012). Thus our results supplement the exist ones. In the coming section, we will prove the main result (i.e. Theorem 2) of this paper.

Global attractivity of a positive equilibrium
In this section, we first introduce several lemmas which will be useful in proving the main result (i.e. Theorem 2) of this paper.
Proof of Theorem 2 Let (x(t), y(t)) T be any positive solution of (3). From condition (C 3 ), we can choose a small enough ε > 0 such that The first equation of (3) yields By applying Lemma 1 to (5) leads to Hence, for above ε > 0, there exists a T 1 > 0 such that (6) together with the second equation of (3) leads to From (7), according to Lemma 1, we can obtain Thus, for above ε, there exists a T 2 ≥ T 1 , such that (4) 2 , for all t ≥ T 2 .
(8) together with the first equation of (3) leads to According to (4), we can obtain Therefore, by Lemma 1 and (9), we have Hence, for above ε, there exists a T 3 ≥ T 2 , such that From (11) and the second equation of system (3), we know that for t ≥ T 3 , Applying Lemma 1 to (12) leads to That is, for above ε, there exists a T 4 > T 3 such that From (6), (8), (11) and (13), for t ≥ T 4 , we have x, for all t ≥ T 2 . (10) 1 , for all t ≥ T 3 .
Let us suppose that for n, By direct computation, one can obtain Therefore, we have that Hence, the limits of M (n) i and m (n) i , i = 1, 2, n = 1, 2, . . . exist. Denote that Hence x ≥ x, y ≥ y. Letting n → +∞ in (26), we immediately It follows from (28) that Multiplying the second equality of (29) by −1 and adding it to the first equality of (29), we have We claim x = x. Otherwise, x � = x and (28)

Conclusion
In this paper, we consider a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge. The structure of equilibria and their linearized stability is investigated. Morever, by using the iterative technique and further precise analysis, sufficient conditions on the global attractivity of a positive equilibrium are obtained. When m = 0 that is there is no prey refuge, (3) we discussed reduces to (2) which was studied by Yu (2012). Yu (2012) have provided a sufficient condition on the global asymptotic stability of a positive equilibrium by employing the Fluctuation Lemma and obtained Theorem 1. By comparing Theorems 1 with Corollary 2, we find that the condition C 2 in Theorem 1 is redundant. Thus our results not only supplement but also improve some existing ones. The numerical simulation of system (33) verify our main results. It follows from Theorem 2 and condition C 3 that increasing the amount of refuge can ensure the coexistence and attractivity of the two species more easily. This is rational, since the existence of alternate prey can prevent the predator from extinction and increasing the amount of refuge could protect more prey from predation and become permanent. Note that for the diffusion/PDE model where refuge can be spatial, whether refuge can change global attractivity of the interior equilibrium? This is a further problem, which can be studied in the future. The author declare that he has no competing interests.