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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.
The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Leslie (1948, 1958) introduced the following two species Leslie–Gower predator–prey model:
where x(t), y(t) stand for the population (the density) of the prey and the predator at time t, respectively. The parameters r1 and r2 are the intrinsic growth rates of the prey and the predator, respectively. b1 measures the strength of competition among individuals of species x. The value
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 predator–prey model with modified Leslie–Gower and Holling-type II schemes:
where r1, b1, r2, a2 have the samemeaning as in system (1). a1 is the maximum value which per capita reduction rate of x can attain; k1 and k2 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. 2007, 2008; Nindjin 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; Liu et al. 2013; Gupta and Chandra 2013; Ji et al. 2009, 2011; Yu 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):
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, 2010, 2012), Ji and Wu (2010), Tao et al. (2011) and the references cited therein.
Although many authors have considered the dynamic behaviors of the modified Leslie–Gower model (Yu 2012; Nindjin et al. 2006; Yafia et al. 2007, 2008; Nindjin 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; Liu et al. 2013; Gupta and Chandra 2013; Ji et al. 2009, 2011; Yu 2014; Yu and Chen 2014; Yue 2015) and predator–prey with a prey refuge (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, 2010, 2012; Ji and Wu 2010; Tao et al. 2011), as far as we know, there are almost no literatures discussing the modified Leslie–Gower model with a prey refuge. Stimulated by the works of 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, 2010, 2012), Ji and Wu (2010), Tao et al. (2011), we will extend model (2) by incorporating a refuge protecting mx of the prey, where m ∈ [0, 1) is constant. This leaves (1 - m)x of the prey available to the predator, and modifying system (2) accordingly to the system:
holds, then system (3) has a unique positive equilibrium(x∗, y∗)which is globally attractive.
Theorem 2 shows that limt→∞x(t) = x∗, limt→∞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:
Suppose thatC3holds, then system (2) is permanent.
Suppose thatC1holds, then system (2) has a unique positive equilibrium which is globally attractive.
Comparing with Theorem 1, it follows from Corollary 2 that C2 is superfluous, so our results improve the main results in Yu (2012). Moreover, when consider the case of no alternate prey, so k2 = 0 (this is often called the Holling-Tanner model), by the similar proof of Theorem 2, we can obtain:
holds, then system (3) withk2 = 0has a unique positive equilibrium(x∗, y∗)which is globally attractive.
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.
Obviously, (3) has three boundary equilibria, E0 = (0, 0),
Case 1. Suppose one of the following conditions holds.
Then (3) has a unique positive equilibrium E3,1 = (x3,1, y3,1) with
Case 2. If
Case 3. If no condition in Case 1 or Case 2 holds, then (3) has no positive equilibrium.
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.
In this section, we first introduce several lemmas which will be useful in proving the main result (i.e. Theorem 2) of this paper.
(Chen et al. 2007) Ifa > 0, b > 0and
Ifa > 0, b > 0and
Now, we prove the main result of this paper.
Let (x(t),y(t))T be any positive solution of (3). From condition (C3), we can choose a small enough ε > 0 such that
The first equation of (3) yields
Hence, for above ε > 0, there exists a T1 > 0 such that
Thus, for above ε, there exists a T2 ≥ T1, such that
According to (4), we can obtain
Hence, for above ε, there exists a T3 ≥ T2, such that
That is, for above ε, there exists a T4 > T3 such that
Applying Lemma 1 to the above inequality leads to
Thus, for above ε, there exists a T6 ≥ T5, such that
Thus, similarly to the above analysis, for above ε, there exists a T7 ≥ T6, such that
Thus, similarly to the above analysis, for above ε, there exists a T8 ≥ T7, such that
Repeating the above procedure, we get four sequences
Now, We go to show that the sequences
Let us suppose that for n,
By direct computation, one can obtain
Therefore, we have that
Hence, the limits of
It follows from (28) that
Simplifying (31), one can get
where D = a2(a1(1-m)2r1r2 + a1(1 - m)b1r2k2 - a2b1r1k1) + a1r2(1 - m)(a1r2(1-m)2 - a2b1k1). (H1) shows that a1(1-m)2r1r2 + a1(1 - m)b1r2k2 - a2b1r1k1 < 0 and a1r2(1-m)2 - a2b1k1 < 0. Hence, D < 0, that is, Eq. (31) does not have two positive solutions. So
and this completes the proof. □
Consider the following example:
In this case, we have r1 = 11, b1 = 5, a1 = 4, m = 0.4, k1 = 6.5, r2 = 8, a2 = 2, k2 = 2 and B = a1r2(1-m)2 - a2r1(1 - m) + a2b1k1 = 63.32, Δ = B2 - 4(1 - m)a2b1[(1 - m)a1r2k2 - a2r1k1] = 6519.8, so
By simple computation, we also have
Thus, conditions in Theorem 2 are satisfied, hence, system (33) has a unique positive equilibrium E∗ = (x∗, y∗) which is globally attractive. Numerical simulation also confirms our result (see Fig. 1).
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 C2 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 C3 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 would like to thank the anonymous referees and the editor for their constructive suggestions on improving the presentation of the paper. Also, this research was supported by Anhui Province College Excellent Young Talents Support Plan Key Projects (No. gxyqZD2016240).
The author declare that he has no competing interests.