Stability for a New Discrete Ratio-Dependent Predator–Prey System

The stability of a new two-species discrete ratio-dependent predator–prey system is considered. By using the linearization method, we obtain some sufficient conditions for the local stability of the positive equilibria. We also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper (Chen and Zhou in J Math Anal Appl 27:7358–7366, 2003) has done. The method given in this paper is new and very resultful comparing with articles (Damgaard in J Theor Biol 227:197–203, 2004; Edmunds in Theor Popul Biol 72:379–388, 2007; Fan and Wang in Math Comput Model 35:951–961, 2002; Muroya in J Math Anal Appl 330:24–33, 2007; Huo and Li in Appl Math Comput 153:337–351, 2004; Liao et al. in Appl Math Comput 190:500–509, 2007) and it can also be applied to study other global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present two open questions.


Introduction
In recent years, the dynamical behaviors of the discrete-time predator-prey systems have been widely investigated. Many important and interesting results can be found in many articles, such as in  and the references cited therein. Particularly, the discrete two-species predator-prey systems with ratio-dependent functional responses were studied in [10][11][12][13][14][15][16][17]23,25]. What interested them are the dynamical behaviors, such as, the study for the local and global stability of the equilibria, the persistence, permanence and extinction of species, the existence of positive periodic solutions and positive almost periodic solutions, the bifurcation and chaos phenomenon, etc.. Recently, Chen and Zhou [17] discussed the global stability for a nonautonomous two species discrete competition system. However, the conditions of their results in [17] is strong and complicated. Therefore, as an extension and improvement, we discuss in the present paper the following discrete-time two-species competition system: (1.1) where x(k) and y(k) represent the sizes or the densities of species x and y at kth generation, respectively. Parameters r i , K i and μ i (i = 1, 2) are positive constants and represent the intrinsic growth rates, the carrying capacities, and the competition coefficients of species x and y, respectively. m and n are arbitrary positive integer.
In this paper, we will introduce a new method to discuss the global asymptotic stability of system (1.1). The main results of this paper is to establish the criteria on the existence and local asymptotic stability of equilibria for system (1.1) by using the linear approximation method, and obtain some new sufficient conditions on the global stability of the positive equilibrium for system (1.1) by using the iterative scheme method and the comparison principle of difference equations.

Preliminary Lemmas
Let (x(k), y(k)) be any solution of system (1.1) satisfying the initial value x(0) > 0 and y(0) > 0 considered the biological background of system (1.1). It is clear that any solution (x(k), y(k)) of system (1.1) is defined on Z + and always remains positive, where Z + denotes the set of all nonnegative integers.
What interested us is the positive equilibrium of system (1.1). By a simple computation, we directly obtain the following results.
Further, we need the following lemma, which can be easily proved by the relations between roots and coefficients of a quadratic equation.
Proof Conclusion (1) can be proved using Theorem 2.8 in [4], so we omit it. Note that the function x exp(α − βx n ) has a unique maximum in x = n 1 βn , then then conclusion (2) is proved. This ends the proof. Lemma 2.5 (see [23]) Assume that functions f, g : x)( f (n, x) ≥ g(n, x)) for n ∈ Z + and x ∈ [0, ∞), g(n, x) is nondecreasing for x > 0. Let sequences {x(n)} and {u(n)} be the nonnegative solutions of the following difference equations

Local Stability
In this section, we use the eigenvalues of the variational matrix of system (1.1) at the equilibria E + (x 0 , y 0 ) to study its local stability.
The corresponding characteristic equation of J (E + ) can be written as Then we have the following result.
is a sink if one of the following conditions holds: .
is a source if one of the following conditions holds:

non-hyperbolic if one of the following conditions holds:
(a) r 1 = t 2 and r 2 = t 1 ; is a saddle if one of the following conditions holds: (a) r 2 < t 1 and r 1 > t 2 ; Proof Here, we only prove conclusion (1) of Theorem 3.1. The others can also be proved by the same way. From (3.1), we have If Hence, if condition (a) or (b) of conclusion (1) of Theorem 3.1 holds, then we have F(−1) > 0 and q < 1. From Lemma 2.2, we can obtain E + (x 0 , y 0 ) in system (1.1) is a sink.
On the other hand, if Since r 2 > t 4 , a similar argument as in above we have q < 1 if r 1 < t 3 . Hence, if condition (c) of conclusion (1) of Theorem 3.1 holds, then we have F(−1) > 0 and q < 1. From Lemma 2.2, we obtain E + (x 0 , y 0 ) in system (1.1) is also a sink. This completes the proof.

Global Stability
In this section, we will use the method of iteration scheme and the comparison principle of difference equations to study the global stability of the positive equilibrium of system (1.1).
Proof Assume that (x(k), y(k)) is any a solution of system (1.1) with initial value x(0) > 0 and y(0) > 0. Let In the following, we will prove that U 1 = V 1 = x 0 and U 2 = V 2 = y 0 . From the first equation of system (1.1) we obtain

Consider the auxiliary equation
Hence, for any ε > 0 small enough, there exists a N 1 > 2 such that if n ≥ N 1 , then From the second equation of system (1.1) we have By the same way, we can obtain Hence, for any ε > 0 small enough, there exists a N 2 > N 1 such that if k ≥ N 2 , then y(k) ≤ M y 1 + ε. From the first equations of system (1.1) again, we further have Consider the auxiliary equation From the arbitrariness of ε, we can let ε < From the arbitrariness of ε > 0, we have Hence, for ε > 0 small enough, there exists a N 3 > N 2 such that if k ≥ N 3 , then x(k) ≥ N x 1 − ε. From the second equations of system (1.1) we further have By the same way, we can obtain From the arbitrariness of ε > 0, we get Hence, for ε > 0 small enough, there exists a N 4 ≥ N 3 such that if k ≥ N 4 , then y(k) ≥ N y 1 − ε > 0. Further, from the first equations of system (1.1) we have Using the similar argument as in above, we can get From the arbitrariness of ε > 0, we claim that Hence, for any ε > 0 small enough, there exists a N 5 ≥ N 4 such that if k ≥ N 5 , then x(k) ≤ M x 2 + ε. From the second equations of system (1.1) we further obtain Similarly to the above argument, we can obtain From the arbitrariness of ε > 0, we obtain U 2 ≤ M y 2 , where Hence, for ε > 0 small enough, there exists a N 6 > N 5 such that if k ≥ N 6 , y(k) ≤ M y 2 + ε.
Further, from the first equations of system (1.1) we obtain Using a similar argument, we again can obtain From the arbitrariness of ε > 0, we get that Hence, for any ε > 0 small enough, there exists a N 7 > N 6 such that if k ≥ N 7 , x(k) ≥ N x 2 − ε > 0. From the second equations of system (1.1) we further have Using a similar discussion, we again can obtain From the arbitrariness of ε > 0, we claim that Repeating the above process, we can finally obtain four sequences and Clearly, we have for any integer k > 0 For k(k ≥ 2),we assume that M x k ≤ M x k−1 and N x k ≥ N x k−1 , then we further have and From (4.6) and (4.7) we have } are monotonically increasing. Therefore, by the criterion of monotone bounded, we have proved that every one of this four sequences has a limit.

From (4.3) and (4.4), we can obtain
Taking k → ∞ in both sides of the above two equations, respectively, then we have By the same way, we also can obtain It follows from (4.5) that Therefore, we finally have This shows that equilibrium E + (x 0 , y 0 ) of system (1.1) is globally attractive. From Theorem 3.1, we can obtain that equilibrium E + (x 0 , y 0 ) of system (1.1) is locally asymptotically stable. Therefore, we finally obtain that E + (x 0 , y 0 ) is globally asymptotically stable. This completes the proof.

Remark 1
The main results obtained in the present paper is for any positive integer m and n, which generalizes what paper [7] has obtained. The method given in this paper is new and very resultful comparing with articles [6,9,10,14,16,19,22] on the study of global stability for discrete predator-prey systems. Note that our conditions is more better than the conditions of theorem 3 in paper [7]. For example, the conditions of theorem 3 in paper [7] has been obtained as follows: .

Remark 3 Another important and
interesting open question is whether we can also obtain the same inequality (4.5) but do not apply the comparison principle. If it is possible, then the conditions on the global stability of positive equilibrium of system (1.1) may be extended.

Remark 4
The condition in Theorem 3.1 is to guarantee the existence of positive equilibrium E + (x 0 , y 0 ) of system (1.1), and the possibility of how the two species can coexist. If the conditions in conclusion (1) of Theorem 3.1 do not hold, then the positive equilibrium of system (1.1) will be unstable.

Remark 5
The approach can also be devoted to studying the global asymptotic stability of positive equilibrium for the other general multiple species discrete population systems. We would like to do some valuable research about the subject.