Abstract
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.
Highlights
In recent years, the dynamical behaviors of the discrete-time predator–prey systems have been widely investigated
As an extension and improvement, we discuss in the present paper the following discrete-time two-species competition system:
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
Summary
The dynamical behaviors of the discrete-time predator–prey systems have been widely investigated. 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. Chen and Zhou [17] discussed the global stability for a nonautonomous two species discrete competition system. 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
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