GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR-PREY SYSTEM

The dynamical behaviors of a two-species discrete ratio-dependent predator-prey system are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, 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 [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator-prey system, Indian J. Pure Appl. Math. 42(1) (2011) 1-26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator-prey system with delays, Appl. Math. Comput. 153 (2004) 337-351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin-Ayala competition predator-prey discrete system, Appl. Math. Comput. 190 (2007) 500-509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.