Sufficient and Necessary Conditions for the Permanence of a Discrete Model with Beddington-DeAngelis Functional Response

Abstract

We give a sufficient and necessary condition for the permanence of a discrete model with Beddington-DeAngelis functional response with the form $x(n+\mathrm{1})$ = $x(n)\mathrm{\text{e}}\mathrm{\text{x}}\mathrm{\text{p}}\{a(n)-b(n)x(n)-c(n)y(n)$/$(\alpha (n)+\beta (n)x(n)+\gamma (n)y(n))\},\mathrm{}y(n+\mathrm{1})=y(n)\mathrm{\text{e}}\mathrm{\text{x}}\mathrm{\text{p}}\{-d(n)+f(n)x(n)/(\alpha (n)+\beta (n)x(n)+\gamma (n)y(n))\},$ where $a(n),$ $b(n),$ $c(n),$ $d(n),$ $f(n),$ $\alpha (n),$ $\beta (n)$, and $\gamma (n)$ are periodic sequences with the common period $\omega ;$ $b(n)$ is nonnegative; $c(n),$ $d(n),$ $f(n),$ $\alpha (n),$ $\beta (n)$, and $\gamma (n)$ are positive. It is because of the difference between the comparison theorem for discrete system and its corresponding continuous system that an additional condition should be considered. In addition, through some analysis on the limit case of this system, we find that the sequence $\alpha (n)$ has great influence on the permanence.