Stability analysis of a population model with piecewise constant arguments

In this paper, we study the behavior of positive solutions of differential equation dx/dt = x (t) {r (1 - alpha x(t) - beta(0)x([t]) - beta(1)x([t - 1])) + gamma(1)x([t]) + gamma(2)x([t - 1])} (A) where the parameters r, alpha, beta(0), beta(1), gamma(1) and gamma(2) are positive real numbers and [t] denotes the integer part of t is an element of [0, infinity). We considered the discrete solution of the Eq. (A) to show the global asymptotic stability of this equation. We obtained that the global behavior of the solution of the population model represented by (A) depends on the conditions of the coefficients. In addition, we give a detailed description and conditions of semicycle and damped oscillation of discrete solutions of Eq. (A). (C) 2010 Elsevier Ltd. All rights reserved.