Global stability in a population model with piecewise constant arguments

In this paper, we investigate the global stability and the boundedness character of the positive solutions of the differential equation dx/dt = r . x(t){1 - alpha . x(t) - beta(0)x([t]) - beta(1)x([t - 1])} where t >= 0, the parameters r, alpha, beta(0) and beta(1) denote positive numbers and [t] denotes the integer part of t is an element of [0, infinity). We considered the discrete solution of the logistic differential equation to show the global asymptotic behavior and obtained that the unique positive equilibrium point of the differential equation is a global attractor with a basin that depends on the conditions of the coefficients. Furthermore, we studied the semi-cycle of the positive solutions of the logistic differential equation. (c) 2009 Elsevier Inc. All rights reserved.