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 equationdxdt=r⋅x(t){1−α⋅x(t)−β0x([t])−β1x([t−1])} where t⩾0t⩾0, the parameters r, α  , β0β0 and β1β1 denote positive numbers and [t][t] denotes the integer part of t∈[0,∞)t∈[0,∞). 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.