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 d x d t = r ⋅ x ( t ) { 1 − α ⋅ x ( t ) − β 0 x ( [ t ] ) − β 1 x ( [ t − 1 ] ) } where t ⩾ 0 , the parameters r, α, β 0 and β 1 denote positive numbers and [ t ] denotes the integer part of 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.