Global stability in a population model with piecewise constant arguments.

*(English)*Zbl 1177.34097Summary: We investigate the global stability and the boundedness character of the positive solutions of the differential equation

$$\frac{dx}{dt}=r-x\left(t\right)\{1-\alpha \xb7x\left(t\right)-{\beta}_{0}x\left(\left[t\right]\right)-{\beta}_{1}x\left([t-1]\right)\}$$

where $t>0$, the parameters $r$, $\alpha $, ${\beta}_{0}$ and ${\beta}_{1}$ denote positive numbers and $\left[t\right]$ denotes the integer part of $t\in [0,\infty )$. We consider 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.

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K12 | Growth, boundedness, comparison of solutions of functional-differential equations |

39A12 | Discrete version of topics in analysis |