Stability of nonlinear population systems with individual scale and migration

Abstract

In this paper, we study the stability of a nonlinear population system with a weighted total size of scale structure and migration in a polluted environment, where fertility and mortality depend on the density in different ways. We first prove the existence and uniqueness of the equilibrium point via a contraction mapping and give the expression for the equilibrium point. Some conditions for asymptotic stability and instability are presented by means of a characteristic equation. When the effect of density restriction on mortality is not considered, the threshold value of equilibrium stability can be obtained as Λ=0. When Λ0, the equilibrium is unstable. In addition, the upwind difference method is used to discrete the model, and two examples are given to show the evolution of species.

Keywords

• population model
• polluted environment
• individual scale
• equilibrium solution
• stability