Stability of nonlinear population systems with individual scale and migration

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 $\mathrm{\Lambda }=0.$ When $\mathrm{\Lambda }<0,$ the equilibrium is asymptotically stable, and when $\mathrm{\Lambda }>0,$ 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.