The Solution And Its Stability Of The Population Dynamic System With Nonlinear Migration

Population development is a dynamic process with itself moving rule and affected by people behavior. Because of the birth, death, migration and other factors, population is different with the time in a certain area. The change of the population is bound to have effect on development and balance of the whole ecological system. Therefore, the research of population dynamic system is great important and meaningful. That has appeal hot attention among scientists, so that a great many research papers about the population sprang up. Most of them discussed the linear equation in a closed area. Actually, the birth rate and mortality depend on the population density and the migration is unavoidable in the reality. Therefore the nonlinear model with nonlinear migration is more reasonable and meaningful. This paper investigated the nonlinear population evolution equation with nonlinear migration profoundly. And it gives results about the existence and stability of solution of the population equation.Following is the population evolution equation with the nonlinear migration. Where I(t)=∫₀^∞(?)(r)p(r,t)dr, (?)(r)≥0 represents the contribution ofof the individual at age r to the effect.t∈[0,∞) is time.r∈[0,∞) isage and r_m is the maximum age. p^r,t is density function of the population.μ(r,I(t)≥0, the mortality of age r at t , andβ(r,I(t))≥0 ,the birth rate, are all depend on I(t)=∫₀^∞(?)(r)p(r,t)dr .f(r,I(t)) is the migration, which is also affected by I{t)To the population dynamics system with nonlinear migration f(r,I(t)), this paper proves the existence and uniqueness of solution and the stability of the mild solution through analyzing its equilibrium which are respectively discussed under f(r,I(t)) is Lipschiz continuous to p(r,t) and compact. The results in this paper extend and develop the research of predecessors.