A Study On The Nonlinear Population Models With Age-structure

The differential equation of population dynamics was proposed by Malthus[1]. Verhulst [2, 3] improved it such that the ordinary differential equation ofpopulation dynamics developed very well. But they are only applicable to thelow biologies, because they can't consider the distinction between individuals.To solve the problem, we can use ordinary differential equations or the partialdifferential equation. Obviously, the partial differential equation is better.Lotka [4] proposed the basic ideas of the population dynamical model with age-structure. From then on, people have been studying the population dynamicalmodel with the partial differential equation. Recently, many researchers havebeen interested in the stability of stationary solution to the system. Populationcan develop continuously if only the system is stable.In this paper, we consider a general nonlinear age-structured popula-tion dynamical model, in which fertility and mortality rates depend on bothage and the total population. For the asymptotic stability of zero stationarysolution and positive stationary solution to the system, two suffcient andnecessary conditions are presented, which simplify the method given by M.Farkas. Finally, a more general nonlinear age-structured population dynami-cal model with immigration is considered and the existence and uniqueness ofthe classical solution to the model are given.