Fully Discretizated Numerical Analysis Of Stochastic Age-structured Population Models

Many behaviors of biological population models are age-structured,such as reproduction and growth.Partial differential equations are used to study,because there are interference terms,stochastic partial differential equations are more interesting.Normally,there is no explicit solution for the stochastic age-structured population model,so the numerical approximation scheme is a valuable tool for exploring its properties.So far,for deterministisc problems some significant results have been achieved in numerical analysis of semi-discrete and fully discrete schemes for age-structured population models.However,the numerical analysis of stochastic age-structured population models focuses on the time semi-discrete scheme and the bounded age problem.Therefore,this thesis will mainly study the properties of numerical solutions of several kinds of fully discrete schemes for stochastic age-structured population model with unbounded age.This thesis reviews the application background of age-structured population model,briefly recalls the development of numerical methods for age-structured population model in recent years,especially the application of semi-discrete method in the field of age-structured population model.The basic idea of the discrete method is given,the original equation is transformed by the characteristic line method to obtain a renewal equation,the existence and uniqueness of the solution of the renewal equation and the solution of the original equation are studied,the non-negative and regularity of the solution of the original equation are studied,and the numerical scheme of the fully discretized backward Euler method for age-structured population model is constructed.The convergence of the fully discrete method for linear age-structured population model is analyzed,and the convergence of the reproduction number is proved.Give the relationship between the basic number of regeneration and stability.And numerical examples are given to verify the correctness of the conclusions.Furthermore,the numerical scheme of the fully discretized Euler-Maruyama method for the nonlinear stochastic age-structured population model is constructed.The convergence of the fully discrete Euler-Maruyama method for the nonlinear age-structured population model is discussed.Finally,numerical examples are given to verify the correctness of the conclusions.