Convergence Of Numerical Solutions For A Class Of Stochastic Age-Structured Population Models In A Polluted Environment

Environment pollution due to industrialization has caused many severe ecological problems(e.g.,decreased bio-diversity,and extinction of some species),which has seriously endangered the dynamic balance of the earth’s biological populations and even threatened the survival of human beings.Therefore,it is crucial to use mathematical models to study the effect of pollutants on bio-logical populations.Due to the influence of various uncertain factors,it is more practical to choose an appropriate random process(e.g.Brownian motion,Impulsive,Markov switching)to describe its effect on the biological population model of environmental pollution.In this paper,we mainly stud-ies the numerical methods of toxicant population models with age-structured and uncertain factors,and obtain the existence,uniqueness and convergence of their numerical solutions.The details are as follows:(1)A stochastic age-structured population model with Markovian switching is investigated in a polluted environment.By Ito formula and Markovian property,the boundedness in the qth moment of exact solutions of model are proved.Then,we defined the truncation function and established the truncated Euler-Maruyama(EM)numerical method.Furthermore,the strong convergence criterion of truncated EM numerical approximation in the qth moment is estab-lished,and the rate of convergence is estimated.Numerical simulations are carried out to illustrate the theoretical results.Our results indicates that the truncated EM method can be used for stochastic age-structured population system in a polluted environment.(2)we present a periodic averaging method for impulsive stochastic age-structured population model in a polluted environment.Based on the generalized Lipschitz method,the existence and uniqueness of the solution of the standard impulsive stochastic age-structured population model in a polluted environment are proved.Second,using the idea of the periodic averaging method,a simpler system is approximated to the original system to obtain the average impul-sive stochastic age-structured population model in a polluted environment.We revealed that the averaged impulsive stochastic age-structured population model have a unique solution.Then the mean-square convergence criteria of the numerical solutions produced by the periodic aver-aging method are studied.Finally,a simulation example is given to prove that the algorithm is efficient of our results.