Dissipativity For Number Solution Of Age-dependent Stochastic Population System

The theory of stochastic differential equations(SDEs)has been widely applied to physic-s,biomathematics,economic mathematics,automatic control,communication,etc.In the practical problem,due to the influence of stochastic factors,the model with stochastic pa-rameter which was structured by SDEs could preferably reflect the phenomenon's nature.This paper mainly studied the dynamic behavior of such system,respectively considering three kinds of random disturbance,which are Brown motion,Fractional Brown motion and Poisson process.On the other hand,usually,most of the SDEs do not have exact solutions or exact solutions.especially for the equations with Poisson process.Therefore,numerical methods are the powerful tools to study the solutions of SDEs.This paper discussed dissi-pativity of stochastic age-dependent population model.The main research contents have the following several aspects.The first part,we consider mean-square dissipativity of two numerical methods for s-tochastic age-dependent population equations with jumps.Based on the step length under the condition of limited and unlimited,it is essential for studying the mean-square dissipa-tivity to use backward Euler method and compensated backward Euler method for stochastic age-dependent population equations with jumps.Finally,an example is studied to illustrate the theory with MATLAB packages.The second part,by using Ito's formula,Cauchy-Schwarz inequality and Bellman-Gronwall-type estimates,sufficient conditions are established to guarantee the mean-square dissipativity of stochastic age-dependent population system.Finally,it is shown that the mean-square dissipativity is preserved by the split-step backward Euler method and compen-sated split-step backward Euler method under a step-size constraint.Numerical examples are studied to illustrate the conclusions.The third part,we consider global stability of more general stochastic age-dependent population models.By use of Lyapunov functions,Barbashin-Krasovskii theorem,LaSalle theorem and Ito's formula to cover the stochastic age-dependent population systems having more than one weak solution.And it gives the sufficient condition for the global stability of the zero solution.Finally,the numerical example is provided to demonstrate the effectiveness of our criteria.