Asymptotic Behavior For Stochastic Age-structured Population System

The theory of stochastic differential equations(SDEs) has been widely applied to physics, biomathematics, economic mathematics, automatic control, communication, etc.In the practical problem, due to the influence of stochastic factors, the application of the system is dependent on dynamic behavior. In this paper, we mainly consider the dynamical behavior of such system, dissipativity and stability, respectively considering four kinds of random disturbances, which are Brown motion, Fractional Brown motion, Poisson process and fuzzy. The main research contents have the following several aspects:The first part, by using Ito's formula and Bellman-Gronwall-type estimates, sufficien-t conditions are established to guarantee the mean-square dissipativity of stochastic age-dependent population system with fractional Brown motion. Finally, it is respectively shown that the mean-square dissipativity is preserved by the compensated backward Euler method and split-step backward Euler method under Hurst parameter H constraint, the dissipation characteristics of the original system are preserved. A numerical example is given to illus-trate the conclusions.The second part, by using Ito's formula, Cauchy-Schwarz inequality and the properties of stochastic analysis,some criteria are researched which guarantee the stability in the mean square for stochastic delay age-dependent population system with Poisson jump. Finally, it is shown that the mean-square stability is proved by the compensated stochastic ? method under the restriction of the stepsize At and the parameter ?. An example is studied to illustrate the theory with MATLAB packages.The third part, by constructing proper Lyaunov-Krasovskii function, combining with Ito's formula, Bellman-Gronwall-type estimates and fuzzy set theory, sufficient conditions are established to guarantee the mean-square dissipativity of stochastic fuzzy age-dependent population system in a polluted environment. Finally,a numerical example is given to demonstrate the correctness and validity of the results with MATLAB packages.