Existence,Uniqueness And Exponential Stability For Stochastic Fuzzy Age-dependent Population Equations

In this paper, we include Fractional Brown motion、Poisson jumps and polluted environment in-to the stochastic age-structured population model with diffusion which is the basic model of the paper, respectively. Therefore, three kinds of stochastic age-structured population model with diffusion are estab-lished. Meanwhile, the fuzziness of uncertain factor is considered in the population system, it is obtained that a few class of stochastic fuzzy age-structured population equations with diffusion. The system’s ex-istence and uniqueness are discussed, respectively, and an estimation of error of approximate solution is established for two kinds of population models. Finally, the numerical results are consistent with theo-retical conclusions and validate the feasibility and efficiency of expoential stability. The main research content has the following several aspects.(1). There is based on the stochastic age-structured population model with diffusion, we take advan-tage of Poisson jumps to simulate the sudden disasters such as the fire、flood、earthquake and hurricane, at the same time the fuzzy uncertainty is introduced into the population models. Then we take it into account in the basic model of the paper to establish another stochastic fuzzy age-structured population equations with diffusion and Poisson jumps. Under the coefficients of the equation satisfy Lipschitz con-dition together with boundedness condition(which is weaker than linear growth condition), it can apply the method of successive approximation by constructing Picard iteration sequence to discusse the exis-tence and uniqueness of solutions to stochastic fuzzy age-structured population equations with diffusion and Poisson jumps. Using Gronwall lemma、fuzzy stochastic Ito integral and triangle inequality, the suf-ficient condition for the existence of the strong solution is given and an estimation of error of approximate solution and exponentially mean square stable are established, the sufficient conditions are presented. An example is given for illustration.(2). We research a stochastic fuzzy age-structured population equations with fractional Brown mo-tion and Poisson jumps. By using Gronwall lemma、Ito integral of fractional Brown motion、fuzzy stochastic Ito integral and triangle inequality and so on, the existence、uniqueness and exponential sta-bility of the model are discussed in this paper. At last, the numerical examples validate the feasibility and efficiency of numerical method which is shown in the paper.(3). The fuzziness and randomness, which are two kinds of uncertain factors, and are considered in the polluted environment simultaneously, it is obtained that a class of stochastic fuzzy age-structured pop-ulation equations with diffusion in a polluted environment. Under the coefficients of the equation satisfy Lipschitz condition together with boundedness condition(which is weaker than linear growth condition), it can apply the method of successive approximation by constructing Picard iteration sequence to discusse the existence and uniqueness of solutions to stochastic fuzzy age-structured population equations with diffusion and Poisson jumps. Using Gronwall lemma、fuzzy stochastic Ito integral and triangle inequal-ity, the sufficient condition for the existence of the strong solution is given and an estimation of error of approximate solution and exponentially mean square stable are established, the sufficient conditions are presented.