Optimal Control Of Several Species Of Population Systems With Individual Scale In Polluted Environment

At present,due to the rapid development of modern industry and agriculture in human society and other adverse production activities,serious environmental pollution has been caused.Environmental pollution not only affects the survival of human beings,but also the others creatures.Therefore,it is more valuable to study the population system in polluted environment.Single population is the basic unit of the whole ecosystem.Biological population is composed of individuals,and there are structural differences among individuals,such as age,physiological scale,gender,gene and so on,It is very important to study the biological population system with individual structure differences.In this paper,we consider two kinds of population models with scale structure in polluted environment,and analyze their dynamic behavior and related control problems.The details are as follows:(1)In the real world,the maturation of biological population is not an instantaneous process,but there is a time delay.The time delay can be considered as the time spent from breeding to maturity.In this paper,we consider the incubation time delay.Therefore,we establish a population system model with time delay and individual scale in polluted environment,It is proved that the well posedness of the state system,the continuous dependence of the solution on the control variables and the existence of the optimal control are proved.Finally,the necessary conditions for the optimal control of the system are obtained.(2)Because biological populations often move their positions,in real life,the population system will inevitably be affected by the random environment.Therefore,a nonlinear stochastic population diffusion system model with individual scale in polluted environment is established.By using It(?)formula,the existence and uniqueness of strong solution for stochastic population system is proved,The necessary and sufficient conditions for the existence of optimal control are obtained when the external disturbance is linear.Finally,the necessary conditions of optimal control are derived by using integral partial differential equation and variational inequality,and the optimality system is obtained.