Optimal control of time-delay stochastic population systems in polluted environments

To investigate the optimal control problem for a stochastic population model incorporating individual size structure and time-delay under pollution stress, the method of characteristics is adopted to prove the existence and uniqueness of the system solution and to establish the existence of the optimal control． The first-order necessary conditions for optimal control are derived via the Hamiltonian function, accompanied by corresponding two-dimensional diagrams． The results indicate that a longer gestation delay postpones the optimal harvesting period of the population, whereas increased environmental stochasticity amplifies population oscillations and constrains resource utilization．