Optimal control strategies for a new ecosystem governed by reaction-diffusion equations

An optimal control problem for a new ecosystem described by reaction-diffusion equations is studied in this contribution. Different from the classical prey-predator law, the phenomenon that the growth of predator population gradually reduces due to the after-effects of prey population is ubiquitous in the real world, such as, allelopathy phenomenon in planktonic ecosystems. To model this relationship between predator and prey populations, together with diffusion behaviors, a novel model with general Holling type functional response is presented. After that, the existence and uniqueness of the global positive strong solution to the ecosystem is obtained by the method of semigroups of operators. On this basis, an optimal control problem of the ecosystem is considered. The existence of optimal control strategy is proved by using minimal sequence. Then the first order necessary optimality condition is obtained and the corresponding optimal control is found to be of on-off form. Furthermore, the second order necessary condition for optimal control and sufficient condition for local optimal control are all established. Finally, a numerical example is offered to demonstrate the applications of the results obtained in this work.