Study On Periodic Solutions, Bifurcation And Optimal Control In Predator-Prey Systems

This paper consists of two parts:In the first part, positive periodic solutions for a four-species predator-prey system with harvesting terms is studied. After applying coincidence degree theory and the comparison theorem, the sufficient conditions of sixteen periodic solutions are obtained. Further, an example is given to illustrate the effectiveness of our results. Finally, periodic solutions can be obtained only when it meets the theorem conditions. Obviously, predation has an important influence on the ecological system. Excessive capture certainly will cause damage to ecological system.In the second part, singularity induced bifurcation and optimal control of predator-prey model with stage structure and harvesting terms are discussed in this paper.First, from the bifurcation theory of differential-algebraic system, the stability for internal equilibrium point of the model is investigated. Sufficient conditions of a singular induced bifurcation for the model are obtained.Second, the state feedback controller is designed to eliminate the singular induced bifurcation.Third, by Pontryagin’s maximum principles, optimal control is obtained, and the numerical simulations are given. According to the optimal control for management strategy department can not only ensure the sustainable development of the ecological system, but also can obtain the maximum economic benefit.