The Study On Predator-prey System With Nonlinear Density Restrict And Functional Response

Mathematics bionomics is the most systematic and complete branch of the mathematical biology up to now, its research of the dynamic properties concerning the predator-prey systems have become an important topic, not only for biologists but also for mathematicians. Thus, considering the differences between the nature species, human intervention, in this paper, we construct three kinds of predator-prey system model and discuss it completely, based on the existing results.Firstly, an autonomous predator-prey harvesting system with Holling II functional responses is constructed, which the prey has Smith dependent. Using qualitative theory of ordinary differential equation, the system’s dynamical behaviors are studied. The sufficient condition for the nonexistence of limit cycle is given by mean of the Dulac Function. An optimal taxation policy is obtained using the corresponding optimal control theory.Secondly, an autonomous predator-prey harvesting model is constructed, which density restriction is 2a-bx-cx and the prey is constant release and the functional response is x. By mean of qualitative theory of ordinary differential equation, dynamical behaviors and global asymptotic stability of the system are studied. With the ring domain theorem and Zhang Zhi-fen uniqueness theorem, the existence and uniqueness of the limit cycle about the system are discussed. The correctness of the conclusion is validated by numerical simulation.Finally, a nonautonomous predator-prey model, which a class of prey density restriction is()()()21a t-b t x t and the predator with stage structure and the functional response is Holling IV is given. Using the method of qualitative theory of ordinary differential equation and Brouwer fixed point theory, uniform persistence of the system is obtained, and the sufficient condition, which it can have a global asymptotically and strictly positive periodic solution is obtained.Three types of predator-prey system model in this paper enrich the model theory of population dynamical system, meanwhile the related conclusions provide theoretical basis for biological researchers and an effective way to guide the protection of animals and plants. Though these results are obtained, there are the further problem to research, such as stochastic divisor and chaos in the variety of predator-prey system.