Study On Predator-Prey Models With Fear Effect Based On State Feedback Control

In natural ecosystems,predation relationship is a common biological relationship among species,and it is a key component that determines species behavior,species composition and dynamics in communities.Therefore,the study of predator-prey interaction has become one of the core topics in biomathemics.Predation is the direct effect of one species on another,resulting in a certain number of prey species being killed,which in turn affects the prey population.In addition to direct predation,related experiments have found that the physical and psychological changes caused by prey’s fear of predators may be a more underlying and longlasting evolutionary consequence.Therefore,the research on fear has attracted the attention of many mathematicians.In order to better protect,exploit and utilize fish resources,it is necessary to carry out relevant research from the aspect of mathematical models.In this paper,different forms of fear functions and functional response functions are introduced to establish predatorprey models,then we discuss their dynamics and feedback control problems.The main research work is as follows:1)Aiming at the fear effect in fishery system,a Holling-II predator-prey model with inverse fear function is established.Firstly,based on the qualitative theory of ordinary differential equation,the existence and stability of equilibrium and the influence of fear level on the model are discussed.Secondly,in order to realize the sustainable development of fish resources,a fishing model based on state-dependent strategy is established.The existence and stability of order-1 periodic solution of the system are analyzed by using the correlation theory of semi-continuous dynamic systems,and the existence of order-n n(28))5,4,3,2(periodic solution is discussed.In addition,an optimization model of the fishing process is constructed to determine the optimal harvest threshold of prey by numerical methods.2)Considering the influence of different fear functions on the system,Hassell-Varley predator-prey model with exponential fear function is introduced and established.Firstly,the influence of fear level and HV constant on the positive equilibrium of the model is discussed.Secondly,in order to reduce the adverse effects of zooplankton on the fish pond environment,a fishing model based on state feedback control is proposed.The existence of the order-n n(28))4,3,2,1(periodic solution and the stability of order-1 periodic solution are proved by using methods of successor function,Poincaré map and analogous to the Poincaré criterion.What’s more,the optimal control threshold of prey is discussed by means of numerical optimization.3)In view of the fear and cooperative hunting behaviors in the fishery system and the influence of the fear level on the predation rate of predators,a predator-prey model with an inverse fear function and cooperative hunting is established.Firstly,the effects of fear level and cooperative hunting factors on the existence of positive equilibrium are discussed.Secondly,in order to realize the sustainable development of fish resources and maintain the balance of the ecosystem,the fishing strategy based on state feedback and nonlinear release is adopted.The influence of nonlinear impulse function on fishing system is discussed.The existence of order-n n(28))3,2,1(periodic solution is discussed by constructing Poincaré map.The stability of order-1 periodic solution is analyzed by using analogous to the Poi ncaré criterion.At last,Matlab software is used to simulate the above models to verify the correctness of the theoretical results.