Local Stability And Hopf Biurcation Analysis For Two Classes Of Predator-prey Models With Time Delay

Ecosystems are complex and diverse,and the population ecosystem model is rich in dynamic properties,so exploring the mysteries of population ecosystem theory has become one of the hot research topics at present.It is worth mentioning that factors such as multiple delays,rate dependency and so on exist widely in the population ecosystem,so these factors can be more appropriate description of the natural ecology.Based on this background,this paper focuses on the dynamic characteristics and properties of two predator-prey models with multiple delays.Firstly,the research background,significance and status of predator-prey model are reviewed,and the research content of this work is briefly addressed.Secondly,the Holling III functional response function ecosystem model of rate dependence,feedback and pregnancy delay was explored.By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations,the local stability of each of equilibrium is discussed and the existence of Hopf bifurcations at the endemic equilibrium is established.For illustrating the theoretical results,some numerical simulations are also performed.Next,a predator-prey system model with stage structure,functional response function,selective harvesting and a prey refuge is investigated.In the same way,the bifurcation condition and the periodic solution of themodel are explored by combining the stability theory and the bifurcation theory.Compared with the model which does not consider the ashes evacuation factor before,this model has the practical significance to avoid the extinction of the prey,and data simulation verifies the correctness of theoretical analysis.Finally,the paper summarizes the research work of the paper,and points out the need to improve the paper,and make future prospects for further research.