Study On The Dynamics Of A Predator-prey Model With Hunting Cooperation

The predator-prey system has an important position in ecology,and hunting cooperation has always been a hot topic in the predator-prey system.Hunting cooperation is a common behavior within species in the ecosystem,and many species in nature seek speed catching food faster and with a lower risk of predation will choose group hunting cooperation.For example,lions,wolves,and African wild dogs will cooperate to capture prey,and aquatic creatures,birds,ants,and spiders will also cooperate to capture prey.This paper mainly studies two types of predator-prey models with hunting cooperation,namely the hunting cooperation system with fear effect and Allee effect and the hunting cooperation system with strong Allee effect under the Leslie-Gower predator-prey model.The main content of the article is arranged as follows:The first chapter first introduces the origin,development and research status of biological mathematics at home and abroad,and then introduces the research background and significance of this article;The second chapter studies a kind of hunting cooperation system with fear effect and Allee effect,builds a model and gives the parameter conditions for the existence of equilibrium point by Descartes' theorem,then discusses the stability of the equilibrium point,and finally verifies it by Matlab numerical simulation concluded;The third chapter mainly considers the hunting cooperation system with strong Allee effect under the Leslie-Gower predator-prey model.Using the topological equivalence method,the type of singular point(0,0)is obtained.In addition,The basin of attraction at(0,0)is studied.Then,the stability of all non-negative equilibrium points of the system,including the types of degenerate equilibrium points,is discussed.Then based on Sotomayor's theorem,saddle knot bifurcation is derived the existence of.In order to determine the stability of the limit cycle caused by the Hopf bifurcation,the first Lyapunov coefficient is calculated.At the same time,through rigorous mathematical proof,when the hunting coordination coefficient and the severity of the Allee effect are small enough,the bifurcation limit cycle is stable.Finally,numerical simulations are performed to verify the theoretical results,and further explore the influence of the Allee effect and hunting cooperation.These results show that the strong Allee effect and hunting cooperation have a significant effect on the dynamic behavior of the system significant impact.