On The Stability And Bifurcations Of Two Kinds Of Discrete Predator-prey Models

The study of the dynamics of discrete predator-prey models is one of the important research topics in biomathematics.This thesis analyzes the stability and bifurcations in the two-dimensional discrete predator-prey model with refuge and fear and the three-dimensional discrete predator-prey model with non-monotonic functional response by using eigenvalue theory,central manifold theory and bifurcation theory.The study of these issues will further develop stability theory and bifurcation theory of discrete system.It has an important guiding role in the development,utilization and control of biological resources.This thesis is divided into four Chapters.The first Chapter mainly introduces the significance of this thesis and elaborates the establishment methods of the two types of models which are used in this thesis,then gives the content of this thesis.We study a two-dimensional discrete predator-prey model with shelter and fear at Chap-ter 2.After calculation with maple,we get four fixed points and the stability of every fixed point is determined by evaluating the eigenvalues.Then we study bifurcations of the fixed point which is hyperbolic by using the central manifold theory and the bifurcation theory.And we get E₂gives rise to Flip bifurcation and E₃produces Neimark-sacker bifurcation.Finally,we illustrate the validity of the theoretical results in this thesis by giving numerical simulations of bifurcations at positive fixed point.In the third Chapter,we study a three-dimensional discrete predator-prey model with non-monotonic functional response.Through the calculation by maple,seven fixed points are obtained.And the stability of each fixed point are analyzed by finding the eigenvalues of the jacobian at the fixed point.Then studying bifurcations of non-hyperbolic fixed points by using central manifold theory and bifurcation theory.The results show that transcritical bifurcations are generated in the state of total extinction,Flip bifurcations and transcritical bifurcations are generated in the state of prey only,Flip bifurcations and Neimark-Sacker bifurcations are generated in the state of prey and middle predator coexisting or all coexist-ing.At last,numerical simulations of the Flip bifurcation and Neimark-Sacker bifurcation are given.The fourth Chapter summarizes this thesis and proposes the future research work.