Bifurcation Analysis Of A Time-delayed Predator-prey Diffusion System With Harvest Rates

Ecologic dynamics is a study that uses the theory of the environmental dynamics to analyze the behaviors and quantitative structures of biological groups.The predator-prey system,one of the main topics in ecological mathematics,is also a branch of ecological dynamics.In this thesis,we study a class of time-delayed predator-prey systems with harvest rates first.Under the premise of biological reality,the existence conditions of the positive equilibrium point of the system are discussed.And we use linearized method to analyze the stability of the equilibrium point of the system.It is found by calculation that the Hopf bifurcation exists in the system when the time delay is taken as the branch parameter.And it is verified by numerical simulation that without the delay,the system has a stable focus with appropriate parameters,and when the time delay is considered,the system has a periodic solution.Considering that the dispersal behavior of the population may have an impact on the system,we analyze a predator-prey dispersal model with linear harvest rate and cooperative predation.Based on the Routh-Hurwitz criterion,the existence conditions and stability of the positive equilibrium point of the model are discussed.The conditions of Hopf bifurcation are further analyzed and the Hopf bifurcation direction is judged using the central manifold and normative theory.The conditions of Turing's instability are discussed using branch theory.And numerical simulations are performed on Matlab to show that the system has rich dynamic behaviors.We found that human hunting can modulate stability of system,and population dispersal affects morphology of Turing pattern.The results of this thesis show that appropriate human intervention on predator populations is beneficial to the stability of the ecosystem.Spread of the population and the time-lag of the prey have a huge impact on the stability of the system.