Dynamics Of Two Predator-prey Models With Stage-structure

In natural ecology,there are many complex interactions between organisms that together form a biological population system,and predator-prey population interactions are considered to be one of the most important roles.In general,in order to descript the reality,fear effect,group effect,cooperation effect,disease,time delay and optimal harvesting are considered into the biological population system.Based on these considerations,two kinds of stage-structured predator-prey models with multiple effects are established,the local and global stability of the system,Hopf bifurcation properties and optimal control problems are analyzed.In Chapter 2,a delayed predator-prey system with fear effect,disease and herd behavior in prey incorporating refuge is established.Firstly,the positiveness and boundedness of the solutions is proved,and the basic reproduction number R₀ is calculated.Secondly,by analyzing the characteristic equations of the system,the locally asymptotic stability of the equilibria is discussed.Then taking time delay as the bifurcation parameter,the existence of Hopf bifurcation of such system at the positive equilibrium is given.Thirdly,the global asymptotic stability of the equilibria is discussed by constructing a suitable Lyapunov function.Next,the direction of Hopf bifurcation and the stability of the periodic solution are analyzed based on the center manifold theorem and normal form theory.What's more,the impact of the prey refuge,fear effect and capture rate on system are given.Finally,some numerical simulations are performed to verify the theoretical results.In Chapter 3,a delayed predator-prey model with stage structure for prey,refuge and cooperation is established.First,by analyzing the characteristic equations of the system,the local asymptotic stability of the trivial equilibrium and the predator extinction equilibrium are discussed.Then considering time delay as the bifurcation parameters,the existence of Hopf bifurcation of this system at the positive equilibrium is given.Next,the direction of Hopf bifurcation and the stability of the periodic solutions are analyzed due to the center manifold theorem and normal form theory.In addition,the optimal harvesting policy of the system is showed by using the Pontryagin's maximum principle.Finally,some numerical simulations are given to check the correctness and feasibility of the theoretical results.