Dynamical Analysis In Some Predator-prey System With Allee Effect

Predation plays an important role in regulating the quantity and quality of the predator and the prey.Studying the dynamics of the predator-prey model is helpful to understand the regulation mechanism in the predation process and to make accurate prediction of the predator and prey.When characterizing the population growth of social species,the Allee effect function is important in describing the growth rate of social species.In this paper,the dynamics of several predator-prey models are studied,including the stability analysis,the existence and properties of Hopf bifurcation,Turing-Hopf bifurcation,Hopf-Hopf Bifurcation.The main work is as follows:(?)A diffusive predator-prey model with strong Allee effect and delay is considered.First,by using the parabolic equation theory we study the global existence and uniqueness of the solutions.By analyzing the distribution of the roots of the characteristic equations,we get the stability of the constant steady states and obtain the sufficient conditions for the bistablity of the system.Then by taking delay as bifurcation parameter,we study the existence of Hopf bifurcation,verify that the system may undergo a series of Hopf singularities and calculate the expression of the critical Hopf singularities.The properties of Hopf bifurcation are determined by the normal form theory and the center manifold reduction theorem.Some numerical simulations are carried out for illustrating the theoretical results.Our results suggest that when the initial value of the predator is large enough,the“Over-Predation” phenomenon would occur in the system,thus both the predator and the prey will become extinct eventually.Besides,when the parameter values are given under given conditions,different initial values of the system will tend to different steady states,which indicates that the system has a strong sensitivity to the initial value.In addition,we obtain that under certain conditions,the delay may unstablize the positive constant steady state and cause the periodic oscillation of the system.(?)A diffusive predator-prey model with strong Allee effect and stage structure is established.First we obtain the basic properties of the solutions,analyze the stability and the attractivity basin of the constant steady states.Then by taking the mature age as bifurcation parameter,we study the existence of Hopf bifurcation.On the center manifold,we investigate the criticality of the Hopf bifurcation by the normal form theory.Some numerical simulations are carried out for illustrating the theoretical results.Our results showthat the “Over-Predation” phenomenon and the bistable situation still occur in this model.Under certain conditions,if the mature age is fixed near a positive value,the positive constant steady state is stable;when the mature age is decreasing from this positive value,the Hopf bifurcation may occur and the positive constant steady state is unstable,the system exhibits periodic solutions.As the mature age continues to decrease,the system may also exhibit the transient periodic solutions.(?)A diffusive predator-prey system with strong Allee effect and two delays is studied.By taking two delys as parameters and by using the method the stability switching curves,we study the stability of the positive constant steady state.Then we derive the Hopf and double Hopf bifurcation theorem by defining the crossing directions of the stability switching curves.We also calculate the normal forms near the Hopf-Hopf singularities on the center manifold.The study shows that the system has rich dynamics near the Hopf-Hopf singularities,including the positive constant steady state,the spatially homogeneous and inhomogeneous periodic solutions,etc.Besides,the “Over-Predation”phenomenon still exist in this system.Finally we carry out some numerical simulations to illustrate the theoretical results.(?)A diffusive Leslie-Gower model with weak Allee effect in prey is considered.First we analyze the basic properties of the solutions,the number and the attractivity of the positive constant steady states.Then the influence of two varying parameters on the system are evaluated,by analyzing the distribution of the roots of the characteristic equations,we study the existence of Hopf bifurcation,Turing bifurcation,Turing-Hopf bifurcation and Turing-Turing bifurcation.Our conclusions illustrate that the boundary of the stable region of the positive constant steady state contains a Hopf curve and countable Turing curves.The intersections of these curves include Turing-Hopf singularity and Turing-Turing singularity,the system may produce spatially homogeneous periodic solutions,spatially inhomogeneous periodic solutions or spatially inhomogeneous steady states around these singularities.Finally,we carry out some numerical simulations for illustrating the analytical results.