Study On The Dynamics Of Several Predator-Prey Models

Based on the modified Leslie-Gower model,three predator-prey models are proposed,where Allee effect,prey refuge,time delay and diffusion are considered in these models.The aim of this paper is to explore the effect of every factor and coupled effects of these factors on the dynamics of population growth and distribution.Firstly,we construct a modified Leslie-Gower predator-prey model incorporating Allee effect and prey refuge in Chapter 2.The existence and stability of positive equilibria and multiple bifurcation behaviors are discussed,including Hopf bifurcation and BogdanovTakens bifurcation.Numerical simulations reveal that Allee effect and prey refuge have a significant impact on the dynamics of the model.Secondly,based on the predator-prey model in Chapter 2,a predator-prey model incorporating time delay is established in Chapter 3.Theoretical analysis indicates that time delay can destabilize the stable equilibrium via Hopf bifurcation,and then the corresponding periodic solutions emerge.Sensitivity analysis displays that the coupled effect of time delay,Allee effect and prey refuge can lead to more complex dynamics of predator-prey model.Finally,a diffusive predator-prey model is established based on the work in Chapter 3.We investigate the effect of diffusion and delay on the stability of positive steady-state solution,respectively.The results suggest that,under some conditions,both diffusion and time delay can destabilize the positive steady-state solution,which lead to temporal and spatial oscillations in population density.Therefore,both the effect of every factor and the coupled effect of muliple factors have an important impact on the growth and distribution of populations.