Dynamics On Predator–prey Systems With Diffusion And Ivlev-type Functional Response

Differential equation is applied widely in physics, mechanics, economics, management science, biology and so on, and this paper focuses on application of differential equation in population problems. If each generation overlaps with the next generation,such as people and mammals, the population’s growth is continuous and can be described by differential equations. First, living beings often move from one area to another, and diffusion promotes the succession around the biological community, increases the diversity of the species. Second, interaction between species may also affect the number of species, such as predator and prey. Last, any species have the mature period, so delay in population also plays an important role in the research of species. Therefore, study on dynamics of partial(functional) differential equations is a hot topic. Bifurcation mainly includes changes of the solution’s stability and topological structure, as some parameters(such as population the mature period, the environmental capacity or predation rate and so on) reach a certain threshold.This thesis mainly investigates some predator-prey models with Ivlev-type functional response influenced by diffusion, delay, and strong Allee effect applying the normal form theory, the center manifold reduction for partial functional differential equations,Hopf bifurcation theorems, steady state bifurcation theorems, the upper and lower solution method, maximum principle, comparison principle and so on. The main contents of this thesis are as follows:(1) A diffusive predator-prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. At first, the stability of the positive equilibrium and the existence of spatially homogenous and inhomogenous periodic solutions are investigated by analyzing the distribution of the roots of characteristic equation. And the direction and stability of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Then steady state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.(2) A delayed diffusive predator-prey system with Ivlev-type predator functional response subject to Neumann boundary conditions is considered. The stability of nonnega-tive equilibria and existence of Hopf bifurcation are obtained by analyzing the distribution of the roots of characteristic equation. By the theory of normal form and center manifold,an explicit algorithm for determining the stability and direction of periodic solution bifurcating from Hopf bifurcation is derived.(3) The dynamics of a class of reaction-diffusion predator-prey system with strong Allee effect in the prey population is considered. We prove the existence and uniqueness of the global solution and give a priori bound applying the upper and lower solution method and some results of asymptotically autonomous system. We also describe a situation that if the initial value of predator is more than a critical value, the solution of this system will converge to the steady state(0, 0), i.e. large initial value of predator will lead to extinction. We obtain the stability of equilibriums through analysis of the roots of characteristic equation. Hopf bifurcation and steady state bifurcation are studied. Results show that the Allee effect has significant impact on the dynamics of the system. The strong Allee effect enriches the spatio-temporal dynamics, and the conclutions of system becomes more practical.