Study Of The Effects On Delays And Diffusions For Several Eco-population Models

In this Ph.D.thesis, some Hopf bifurcation theories and approaches related to ordinary differential equations, delay differential equations and partial differential equations are used, and by means of numerical simulation, to investigate dynam-ical behaviors for several ecological population models including the asymptotic stability of the positive equilibrium, the existence of Hopf bifurcation at the pos-itive equilibrium, the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. And the effects of time delay and diffusion on the above-mentioned dynamical behaviors are also discussed. The whole thesis is divided into four chapters.In Chapter 1, we give a survey to the developments of Lotka-Volterra predator-prey system, delayed differential equations and diffusion equations. Then we in-troduce the background of problems and the main results of this dissertation.In Chapter 2, two three-species Lotka-Volterra systems with two delays are investigated. By choosing the sum (?) of two delays and three delays as a bifurcation parameter, and analyzing the associated characteristic equation, we show that in the above systems, Hopf bifurcation at the positive equilibrium can occur as (?) crosses some critical values. And we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem.In Chapter 3, we first propose a ratio-dependent predator-prey system with diffusion. Then we discuss the stability and hopf bifurcation of the positive equi-librium for the diffusion PDE system and the reduced ODE system, and discuss the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. We show that when the parameters satisfy some conditions there exists diffusion-driven instability of the equilibrium solution.In Chapter 4, a delayed Lotka-Volterra predator-prey system with diffusion effects and Neumann boundary conditions is investigated. By choosing the sum (?) of two delays as a bifurcation parameter, and linearizing the system at positive equilibrium and analyzing the associated characteristic equation, we show that Hopf bifurcations can occur as (?) crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability if the bifurcating periodic solutions.