Hopf Bifurcation Of Three - Species Food Chain Diffusion Model

The common existence of food chain model has attracted many mathematical biology workers. Specially, the study of Hopf branch is not only beneficial to explain the problems related to the nature, but also beneficial to predicts the population development trend in the living environment. In this paper, basing on the three species food chain model, the two new biological dynamics models are investigated by increasing diffusion terms.In the first chapter, the Hopf bifurcation of a class of deterministic model, which is three species food chain with diffusion phenomenon and group defense ability subject to Nenumann boundary condition, is investigated. By treating the death rate of predator as bifurcation parameter, the stability of the positive constant equilibrium solution is discussed by use of Hurwitz criterion. Then, the conditions which can raise the Hopf bifurcation are given through the theoretical analysis. And also, the normal form method and center manifold theorem are used to study the Hopf bifurcation direction and stability of bifurcation periodic solutions. Finally with the help of matlab software, numerical simulation is verified and added to the results of theoretical analysis.In the second chapter, a class of hybrid ratio-dependent three species food chain model with diffusion is investigated. By analyzing the characteristic equations, the local stability of the non-negative constant equilibrium solution is studied. The ex-istence of Hopf bifurcation at the positive equilibrium is established. The judgement which is about the Hopf bifurcation direction and stability of bifurcation periodic solutions is also established. Finally, a numerical simulation is given to support the results.