Stabilityand Hopf Bifurcation Analysis In Species Model With Two Delay

Maturation is an important factor in population dynamics, and it should not be neglected. J. Wu and M. Y. Li proposed the age–structured model of a single species living in two patches by employing the basic age structure equation and characteristics in terms of time and age. After that Chunbo Yu had considered the model and derived the property of the equilibrium solution and Hopf bifurcations. For the age–structured model of a single species living in two patches the distance of the two patches is another important factor, so in this paper we improve the model by introducing a delay? to consider the distance of the two patches. It is difficult to analyze the model with two delays but is of great theoretical and practical significance.In this thesis, in the first, we get the equilibrium solution of the system. Secondly, we analyze the stability of the equilibrium by using linearizing stability method. We get that whenτ=0 forσ∈[0,+∞) the equilibrium solution ((x|-),(x|-)) is asymptotic stability. Thirdly, whenτ=0, we improve the method of Beretta and Kuang to investigate the stability of the equilibrium solution and Hopf bifurcations. Then, we derive the explicit formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out by using MATLAB, and the results are consistent with our analysis results.