Third-order Biological Systems With Time Delay, Periodic Solution Branch

Ecology is an important discipline aimed at studying the relationship between living beings and the environment. In recent years, much attention has been paid to biological models by many mathematicians and biologicians. In this paper, we will apply Hassard' s method to the discussion of the positive equilibrium state about a three-dimension biological model with delay under a determined condition. The general formulae for bifurcation direction of Hopf-bifurcation is calculated and the estimating formulae of the period and stability for periodic solution is given. The paper includes five parts. In the first part, the basic methods and basic theorems which will be applied to the paper are presented. In the second part, we introduce the background of the system we will discuss and meanwhile proof the system occurs a Hopf-bifurcation at some value of the parameter x . In the third part, we apply Hassard' s method to calculate the direction of Hopf-bifurcation and periodic solution. In the fourth part, we give a specific example by applying the formulae of the period and stability for periodic solution so as to illustrate the formulae' s feasibility. In the fifth part, we summarize the whole paper and point out some future research direction.