Applications Of Bifurcation Theory In Several Biological Systems

This paper consists of three parts:Firstly, we consider a kind of delayed of prey-predator system with harvesting. We investigate the influence of harvesting; we applying the normal form theory and the center manifold theorem to get the effect of the delayed on the system. What’s more, the conditions of the Hopf bifurcation take place and the stability of the bifurcating the periodic solutions are obtained. Finally, the number simulation results are given to support the theoretical predictions.Secondly, we study a discrete single population system with feedback control. For the first place, we utilize the Euler method discretize the system, and through the Schur-Cohn-Jury lemma obtain the stability of the positive fixed point. Next, we recognize the natural growth rate as a parameter, and by control the natural rate, using the normal form theory get the conditions of the Neirmark-Sacker bifurcation occur, and the direction of the bifurcating periodic solutions. At last, a specific example given to express the accuracy of the results.The last section, we will focus our attention on a population system with Allee effect. For this part, we also utilize the norm theory to investigate Hopf bifurcation and Bautin bifurcation of the system, and we give the condition which Hopf bifurcation and Bautin bifurcation take place. At last, we give a sample to elaborate the results we have got.