Hopf Bifurcation Analysis Of A Class Of Delayed Lotka-volterra Predator-prey Models

In this paper, we frst introduce the background of the impact of time delay andresponse function on the prey-predator ststem. Considering the natureal phenomenonthat species whose individuals have a life history that takes them through two stages,immature and mature, and show diferent characteristics in each growth stage. Notice theobjective factor that the spread of the epidemic in the interaction between populations andthe efect of the combination of population ecology and epidemic dynamics on the controlof infectious disease spread between populations. Comprehensive these phenomena andfactors, we improve the model (1.2.1) and (1.2.3) respectively, put forward a class ofdelayed Lotka-Volterra prey-predator system (1.2.2)and (1.2.4) with Holling II responsefunction in chapter2and3.The main content of chapter2and3is the study and analysis of the model (1.2.2)and(1.2.4). By applying the theorem of Hopf bifurcation and regarding the delay τ as thebifurcation parameter, analyzing the linearized equation of the system at the positive equi-librium and the associated characteristic equation, we have investigated the asymptoticstability of the positive equilibrium and Hopf bifurcation. Furthermore, we have madethe formula of confrming the direction of Hopf bifurcation and the stability of the bifur-cating periodic solutions by the normal form theory and the center manifold theorem forfunctional diferential equations. Finally, some numerical simulations are carried out fordemonstrating the theoretical results.