Hopf Bifurcation In A Delayed Population System With Group Defence And Network

In recent years,with the deepening of the research on the biological population system and the wide application of the Hopf bifurcation,the Hopf bifurcation of the population systems has become one of the most popular research topics in the field of biology and applied mathematics.In the real world,the practical problems in many fields can be described by delay differential equations,such as ecosystems and epidemiology.In particular,time delay has a significant impact on population systems and network systems.Therefore,the researches on population systems with time delay are becoming more and more important.In this paper,we mainly studied the stability,the Hopf bifurcation and bifurcating period solutions of the delay population system with group defense and network structure by using the stability theorem,the center manifold reduction and the normal form theorem of the qualitative theory of differential equations.It will be explored from two aspects,the specific situation is as follows:1.We studied the Hopf bifurcation in a delay predator-prey system with general group defence for prey.Firstly,the stability of the positive equilibrium and the existence condition of the Hopf bifurcation are analyzed by using the stability analysis theory of differential equations.Meanwhile,the direction of the Hopf bifurcation is explored by using the first Lyapunov exponential method.By applying the center manifold reduction and the normal form theorem,we researched direction of the Hopf bifurcation in the present of delay and diffusion.Our numerical results indicated that both in the system without delay and diffusion and in the delay-diffusion system,the aggregation efficiency? could induce instability,bifurcation and nonhomogeneous solutions.However,from the analysis,when the aggregation efficiency ?=1 of prey,no Hopf bifurcations will occur for the system.2.We investigated the Hopf bifurcation in a delayed single species network system.A network structure was constructed to express the movement behavior of a single population in different patches.Based on the stability analysis of a single population model with time delay in a network,it is found that the network can generate temporal patterns only when the time delay is present.It can be seen from the numerical simulation results,for the single population model with network structure and time delay,the system presents regular temporal periodic solutions and the space inhomogeneous structure will appear at different time moments due to the Hopf bifurcation induced by time delay.For the predator-prey model with group defense for prey,the scholars mainly studied the special case of prey aggregation efficiency ?=1/2.However,there are few researches on the effects of ??1/2 and ??(0.1)itself in the existing literature.In the second chapter of this paper,the effects of the more general prey aggregation efficiency are explored in which ??(0,1),and compared with the influence of ?=1/2.In addition,introducing the time delay into the predator-prey model will produce more abundant dynamics.For the single population system,it is shown that time delay and diffusion can induce system instability.In the third chapter,the network structure is introduced into the single population system to disperse the diffusion term,and explore whether the network structure and time delay can induce the Hopf bifurcation and other dynamics.