| How to describe population dynamics behavior more accurately through mathematical models is one of the most important research topics of biomathematics.Since in real-life problems,time delay is usually required between individual behaviors and responses to produce an effect,the present state and future state of an organism are often related to certain states in the past.Therefore,the time-delay models can better reflect the fluctuation and development of the population of this species.In this paper,we start with the intraguild predation model(IGP model).There have been many research results on IGP model with the discrete time-delay,but there are few studies of the IGP model with distributed delay,and many problems remain to be solved.Therefore,this thesis shall study the dynamics of a class of intraguild predation model(IGP model)with distributed delay,and we obtain some new results.The thesis is divided into the following parts.Firstly,we introduce the research background and research status of the time-delay model and the IGP model respectively,and the main work of this paper also be introduced.Secondly,an intraguild predation model with distributed delay is proposed.We give the existence of trivial equilibrium,semi-trivial equilibrium,boundary equilibrium and positive equilibrium of the system,respectively.By analyzing the eigenvalues of the linearized system,the stability of the trivial equilibrium,semi-trivial equilibrium and boundary equilibrium of the system under strong and weak kernels can be obtained.Then the stability and the local Hopf bifurcation of the positive equilibrium are investigated,while the direction,stability of periodic solutions bifurcating from Hopf bifurcations are analyzed.And the cases of the strong and weak cores are discussed.Taking the average delay as a bifurcation parameter,we give the conditions for generating the Hopf bifurcation at the positive equilibrium point.Finally,the numerical simulation of the above theoretical results is carried out with Matlab software,and some dynamic properties of the model are revealed. |
