Bifurcation Analysis In A Time-delay Model For Predator-prey With Food Subsidies

To protect the diversity of species and maintain ecological balance,we need to study the population dynamics in depth,and then reveal the interaction between populations.Predator-prey mechanism has been paid close attention by many scholars because of its importance in population dynamics.When describing the population evolution,we need to take into account the maturity of the species and the energy conversion time.So it is necessary to introduce a delay into the system in order to reflect the practical situation better.Therefore,in this paper we discuss a class of predator-prey model with time delay,and introduce the influence of food subsidy in the model.Firstly,we discuss the existence and uniqueness of the positive equilibrium point of the system.Based on this,according to the distribution analysis method of roots of the characteristic equation,the stability of the system is analyzed.We obtain the sufficient conditions for the local Hopf at the positive equilibrium point.Then,by using the center manifold theorem and normal form theory,the properties of the Hopf bifurcation at the positive equilibrium point are analyzed,including the direction of the branch,the stability of the periodic solution and the periodic solution,etc.Furthermore,on the basis of the local Hopf bifurcation analysis,we further study the global existence problem of periodic solutions.By the global Hopf branch theorem,we can obtain each connected branch is unbounded,then we prove that the solution of the system is positive.The high-dimensional Bendixson theorem of the ordinary differential equation is used to prove that the system has no nonconstant periodic solutions with period ?,then we obtain the global existence of periodic solution.Finally,the numerical simulations are divided into two parts.In the first part,we use the time delay as the parameter to observe the stability of the system at different time delays and the existence of the global Hopf branch,and give an example support for the previous theoretical results.In the second part,the rate at which the subsidy appears,carrying capacity,the maximum rate at which predators can consume prey and conversion factor are taken as parameters.The effect of each parameter on the stability of the system is obtained by simulating its influence on the first bifurcation value,and the biological implication of each condition is explained.