Stability And Bifurcation Analysis Of A Predator-prey Model With Age Structure

Predator-prey relationship,as one of the basic relationships of interactions among biological populations,has become a major topic in mathematics and ecology in many years.Because species can only reproduce after reaching mature age,it is of great value and theoretical significance to introduce age structure to study models of this kind of biological significance in view of th e different characteristics of species in their infancy and adulthood.In this paper,we study a two-delay predator-prey model with age structure and Beddington-De Angelis function response.One of the delays occurs in the maturity function of predators,characterizing the reproductivity of predators,and the other occurs in the birth function of predators,describing the time required for the predator to give birth to a new individual.Firstly,the model is transformed into an abstract Cauchy problem by using the integral semigroup theory.The conditions for the system to produce a unique positive equilibrium point and the explicit representation of the positive equilibrium point are obtained.Then,the characteristic equation of the linear system at the positive equilibrium point is deduced by using the spectral theory.Secondly,the algebraic method is used to study the distribution of eigenvalues with equal delay.By Hurwitz criterion,the conditions for the stability of the positive equilibrium point of the system with zero delay are obtained,and the conditions for the existence of pure imaginary roots and the equivalent forms for judging the transversal conditions are derived.When the above conditions are satisfied,Hopf bifurcation theorem shows the system undergoes Hopf bifurcation at the positive equilibrium point.The explicit formula of Hopf bifurcation value is further obtained,and the conclusion is explained by numerical simulation.Finally,the geometric method is used to study the distribution of two unequal time-delay eigenvalues.Through the transformation analysis of the equation,the feasible region of pure imaginary root,the explicit expression of time delay and transversal curve are obtained.Then the equivalent form of judging the transverse direction is deduced from the implicit function theorem,and the stability region of the positive equilibrium point is obtained.When the system satisfies certain conditions,the system undergoes Hopf bifurcation at equilibrium point.From the numerical simulation,it can be observed that when two time delays pass through the continuous transversal curve on the parameter plane,a stability switch appears.