Research On The Number Of Hopf Bifurcations And Limit Cycles Of The Three-dimensional Competitive Ricker System

This paper studies Hopf bifurcation of three dimensional competitive Ricker systems We give the necessary and sufficient conditions for the system to have positive Hopf bifur-cation values,and based on this strictly prove that under one of the conditions(C1)-(C5)which is satisfied by the competitive coefficients of species,Hopf bifurcations occur in the classes 26-31 of Zeeman's 33 nullcline stable classes for three dimensional competitive Ricker system,and the direction of bifurcation is pointed out,among these conditions it contains a new type which is that the three principal minors of the coefficient matrix are all non-negative.We present the formula of the first focal value by center manifold the-ory to guarantee that in all concrete systems the first focal values by manual calculation are rigorous.Finally,based on the different signs of cross section condition and the first focal value,we construct concrete systems to illustrate that Hopf bifurcations occur in the classes 26-31 and verify these results by numerical simulation.Moreover,using Poincare-Bendixson theory it is demonstrated that there exist at least two limit cycles for the systems of the classes 27-31Besides,this paper presents a three-dimensional competitive Ricker system with two perturbation parameters.By center manifold theory and computing the simplest normal forms,it is proved that there exist parameter values for which this system belongs to the class 28 of Zeeman's classification and possesses three limit cycles,the inner two of them are small-amplitude limit cycles from Hopf bifurcation,and the outer one enclosing all the two small-amplitude limit cycles is generated by the dynamics on the boundary of the carrying simplex for the class 28 and Poincare-Bendixson theoryFinally,this paper investigates a three-dimensional mixing competitive system with one exponential growth rate and two rational growth rates,whose nullclines are linearly determined.In total,33 stable nullcline classes exist.Hopf bifurcations occur in the classes 26-31.We prove that there are systems possessing at least two limit cycles in each of the classes 27-31.