On the validity of Zeeman's classification for three dimensional competitive differential equations with linearly determined nullclines

Abstract

We study three dimensional competitive differential equations with linearly determined nullclines and prove that they always have 33 stable nullcline classes in total. Each class is given in terms of inequalities on the intrinsic growth rates and competitive coefficients and is independent of generating functions. The common characteristics are that every trajectory converges to an equilibrium in classes 1–25, that Hopf bifurcations do not occur within class 32, and that there is always a heteroclinic cycle in class 27. Nontrivial dynamical behaviors, such as the existence and multiplicity of limit cycles, only may occur in classes 26–33, but these nontrivial dynamical behaviors depend on generating functions. We show that Hopf bifurcation can occur within each of classes 26–31 for continuous-time Leslie/Gower system and Ricker system, the same as Lotka–Volterra system; but it only occurs in classes 26 and 27 for continuous-time Atkinson/Allen system and Gompertz system. There is an apparent distinction between Lotka–Volterra system and Leslie/Gower system, Ricker system, Atkinson/Allen system, and Gompertz system with the identical growth rate. Lotka–Volterra system with the identical growth rate has no limit cycle, but admits a center on the carrying simplex in classes 26 and 27. But Leslie/Gower system, Ricker system, Atkinson/Allen system, and Gompertz system with the identical growth rate do possess limit cycles. At last, we provide examples to show that Leslie/Gower system and Ricker system can also admit two limit cycles. This general classification greatly widens applications of Zeeman's method and makes it possible to investigate the existence and multiplicity of limit cycles, centers and stability of heteroclinic cycles for three dimensional competitive systems with linearly determined nullclines, as done in planar systems.