On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points

Abstract

We study the fixed point index on the carrying simplex of the competitive map. The sum of the indices of the fixed points on the carrying simplex for the three-dimensional competitive map is unit. Based on that, we analyze the asymptotic behavior of the three-dimensional competitive Atkinson/Allen model. We present all the equivalence classes relative to the boundary of the carrying simplex of the low-dimensional (two or three) map, depending upon relationship among the model coefficients. For the two-dimensional case, there are only three dynamic scenarios, and every orbit converges to some fixed point. For the three-dimensional case, there are total $33$ stable equivalence classes, and in $18$ of them all the compact limit sets are fixed points. Further, we focus on the analysis of the dynamics of the other $15$ cases. Hopf bifurcation is studied and a necessary condition for it occurring is given, which implies that the classes $19$-$25$, $28$, $30$ and $32$ do not have any Hopf bifurcation. However, the class $26$ and class $27$ do admit Hopf bifurcations, which means that these two classes may have isolated invariant closed curves in their carrying simplex, and such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous time systems. Each system in class $27$ has a heteroclinic cycle and the numerical simulation also reveals that there exist systems having May-Leonard phenomenon: the existence of nonquasiperiodic oscillation.