On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex

Abstract

We propose the generalized competitive Atkinson-Allen map \begin{document}$T_i(x)=\frac{(1+r_i)(1-c_i)x_i}{1+\sum_{j=1}^nb_{ij}x_j}+c_ix_i, 0 0, i, j=1, ···, n, $ \end{document} which is the classical Atkson-Allen map when \begin{document}$r_i=1$\end{document} and \begin{document}$c_i=c$\end{document} for all \begin{document}$i=1, ..., n$\end{document} and a discretized system of the competitive Lotka-Volterra equations. It is proved that every \begin{document}$n$\end{document} -dimensional map \begin{document}$T$\end{document} of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map differs from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have not yet been found for all other three-dimensional competitive mappings via the carrying simplex. In one class every map admits a heteroclinic cycle; we provide a stability criterion for heteroclinic cycles. Besides, the generalized Atkinson-Allen model is not dynamically consistent with the Lotka-Volterra system.