On the dynamics of multi-species Ricker models admitting a carrying simplex

We study the dynamics of the Ricker model (map) T. It is known that under mild conditions, T admits a carrying simplex
, 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 3D Ricker models admitting carrying simplices.
There are a total of 33 stable equivalence classes. We list them in terms of simple inequalities on the parameters, and draw each one's phase portrait on Σ. Classes 1-18 have trivial dynamics, i.e. every orbit converges to some fixed
point. Each map from classes 19-25 admits a unique positive fixed point with index -1, and Neimark-Sacker bifurcations do not occur in these 7 classes. In classes 26-33, there exists a unique positive fixed point with index 1. Within each of classes 26 to 31, there do exist Neimark-Sacker bifurcations, while in class 32 Neimark-Sacker bifurcations can not occur. Whether there is a Neimark-Sacker bifurcation in class 33 or not is still an open problem. Class 29 can admit Chenciner bifurcations, so two isolated closed invariant curves can coexist on the carrying simplex in this class. Each map in class 27 admits a heteroclinic cycle, i.e. a cyclic arrangement of saddle fixed points and heteroclinic connections. As the growth rate increases the carrying simplex will break, and chaos can occur for large growth rate. We also numerically show that the 4D Ricker map can admit a carrying simplex containing a chaotic attractor, which is found in competitive mappings for the first time.