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

We study the dynamics of the multi-species Ricker model given by the map T : R-+(n) -> R-+(n) defined by T-i(X) = X-i exp (nu(i) (1 - Sigma(n)(j=1)mu X-ij(j))), nu(i), mu(ij) > 0, i, j = 1, ..., n. It is known that under mild conditions (for small nu(i)), T admits a carrying simplex Sigma, 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 Sigma on the space of all 3D Ricker models admitting a carrying simplex. There are a total of 33 stable equivalence classes. We list them in terms of simple inequalities on the parameters nu(i) and mu(ij), and draw the phase portrait on Sigma of each class. Classes 1-25 and 33 have trivial dynamics, i.e. every orbit converges to some fixed point, and in particular, the unique positive fixed point in class 33 is globally asymptotically stable. Within each of classes 26 to 31, there exist Neimark-Sacker bifurcations, while in class 32 Neimark-Sacker bifurcations cannot occur. 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 nu(i) increases the carrying simplex will break, and chaos can occur for large nu(i). 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.