Global stability of a tridiagonal competition model with seasonal succession.

In this paper, we investigate a tridiagonal three-species competition model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. Together with the Brouwer degree theory, sufficient conditions for existence and uniqueness of the positive periodic solution are given. We further obtain the global dynamics of coexistence and extinction for three competing species in this periodically forced environment. Finally, some numerical examples are presented to illustrate the effectiveness of our theoretical results.

Chaotic attractors in Atkinson-Allen model of four competing species.

We study the occurrence of chaos in the Atkinson-Allen model of four competing species, which plays the role as a discrete-time Lotka-Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson-Allen dynamics can lead to chaos.

The balance simplex in non-competitive 2-species scaled Lotka-Volterra systems.

Explicit expressions in terms of Gaussian Hypergeometric functions are found for a 'balance' manifold that connects the non-zero steady states of a 2-species, non-competitive, scaled Lotka-Volterra system by the unique heteroclinic orbits. In this model, the parameters are the interspecific interaction coefficients which affects the form of the solution used. Similar to the carrying simplex of the competitive model, this balance simplex is the common boundary of the basin of repulsion of the origin and infinity, and is smooth except possibly at steady states.

On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex.

We study the asymptotic behavior of the competitive Leslie/Gower model (map) [Formula: see text]It is shown that T unconditionally admits a globally attracting 1-codimensional invariant hypersurface [Formula: see text], called carrying simplex, such that every nontrivial orbit is asymptotic to one in [Formula: see text]. More general and easily checked conditions to guarantee the existence of carrying simplex for competitive maps are provided. An equivalence relation is defined relative to local stability of fixed points on [Formula: see text] (the boundary of [Formula: see text]) on the space of all three-dimensional Leslie/Gower models. Using a formula on the sum of the indices of all fixed points on the carrying simplex for three-dimensional maps, we list the 33 stable equivalence classes in terms of simple inequalities on the parameters [Formula: see text] and [Formula: see text] and draw their orbits on [Formula: see text]. In classes 1-18, every nontrivial orbit tends to a fixed point on [Formula: see text]. In classes 19-25, each map possesses a unique positive fixed point which is a saddle on [Formula: see text], and hence Neimark-Sacker bifurcations do not occur. Neimark-Sacker bifurcation does occur within each of classes 26-31, while it does not occur in class 32. Each map from class 27 admits a heteroclinic cycle, which forms the boundary of [Formula: see text]. The criteria on the stability of heteroclinic cycles are also given. This classification makes it possible to further investigate various dynamical properties in respective class.

On heteroclinic cycles of competitive maps via carrying simplices.

We concentrate on the effects of heteroclinic cycles and the interplay of heteroclinic attractors or repellers on the boundary of the carrying simplices for low-dimensional discrete-time competitive systems. Based on the existence of the carrying simplex for the competitive mapping, we provide the criteria on stability of the heteroclinic cycle. This result can be seen as a discrete counterpart of that for the continuous-time systems. Several concrete discrete-time competition models are further analyzed, which do admit heteroclinic cycles. The criteria on the stability of the heteroclinic cycle for each model are also given, which are comparable with the corresponding continuous-time models.