We investigate the dynamics of the Poincaré-map for an n-dimensional Lotka-Volterra competitive model with seasonal succession. It is proved that there exists an (n − 1)-dimensional carrying simplex Σ which attracts every nontrivial orbit in R n + . By using the theory of the carrying simplex, we simplify the approach for the complete classification of global dynamics for the two-dimensional Lotka-Volterra competitive model with seasonal succession proposed in [Hsu and Zhao, J. Math. Biology 64(2012), 109-130]. Our approach avoids the complicated estimates for the Floquet multipliers of the positive periodic solutions.

In this paper, we focus on a three-species Lotka-Volterra cooperation 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. By Brouwer fixed point theorem and the connecting orbits theorem, it is proved that there admits a unique positive periodic solution under appropriate conditions. Furthermore, sharp global asymptotical stability criteria for extinction and coexistence are established. Compared to the classical three-species Lotka-Volterra cooperation model, the introduction of seasonal succession may lead to species' extinction. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.

In this paper, we are concerned with the stability of heteroclinic cycles of the symmetric May-Leonard competition model with seasonal succession. Sufficient conditions for stability of heteroclinic cycles are obtained. Meanwhile, we present the explicit expression of the carrying simplex in a special case. By taking ϕ as the switching strategy parameter for a given example, the bifurcation of stability of the heteroclinic cycle is investigated. We also find that there are rich dynamics (including nontrivial periodic solutions, invariant closed curves and heteroclinic cycles) for such a new system via numerical simulation. Biologically, the result is interesting as a caricature of the complexities that seasonal succession can introduce into the classical May-Leonard competitive model.