Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications

Abstract

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.

Heteroclinic Cycles of A Symmetric May-Leonard Competition Model with Seasonal Succession

“…These three line segments are written as Γ 12 , Γ 23 and Γ 13 , respectively. By the invariance of Σ, one can obtain that Γ 12 , Γ 23 and Γ 13 are invariant under the map S. In view of h 2 < βh 1 and h 1 > αh 2 , it follows from Niu et al [26,Theorem 3.6(ii)] that there are no interior fixed points on Γ 12 . Note that the restriction of S to the segment Γ 12 must be a monotone one-dimensional map, all orbits on Γ 12 have one fixed point as the α-limit set and the other as the ω-limit set.…”

“…By Hsu and Zhao [10, Lemma 2.1], if h i > 0 (i = 1, 2, 3), then S has three axial fixed points, written as R 1 := (x * 1 , 0, 0), R 2 := (0, x * 2 , 0) and R 3 := (0, 0, x * 3 ). Besides, there are three planar fixed points for S in three coordinate planes under appropriate conditions (see [10, Theorem 2.2-2.4] or [26,Theorem 3.6]). First, we provide the existence result of carrying simplex for system (1.2).…”

“…3), a 12 = a 23 = a 31 = α and a 32 = a 21 = a 13 = β, where b i , a ij , i, j = 1, 2, 3 are defined in [26].…”

“…Note that the restriction of S to the segment Γ 12 must be a monotone one-dimensional map, all orbits on Γ 12 have one fixed point as the α-limit set and the other as the ω-limit set. Meanwhile, Niu et al [26,Theorem 3.6(ii)] ensures that all orbits on Γ 12 take R 2 as the α-limit set and R 1 as the ω-limit set under the condition ( H). By the cyclic symmetry, we also obtain that Γ 23 and Γ 13 have similar properties as Γ 12 under the condition ( H).…”

“…Note that system (1.2) is a time-periodic Kolmogorov systems and the the Poincaré map S is C 1 -diffeomorphism onto its image (see the proof of Theorem 2.3 in Niu et al [26]), we can rewrite S as:…”