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

Lin Niu; Yi Wang; Xizhuang Xie

doi:10.3934/dcdsb.2021014

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2021, Volume 26, Issue 4: 2161-2172. Doi: 10.3934/dcdsb.2021014

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

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China
2.
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, China

^* Corresponding author: Xizhuang Xie
^* Corresponding author: Xizhuang Xie

Received: October 2020

Revised: December 2020

Early access: December 2020

Published: April 2021

This work is supported by NSF of China No. 11825106, 11871231 and 11771414, CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China.

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Abstract

We investigate the dynamics of the Poincar$ \acute{\rm e} $-map for an $ n $-dimensional Lotka-Volterra competitive model with seasonal succession. It is proved that there exists an $ (n-1) $-dimensional carrying simplex $ \Sigma $ which attracts every nontrivial orbit in $ \mathbb{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.

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Mathematics Subject Classification: Primary: 34C25, 37C65; Secondary: 92D25.

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