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Classification of permanence and impermanence for a Lotka-Volterra model of three competing species with seasonal succession

Lei Niu, Xizhuang Xie

Abstract

In this paper, we are concerned with the permanence of a Lotka-Volterra model of three competing species with seasonal succession. Based on the existence of a carrying simplex, that is a globally attracting hypersurface of codimension one, we provide a complete classification of the permanence and impermanence in terms of inequalities on the parameters of this model. Moreover, we numerically show that invariant closed curves can occur in the permanent classes, which means that the positive fixed point of the associated Poincare map in the permanent classes is not always globally asymptotically stable.

Classification of permanence and impermanence for a Lotka-Volterra model of three competing species with seasonal succession

Abstract

In this paper, we are concerned with the permanence of a Lotka-Volterra model of three competing species with seasonal succession. Based on the existence of a carrying simplex, that is a globally attracting hypersurface of codimension one, we provide a complete classification of the permanence and impermanence in terms of inequalities on the parameters of this model. Moreover, we numerically show that invariant closed curves can occur in the permanent classes, which means that the positive fixed point of the associated Poincare map in the permanent classes is not always globally asymptotically stable.
Paper Structure (4 sections, 9 theorems, 34 equations, 3 figures)

This paper contains 4 sections, 9 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The characterisation of permanence
  4. Discussion

Key Result

Lemma 2.1

$F_i(x)>0$ for all $x\in \mathbb{R}_{+}^{3}$, $i=1,2,3$.

Figures (3)

  • Figure 1: The attracting invariant closed curve can occur in class 27.
  • Figure 2: The attracting invariant closed curve can occur in class 29.
  • Figure 3: The attracting invariant closed curve can occur in class 31.

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Definition 2.1: Hofbauer1988Kon2004Smith2010
  • Definition 2.2: hofbauer1987coexistenceHutson1982Gyllenberg2020b
  • Lemma 2.2: Lemma 2.1 in Hsu-Zhao
  • Lemma 2.3: Theorem 2.3 in Niu-Wang-Xie
  • Lemma 3.1: NWX2023
  • Proposition 3.1: NWX2023
  • Proposition 3.2
  • proof
  • ...and 8 more