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Permanence for continuous-time competitive Kolmogorov systems via the carrying simplex

Lei Niu, Yuheng Song

Abstract

In this paper we study the permanence and impermanence for continuous-time competitive Kolmogorov systems via the carrying simplex. We first give an extension to attractors of V. Hutson's results on the existence of repellors in continuous-time dynamical systems that have found wide use in the study of permanence via average Liapunov functions. We then give a general criterion for the stability of the boundary of carrying simplex for competitive Kolmogorov systems of differential equations, which determines the permanence and impermanence of such systems. Based on the criterion, we present a complete classification of the permanence and impermanence in terms of inequalities on parameters for all three-dimensional competitive systems which have linearly determined nullclines. The results are applied to many classical models in population dynamics including the Lotka-Volterrra system, Gompertz system, Leslie-Gower system and Ricker system.

Permanence for continuous-time competitive Kolmogorov systems via the carrying simplex

Abstract

In this paper we study the permanence and impermanence for continuous-time competitive Kolmogorov systems via the carrying simplex. We first give an extension to attractors of V. Hutson's results on the existence of repellors in continuous-time dynamical systems that have found wide use in the study of permanence via average Liapunov functions. We then give a general criterion for the stability of the boundary of carrying simplex for competitive Kolmogorov systems of differential equations, which determines the permanence and impermanence of such systems. Based on the criterion, we present a complete classification of the permanence and impermanence in terms of inequalities on parameters for all three-dimensional competitive systems which have linearly determined nullclines. The results are applied to many classical models in population dynamics including the Lotka-Volterrra system, Gompertz system, Leslie-Gower system and Ricker system.
Paper Structure (8 sections, 14 theorems, 53 equations, 2 figures)

This paper contains 8 sections, 14 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Topological results on repellor and attractor
  4. Permanence for Kolmogorov systems
  5. Permanence for dissipative systems
  6. Permanence for competitive systems
  7. Applications to ecological systems
  8. Discussion

Key Result

Lemma 3.1

Let $M$ be a compact metric space and consider the continuous dynamical system $(M,\mathbb{R}_+,\varPsi)$. Assume that $S$ is a compact subset of $M$ with empty interior such that $S$ and $M\setminus S$ are positively invariant under $\varPsi$. Suppose that there is a continuous function $V: M\to \m Then $S$ is a repellor.

Figures (2)

  • Figure 1: A carrying simplex $\Sigma$ with a repelling heteroclinic cycle $\partial\Sigma$.
  • Figure 2: The dynamics on $\Sigma$ for class $29$. An equilibrium is represented by an open dot $\circ$ if it is a repellor on $\Sigma$, while if the equilibrium is a saddle on $\Sigma$, we do not use any symbol.

Theorems & Definitions (32)

  • Lemma 3.1: Criterion for repellor Hutson1984
  • Remark 3.1
  • Theorem 3.1: Criterion for attractor
  • proof
  • Definition 4.1: hofbauer1987coexistenceHutson1984Hutson1982
  • Remark 4.1
  • Theorem 4.1
  • proof
  • Corollary 4.1
  • proof
  • ...and 22 more