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Structural stability of invasion graphs for Lotka--Volterra systems

Pablo Almaraz, Piotr Kalita, José A. Langa, Fernando Soler-Toscano

Abstract

In this paper, we study in detail the structure of the global attractor for the Lotka--Volterra system with a Volterra--Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in [19] and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.

Structural stability of invasion graphs for Lotka--Volterra systems

Abstract

In this paper, we study in detail the structure of the global attractor for the Lotka--Volterra system with a Volterra--Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in [19] and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.
Paper Structure (14 sections, 15 theorems, 44 equations, 3 figures, 2 algorithms)

This paper contains 14 sections, 15 theorems, 44 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction. Invasion graphs and ecological assembly
  2. Lotka--Volterra systems and their global attractors.
  3. Equilibria and the local dynamics
  4. Linearization and its properties.
  5. Invasion graphs and information structures
  6. Invasion rates and invasion graphs.
  7. Finding the connections between equilibria
  8. Structural stability of invasion graphs
  9. Local structural stability
  10. The regions of structural stability.
  11. Appendix A. The dynamical system generated by \ref{['lv']} is not Morse--Smale.
  12. Appendix B. Questions and open problems
  13. Wider classes of stable matrices
  14. Symmetric case

Key Result

Theorem 5

If $A$ is Volterra--Lyapunov stable then for every $b\in \mathbb{R}^n$ there exists a unique equilibrium $u^*\in \overline{C}_+$ of lv which is globally asymptotically stable in the sense that for every $u_0\in C_+^{J(u^*)}$ the solution $u(t)$ of lv with the initial data $u_0$ converges to $u^*$ as

Figures (3)

  • Figure 1: Information Structure for the problem given in Example \ref{['ex:2']}.
  • Figure 2: The Invasion Graph for the system \ref{['eq:may']}
  • Figure 3: Invasion graph for the example given by \ref{['exmorse']} for an ecological community with three species. The GASS represents a feasible community (all species are present), $u^* = (0.2633778, 0.1695335, 0.377100)$. While the connections $\{3\}\to \{1,3\}$ and $\{1,3\}\to \{1,2,3\}$ are present in the graph, there is no connection $\{3\}\to \{1,2,3\}$

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5: Takeuchi, Theorem 3.2.1
  • Corollary 6
  • Definition 7
  • Theorem 8
  • proof
  • Definition 9
  • ...and 32 more