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 LotkaVolterra system with a Volterra–Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in Hofbauer and Schreiber (J Math Biol 85:54, 2022) 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.

Important figure

This figure should have appeared in the published paper, but it didn’t. A mystery.

Figure : Global attractor for the example of an ecological community with three species. Given that the matrix is Volterra–Lyapunov stable, the global attractor is identical to the invasion graph. The open circle is the empty community, u∗ = (0, 0, 0), and the circle surrounded by a red ring is the GASS. The three different colors of the solutions denote the identity of each species, and the pie chart represent the relative abundance of each species in that particular stationary solution. In this case, the GASS represents a feasible community (all species are present), u∗ = (0.2633778, 0.1695335, 0.377100).

Citation

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@article{Almaraz2024,
  title = {Structural Stability of Invasion Graphs for {{Lotka}}--{{Volterra}} Systems},
  author = {Almaraz, Pablo and Kalita, Piotr and Langa, Jos\'e A and Soler--Toscano, Fernando},
  date = {2024-06-17},
  journaltitle = {Journal of Mathematical Biology},
  volume = {88},
  number = {6},
  pages = {64},
  doi = {10.1007/s00285-024-02087-8}
}