On classification of a 4D competitive LV system

Wu, Wenxi; Jiang, Jifa

doi:10.1090/proc/16601

On classification of a 4D competitive LV system
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by Wenxi Wu and Jifa Jiang;

Proc. Amer. Math. Soc. 152 (2024), 1983-1997

DOI: https://doi.org/10.1090/proc/16601

Published electronically: March 1, 2024

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Abstract:

This paper classifies the global dynamics of a 4D competitive Lotka-Volterra system (1.3) with two positive parameters $k_1,\ k_2$ via carrying simplex. It is proved that the interior of the carrying simplex is filled with periodic orbits except equilibria and each interior trajectory is persistent and tends to either a periodic orbit or an equilibrium if $k_1/k_2=2$. Otherwise, the system admits two 2D carrying simplices $\Delta _i$ on $x_1=i$ for $i=0,\ 1$, which are filled with periodic orbits surrounding an equilibrium. All interior orbits go in the long run to $\Delta _1$ if $k_1/k_2>2$. If $0<k_1/k_2<2$, then the system admits a bistable structure: $\Delta _0$ and the equilibrium $R_1(1+k_2/k_1,\ 0,\ 0,\ 0)$ are locally asymptotically stable, the attracting set of $\Delta _1$ separates the attracting basins of $\Delta _0$ and $R_1$.

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