On classification of a 4D competitive LV system

Abstract

This paper classifies the global dynamics of a 4D competitive Lotka-Volterra system (1.3) with two positive parameters k 1 , k 2 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 k_1/k_2=2 . Otherwise, the system admits two 2D carrying simplices Δ i \Delta _i on x 1 = i x_1=i for i = 0 , 1 i=0,\ 1 , which are filled with periodic orbits surrounding an equilibrium. All interior orbits go in the long run to Δ 1 \Delta _1 if k 1 / k 2 > 2 k_1/k_2>2 . If 0 > k 1 / k 2 > 2 0>k_1/k_2>2 , then the system admits a bistable structure: Δ 0 \Delta _0 and the equilibrium R 1 ( 1 + k 2 / k 1 , 0 , 0 , 0 ) R_1(1+k_2/k_1,\ 0,\ 0,\ 0) are locally asymptotically stable, the attracting set of Δ 1 \Delta _1 separates the attracting basins of Δ 0 \Delta _0 and R 1 R_1 .