Permanence in Lotka-Volterra equations: Linked prey-predator systems
\textit{V. Hutson} and \textit{G. T. Vickers}, Math. Biosci. 63, 253-269 (1983; Zbl 0524.92023), have shown that a bistable Lotka-Volterra system of two competing species can not be stabilized by the introduction of one predator. In the present paper, two predators are introduced one feeding on one prey, the other one on the second prey. An intuitively meaningful inequality condition is established implying that the four-dimensional system is permanent provided that a fixed point exists in the interior of the positive orthant of the four-dimensional space. Interesting numerical examples show that the sufficient condition for permanence may hold even when the interior equilibrium is not stable, and conversely, a system may not be permanent in spite of the fact that the interior equilibrium is (locally) asymptotically stable. The results are extended to six-dimensional systems (three predators feeding on three preys), and to two predators two prey systems with interspecific competition between the predators. The fairly sophisticated proof of the first theorem is lucidly presented.
scientific article; zbMATH DE number 3982986
scientific article