Boundedness of three-species cyclic predator-prey systems which lack some self-limiting terms

Abstract

We consider the three species cyclic predator-prey systems of Lotka-Volterra type where the first species feeds on the second, the second on the third, the third on the first, cyclically. Each of the self-limiting terms may be either present or absent. However, we impose the condition that each species remains bounded in the absence of the other two. Under these settings, we give a necessary and sufficient condition for every solution of the system to remain bounded as time passes. The main idea for the proof is to construct a family of invariant (confinement) regions, in other words, to construct a kind of Liapunoff function (non-differentiable at some points).