The complete classification on a model of two species competition with an inhibitor (Q928456)

The authors investigate a three-dimensional quadratic vector field that represents a two-species Lotka-Volterra competition model in which one of the species produces a toxin that attacks the second one. Answering questions raised in a previous paper by \textit{G. Hetzer} and \textit{W. Shen} [Discrete Contin. Dyn. Syst. 12, No. 1, 39--57 (2005; Zbl 1073.34053)], they prove that the system has neither periodic orbits nor cycles connecting equilibria in the open positive octant \(P\), and, moreover, if there are two equilibria in \(P\), then both of them are hyperbolic. Further results yield a complete phase-portrait of the flow depending on the six essential parameters, including also the two nontrivial equilibria on the boundary of \(P\), and the cases when there exists at most one equilibrium in \(P\). These results particularly imply that each orbit tends to an equilibrium. The proofs crucially depend on ordering and monotonicity properties of the given system. Three numerical examples illustrate the theoretical results and, in particular, the role of the toxin.