Stability of a three species ecological system consisting of prey- predator species and a third species which is a host to the prey and enemy to the predator

Abstract

Prey-Predator ecological system was presented by Lotka and Volterra in their classical model. Inspired by that, several researchers made significant contributions in this area by considering various special types of interactions between the prey and the predator. This has been the motivation for others in bringing a third species into the system thus forming a three species ecological system. Recently, some researchers worked on this three species system by considering interactions like Prey-Predator, Commensal -Host, Ammensal-Enemy, between the three species, which motivated the present paper. In this paper we discussed a three species ecological system consisting of a Prey (S1), a Predator (S2) and a third species (S3) which is a host to the prey and enemy to the predator.  Hence the prey plays Commensal for the third species and the predator plays Ammensal for the same. The mathematical model consists of three equations which constitute a set of three first order non-linear simultaneous differential equations in N1, N2 and N3, which are respective populations of the species S1, S2 and S3.The Equation for the third species is non-linear but decoupled with the prey-predator pair. Totally, eight equilibrium points of the model are identified and the criteria for their local stability are discussed. Solutions for the linearized perturbed equations are found and the results are illustrated.