Discrete Time Stability Analysis of a Three Species Eco System

In this paper, a three species eco system, involving three pairs is considered to examine the local asymptotic stability: A Prey-Predator, a Commensal-Host and an Ammensal-Enemy. Among the three species, one plays a dual role: A host and an enemy. Time is considered as a discrete unit and the system is modeled as a set of three difference equations. All the equilibrium states are identified and the local asymptotic stability of some of the equilibrium states is examined by considering the perturbation equations. It is observed that among the states, the state in which the Prey and its Host species are washed out(extinct), is spectrally stable and the state where the Predator/Ammensal species is washed out, is asymptotically stable. The results are illustrated with two dimensional plots as well as surface plots.