A Mathematical model of four species syn-ecosymbiosis comprising of prey-predation, mutualism and commensalisms-V(the co-existent state)

Abstract

This investigation deals with a mathematical model of a four species (S1, S2, S3 and S4) Syn-Ecological system (The Co-existent State). S2 is a predator surviving on the prey S1: the prey is a commensal to the host S3 which itself is in mutualism with the fourth species S4; S2 and S4 are neutral. The model equations of the system constitute a set of four first order non-linear ordinary differential coupled equations.  In all, there are sixteen equilibrium points.  Criteria for the asymptotic stability of one of the sixteen equilibrium points: The Co-existent State only is established in this paper.  The Co-existent State is found to be stable.  The linearized equations for the perturbations over the equilibrium points are analyzed to establish the criteria for stability and the trajectories illustrated. Further the global stability is discussed using Liapunov’s method.