Stability Analysis of a Three Species Syn-Eco-System with Mortality Rate for Commensal

The main objective of the present paper is to examine the stability of three species syn-eco-system. The system comprises of a commensal (S1) common to two hosts S2 and S3 with mortality rate for commensal. Further S2 is a commensal of S3 and S3 is a host of both S1, S2. Here all three species are having limited resources. The mathematical model equations constitute a set of three first order non- linear simultaneous coupled differential equations in the strengths N1, N2, N3 of S1, S2, S3 respectively. Criteria for the asymptotic stability of all the eight equilibrium points are established. Trajectories of the perturbations over the equilibrium points are illustrated. Further the global stability of a three species syn-eco system is established with the aid of suitably constructed Liapunovs function-pair. Also the growth rates of the species are numerically estimated using Runge-Kutta fourth order scheme.