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On The Stability of a Four Species: A Prey-Predator-Host-Commensal-Syn Eco-System-VIII(Host of the Predator Washed Out States)

B. Hari Prasad and N.Ch. Pattabhi Ramacharyulu

This paper deals with an investigation on a Four Species Syn-Ecological System (Host of the Predator washed out states). The System comprises of a Prey (S1), a Predator (S2) that survives upon S1, two Hosts S3 and S4 for which S1, S2 are commensal respectively i.e., S3 and S4 benefit S1 and S2 respectively, without getting effected either positively or adversely. Further S3 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 four of these sixteen equilibrium points: Host of the Predator washed out states only are established in this paper. The linearized equations for the perturbations over the equilibrium points are analyzed to establish the criteria for stability and the trajectories are illustrated.