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Abstract

The present paper is devoted to an investigation on a Four Species (S₁, S₂, S₃, S₄) Syn Eco-System with Commensal Prey-Predator pair with Prey-Predator pair of Hosts (Host of S₁ 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 is Prey for S₄ and S₄ is Predator for S₃. The pair (S₁, S₂) may be referred as 1^st level Prey-Predator and the pair (S₃, S₄) the 2^nd level Prey-Predator. 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 S₁ washed out states is established. The system would be stable if all the characteristic roots are negative, in case they are real, and have negative real parts, in case they are complex. The linearized equations for the perturbations over the equilibrium points are analyzed to establish the criteria for stability and the trajectories are illustrated.