Stability analysis of mathematical Syn-Ecological model comprising of prey-predator, host-commensal, mutualism and neutral pairs-III (Two of the four species are washed out states)

This investigation deals with a mathematical model of a four species (S₁, S₂, S₃ and S₄) Syn-Ecological system (Two of the four species are washed out states). S₂ is a predator surviving on the prey S₁. The predator S₂ is a commensal to the host S₃. The pairs S₂ and S₄, S₁ and S₃ are neutral. The mathematical model equations characterizing the syn-ecosystem constitute a set of four first order non-linear coupled differential equations. There are in all sixteen equilibrium points. Criteria for the asymptotic stability of six of the sixteen equilibrium points: Two of the four species are 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 illustrated.