On the stability of a four species syn eco-system with commensal prey-predator pair with prey-predator pair of hosts-III (2nd level prey-predator washed out states)

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 (Both the Hosts are 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 1st level Prey-Predator and the pair (S3, S4) the 2nd 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 three of these sixteen equilibrium points: 2nd  Level Prey-Predator Washed Out States are 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.