On the Stability of a Four Species Syn Eco-System with Commensal Prey-Predator Pair with Prey-Predator Pair of Hosts-VI (Normal Steady State)

Abstract

The present paper is devoted to an investigation on a Four Species $(S_1, S_2, S_3, S_4)$ Syn Eco-System with Commensal Prey-Predator pair with Prey-Predator pair of Hosts (Normal Steady state). The System comprises of a Prey $(S_1),$ a Predator $(S_2)$ that survives upon $S_1,$ two Hosts $S_3$ and $S_4$ for which $S_1, S-2$ are Commensal respectively i.e., $S_3$ and $S_4$ benefit $S_1 and $S_2$ respectively, without getting effected either positively or adversely. Further $S_3$ is Prey for $S_4$ and $S_4$ is Predator for $S_3.$ The pair $(S_1, S_2)$ may be referred as 1st level Prey-Predator and the pair $(S_3, S_4)$ 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 one of these sixteen equilibrium points : Normal Steady State 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. Keywords: Commensal; Eco-System; Equillibrium point; Host; Prey; Predator; Quasi-linearization; Stable; Trajectories. 2010 Mathematics Subject Classification: 92D25; 92D40.