**InternationaleJournals**

**On The Stability Of A Four Species: A Prey-Predator-Host- Commensal-Mutual-Syn Eco-System-Iv (Prey Washed Out State)**

**N. Shanker ^{1} K. Lakshmi Narayan^{2} and N.Ch.Pattabhi Ramacharyulu**

^{1} Department. of Mathematics, C.M.R. College of Engineering & Technology, Secunderabad, Andhra Pradesh - 501 401, India.

Email: shankermaths@yahoo.co.in

^{2} Department of Mathematics, SLC’s Institute of Engineering & Technology, Hyderabad-501 512, India.

^{3} Former Faculty, Dept of Mathematics, National Institute of Technology, Warangal-506004, India.

**Abstract:**

This paper deals with an investigation on a four Species Syn-Ecological System (Prey Washed out State). 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 mutual. 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 the sixteen equilibrium points: the prey washed out 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 linearised equations for the perturbations over the equilibrium point are analyzed to establish the criteria for stability and the trajectories illustrated.

**Keywords:** Commensal, Eco-system, Equilibrium points, Host, Mutual, Prey, Predator, Stability.

**International
eJournal of Mathematics and Engineering**

Volume 2, Issue 3, Pages: 1082 - 1090