**
InternationaleJournals**

**THE
STABILITY OF A FOUR SPECIES: A PREY-PREDATOR-HOST-COMMENSAL-SYN ECO-SYSTEM-IV**

**(Both the
Hosts are washed out states)**

**
B. Hari Prasad ^{1}, N. Ch. Pattabhi Ramacharyulu^{2}**

Department. of Mathematics, Chaitanya Degree & P.G. College (Autonomous), Hanamkonda, Warangal A.P, India.^{1}Email :sumathi_prasad73@yahoo.com

Former Faculty, Department. of Mathematics , NIT Warangal, India. E-mail: pattabhi1933@yahoo.com^{2}

**ABSTRACT**

This paper deals with an investigation on a Four Species Syn-Ecological System
(Both the Hosts are washed out states). 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} and S_{4} are neutral. 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 the sixteen equilibrium points: Both the Hosts
are washed out states only are established in this paper. 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 illustrated.

**
International eJournal of Mathematics and Engineering**

**
Volume
1, Issue ****2**,
**Pages:** 277 - 292