**
InternationaleJournals**

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

**
(PREDATOR 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

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

**ABSTRACT**

This paper deals with an investigation on a Four Species Syn-Ecological System
(Predator 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 four of the sixteen equilibrium points : the Predator
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.

**Key words:** Equillibrium
point, Host, Prey, Predator, Unstable.

**
International eJournal of Mathematics and
Engineering**

**
Volume 1, Issue 1II**,
**Pages: **398 - 410