**
InternationaleJournals**

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

**
(PREY WASHED
OUT STATES)
**

**
B. Hari Prasad ^{1}, N. Ch. Pattabhi Ramacharyulu^{2} Pages:**
324 - 339

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 (Prey 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 Prey 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:**
Asymptotic, Equillibrium point, Host, Prey, Predator, Stability, Neutrally
stable.

**
International eJournal of Mathematics and
Engineering**

**
Volume 1, Issue 1II**,
**Pages: **324 - 329