**On The
Stability of A Four Species: A Prey-Predator-Host-Commensal-Syn Eco-System-I **

**(Fully Washed
Out State)**

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

** ^{1} **Department. of Mathematics,

Hanamkonda,

** ^{2} **Former Faculty, Department. of Mathematics , NIT Warangal,

**Abstract**

This paper
deals with an investigation on A Four Species Syn-Ecological
System (Fully Washed Out State). The System comprises of a prey (S_{1}),
a predator (S_{2}) that survives upton
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 one of the sixteen equilibrium points
: the fully 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 linearized equations for the perturbations
over the equilibrium point are analyzed to establish the criteria for stability
and the trajectories illustrated.

**International eJournal of
Mathematics and Engineering**

Volume 1, Issue 2, Pages 122-132