**InternationaleJournals**

**A Mathematical
Model of Four Species ****Syn-Ecosymbiosis
comprising of Prey-Predation, Mutualism and Commensalisms-II**

**(Three of the
Four Species are Washed out States)**

**R. Srilatha ^{1} and **

^{1}JNTUH, Kukatpally, Hyderabad-500085, India.

^{2}Faculty of Mathematics (Retd.),
NIT, Warangal – 506004, India. E-mail: bsrilatha82@gmail.com

**Abstract**

This investigation deals with a mathematical model of a four
species (S_{1,} S_{2,} S_{3 }and S_{4}) Syn-Ecological system (Three of the four
species are washed out states).** **S_{2
}is a predator surviving on the prey S_{1}: the prey is a
commensal to the host S_{3 }which itself is in mutualism with the
fourth species S_{4}. Further S_{2} 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: Three of the four species
are washed out states only are established in this paper. The linearized equations for the
perturbations over the equilibrium points are analyzed to establish the
criteria for stability and the trajectories illustrated.

**Key words:** Equilibrium state,
stability, Mutualism, Commensalisms

**International
eJournal of Mathematics and Engineering**

**Volume 3,
Issue ****1,** **Pages: 1322 - 1330**