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

(The Co-Existent State)


B. Hari Prasad1, N. Ch. Pattabhi Ramacharyulu2


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

2 Former Faculty, Department. of Mathematics , NIT Warangal, India. Email: pattabhi1933@yahoo.com





This paper deals with an investigation on a Four Species Syn-Ecological System (The Co-existent State).  The System comprises of a Prey (S1), a Predator (S2) that survives upon S1, two Hosts S3 and S4 for which S1, S2 are commensal respectively i.e., S3 and S4 benefit S1 and S2 respectively, without getting effected either positively or adversely.  Further S3 and S4 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 Co-existent State only is established in this paper.  The Co-existent State is found to be conditionally stable.  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 1I, Pages 163-173