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

(PREDATOR WASHED OUT STATES)
 

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. 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 (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 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