Hopf Bifurcation Analysis for a Two Species Periodic Chemostat Model with Discrete Delays
Jane Ireri *
Catholic University of Eastern Africa, P.O. Box 62157-00200, Nairobi,
Ganesh Pokhariyal
University of Nairobi, P.O. Box 30197-00200, Nairobi, Kenya.
Stephene Moindi
University of Nairobi, P.O. Box 30197-00200, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
Abstract
In this paper we analyze a Chemostat model of two species competing for a single limiting nutrient input varied periodically using a Fourier series with discrete delays. To understand global aspects of the dynamics we use an extension of the Hopf bifurcation theorem, a method that rigorously establishes existence of a periodic solution. We show that the interior equilibrium point changes its stability and due to the delay parameter it undergoes a Hopf bifurcation.
Numerical results shows that coexistence is possible when delays are introduced and Fourier series produces the required seasonal variations. We also show that for small delays periodic variations of nutrients has more influence on species density variations than the delay.
Keywords: Coexistence, competition, competitive exclusion, DDE, Periodic Chemostat, Fourier series, Hopf Bifurcation, stability.