International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 25619, 28 pages

Effect of time delay on a detritus-based ecosystem

Nurul Huda Gazi1 and Malay Bandyopadhyay2

1Department of Applied Mathematics, University of Calcutta, 92 APC Road, Kolkata 700 009, India
2Department of Mathematics, Scottish Church College, Kolkata 700 006, India

Received 22 April 2005; Revised 15 July 2006; Accepted 18 July 2006

Copyright © 2006 Nurul Huda Gazi and Malay Bandyopadhyay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Models of detritus-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with the dynamical analysis of a nonlinear model of a detritus-based ecosystem involving detritivores and predator of detritivores. We have obtained the criteria for local stability of various equilibrium points and persistence of the model system. Next, we have introduced discrete time delay due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. With delay differential equation model system we have studied the effect of time delay on the stability behaviour. Next, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Finally, the existence of Hopf-bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter.