Abstract
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.
Highlights
The dynamical theory of population biology has started to take a formal shape after the pioneering work of Jorgensen [19]
It is quite difficult to find out the condition for which the inequality in (5.12) will be satisfied for all real s and the stability condition independent of length of discrete time delay cannot be obtained for the present model system
In this paper we have considered the dynamical behaviour of a homogeneous model of detritus-based ecosystem comprised with detritivores and predator of detritivores within deterministic environment
Summary
The dynamical theory of population biology has started to take a formal shape after the pioneering work of Jorgensen [19]. Some algal species are always found in these sand flats and mud They always contribute some amount of detritus which is formed by the action of decomposers on dead bodies of these species. The supralittoral zones of this ecosystem are occupied by large number of mangrove plants and the plant leaves are chief source of detritus. In this food chain some 10 percent or less of net primary production is grazed and 90 percent passes through decomposers [8]. The consumption of detritus by micro-organism biomass (detritivores) is assumed to follow Holling type-II functional response. We have derived the condition for the existence of Hopf-bifurcating periodic solutions by considering the time delay as bifurcation parameter
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