CHEMOSTAT EQUATIONS FOR A PREDATOR-PREY CHAIN WITH DELAYED NUTRIENT RECYCLING

Abstract

The problem of the equilibrium stability for the chemostat equation is considered in the special case of a biotic species feeding on one limiting nutrient and predated by another biotic species. Both the biotic species through the decomposition process can return with delay to the chemostat fraction of dead biomass as new nutrient. The delay kernels of nutrient recycling are assumed to be general L2(0, +∞) non-negative functions which admit up to second order finite moments. Two approaches are adopted: the first one applies when both the biotic species have a self-regulating term in their evolution equations and can be worked out without linearizing the equations. This approach does not require any further constraint on delay kernels, whereas it introduces constraints on the remaining parameters of the model. The second approach applies to the linearized equations when the predator self-regulating term is set equal to zero. In this case the stability condition requires a constraint on the average time delays in the recycling processes. Both the approaches are performed by constructing suitable Krasovskii-Lyapunov functionals for the related functional retarded differential equations.