Great time delay in a system with material cycling and delayed biomass growth

Abstract

In this paper, we consider the effects of time delay on the linear stability of the positive equilibrium of a reaction-diffusion model with time delayed nutrient recycling. We take the two delay kernels as f(s) = δ(s), g(s) = pδ(0) + (1-p)δ(r), s ∈ (0, +∞), r ∈ [0, +∞), 0 < p < 1, where δ is the Dirac delta function. When both N 1 and N 2 are space-independent, we can observe that the increasing time delay destabilizes the positive equilibrium, and stability-switches from stability to instability to stability occur. That is, for some special time delay kernels, the equilibrium is stable for small time delay, bifurcates towards stable oscillations when the time delay is increased, then regains stability for larger values. When both N 1 and N 2 are space-dependent, we are concerned with the effects of the delay kernels as well as diffusion on the dynamics of the system.