Successful biocontrol with stability and rhythms of plankton dynamics in biogeophysical ecology

Abstract

A system of retarded functional differential equations is proposed as a nutrient-based model of phytoplankton-zooplankton interaction with a delayed nutrient recycling. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, asymptotic stability, length of delay preserving stability and Hopf-bifurcation are analyzed. We have shown that the equilibrium is locally asymptotically stable when time delay $\tau $ is suitable small, while a loss of stability by a Hopf-bifurcation can occour as the delay increases. Numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.