An investigation of delay induced stability transition in nutrient-plankton systems

Abstract In this paper, a nutrient-plankton interaction model is proposed to explore the characteristic of plankton system in the presence of toxic phytoplankton and discrete time delay. Anti-predator efforts of phytoplankton by toxin liberation act as a prominent role on plankton dynamics. Toxicity controls the system dynamics and reduces the grazing rate of zooplankton. The toxic substance released by phytoplankton is not an instantaneous process, it requires some time for maturity. So, a discrete time delay is incorporated in the toxin liberation by the phytoplankton. The choice of functional response is important to understand the toxin liberation and it depends on the nonlinearity of the system, which follows the Monod-Haldane type functional response. Theoretically, we have studied the boundedness condition along with all the feasible equilibria analysis and stability criteria of delay free system. We have explored the local stability conditions of delayed system. The existence criterion for stability and direction of Hopf-bifurcation are also derived by using the theory of normal form and center manifold arguments. The essential features of time delay are studied by time series, phase portrait and bifurcation diagram. We perform a global sensitivity analysis to identify the important parameters of the model having a significant impact on zooplankton. Our numerical investigation reveals that the toxin liberation delay switches the stability of the system from stable to limit cycle and after a certain interval chaotic dynamics is observed. High rate of toxic substances production shows extinction of zooplankton. Further, the negative and positive impacts of other control parameters are studied. Moreover, to support the occurrence of chaos, the Poincare map is drawn and the maximum Lyapunov exponents are also computed.