Strong stability with impact of maturation delay and diffusion on a toxin producing phytoplankton–zooplankton model

Abstract

Plankton species is backbone for any healthy aquatic system, but a global increment in toxin phytoplankton blooms is a major threat to aquatic systems. As toxic phytoplankton Microcystis aeruginosa becomes too high, Artemia salina a zooplankton species avoids to consume it and more inclines towards a non-toxic phytoplankton Chaetoceros gracilis for food. Considering this fact, additional food is provided to zooplankton species to survive when non-toxic species is insufficient for the zooplankton and toxin phytoplanktons are harvested linearly to control toxicity level in the system. Also, increasing of phytoplankton may increase competitive behaviour between toxic and non-toxic phytoplanktons due to limited resources. Keeping these in mind, we propose a three populations reaction–diffusion delay model. In this study, existence of feasible equilibria has been studied. Conditions for Turing instability of delayed and non-delayed system have been discussed analytically. Strong stability analysis with respect to both delay and diffusion is performed and excitable nature through Hopf bifurcation is also examined. It has been investigated that if the delay spatial system shows diffusion driven instability, then it is also shown by the delay free system for the same parameters. The study explores that the model depicts strong stability with respect to both delay and diffusion. Movements of the species are responsible for the occurrence of planktonic bloom via Hopf bifurcation due to diffusion and affect the critical value of the time delay. This study suggests that a specific amount of harvesting ensures the long-term sustainability. It is also inspected that considerable amount of alternative food supply supports the zooplankton for its growth. Turing patterns like spots are observed. The behaviour of the system depends on the both variation of delay and choice of initial population. This study says that maturation delay stabilizes the system more quickly in the presence of diffusion than in the absence of diffusion.