Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity

The present paper deals with a nutrient-plankton model in an aquatic environment in the context of phytoplankton bloom. Toxin producing phytoplanktons are assumed to play the key role. To account for the toxication effect of the allelopathic phytoplankton, a type-IV functional response is used to model zooplankton grazing. The main aim of the study is to analyze the role, zooplankton grazing plays, in determining the dynamics of the system. Stability and bifurcation behavior of the different equilibrium points of the ODE model is studied. It has been observed that under certain parameter restrictions, extinction of one or both plankton population can occur. The model system is seen to exhibit oscillatory behavior around the equilibrium point of co-existence for a critical value of zooplankton grazing rate. A spatial extension of the model is considered and analyzed for diffusion-driven instability as well as stability. For the diffusive model also, it is found that diffusion-driven instability takes place when the grazing rate of zooplankton lies within an interval. The analysis also revealed that for sufficiently low diffusivity of phytoplankton and high value of half-saturation constant in zooplankton predation function, the diffusive system exhibit asymptotically stable behavior which signifies disappearance of the phenomenon of bloom formation for the diffusive model. Cross-diffusion of zooplankton under the influence of allelopathic phytoplankton is incorporated into the spatial model. Using matrix stability approach, the cross-diffusive model is analyzed and criteria for Turing bifurcation derived. Numerical simulations are performed to justify analytical findings.