Modeling the depletion of dissolved oxygen in a lake due to algal bloom: Effect of time delay

We propose and analyze a non-linear mathematical model for algal bloom in a lake to account for the delay in conversion of detritus into nutrients. It is assumed that there is a continuous inflow of nutrients in the lake due to agricultural run off. The model involves four variables, namely nutrient concentration, algal population density, detritus density and dissolved oxygen concentration. The dynamics of the model is studied in terms of local stability analysis and Hopf-bifurcation analysis. It is found that the positive equilibrium of the model may switch from stability to instability to stability, and eventually instability sets in under certain conditions. The numerical simulation is performed to support the analytical results