Effect of cross--diffusion on the patterns of algal bloom in a lake: A nonlinear analysis

Shukla et al. [Appl. Math. Comp. 196(2) 782-790 (2008)], have proposed and analyzed a nonlinear model for the algal bloom in a lake caused by excessive flow of nutrients from domestic drainage and water runoff from agricultural fields. The model was purely time dependent. To model the scenario more realistically, we study here a reaction-diffusion system, taking into account diffusion in two spatial dimensions. Concentration of nutrients,density of algae and density of detritus are considered as dynamical variables. Model equations are analyzed using the theory of partial differential equations and dynamical systems. The analytically obtained results show that self-diffusion cannot alter the local asymptotical stability of the uniform steady state but the unstable uniform steady state can be made stable by increasing diffusion coefficients in the case of global stability. The modified model cannot explain the patterns of algal bloom in Turing’s sense. We modify the model with taking into accountcross-diffusion. We observe that this newly modified model can explain different kinds of patterns of algal blooms.