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

  • Pankaj Kumar Tiwari Department of Mathematics, Faculty of Science, Banaras Hindu University Varanasi - 221 005, India
  • Sourav Rana Agricultural and Ecological Research Unit, Indian Statistical Institute 203, B. T. Road , Kolkata 700108, India
  • Arvind Kumar Misra Department of Mathematics, Faculty of Science, Banaras Hindu University Varanasi - 221 005, India
  • Joydev Chattopadhyay Agricultural and Ecological Research Unit Indian Statistical Institute, 203 B. T. Road Kol-108, India

Abstract

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 account
cross-diffusion. We observe that this newly modified model can explain different kinds of patterns of algal blooms.
Published
2014-08-27
Section
Articles