Optimal control of a nutrient-phytoplankton-zooplankton-fish system

We consider the mathematical formulation, analysis, and numerical solution of an optimal control problem for a nonlinear " nutrient- phytoplankton- zooplankton-.fish" reaction- di.ffusion system. We study the existence of optimal solutions, derive an optimality system, and determine optimal solutions. In the original spatially homogeneous formulation [ M. Scheffer, Oikos, 62 ( 1991), pp. 271 - 282] the dynamics of plankton were investigated as a function of parameters for nutrient levels and.fish predation rate on zooplankton. In our paper the model is spatially extended and the parameter for. fish predation treated as a multiplicative control variable. The model has implications for the biomanipulation of food- webs in eutrophic lakes to help improve water quality. In order to illustrate the control of irregular spatiotemporal dynamics of plankton in the model we implement a semi- implicit ( in time). finite element method with " mass lumping" and present the results of numerical experiments in two space dimensions.