Figure 7: Interactions incorporated in a fish-nutrients-plankton modelArrows indicate positive effects, circles indicate negative effects (redrawn and modified from OIKOS Vol. 62, pp.271-282, 1991).
Figure 8: Snapshot at T = 100Approximate phytoplankton densities for the controlled and uncontrolled problem evolving from an initial chaotic state, with the target and control 'f' (bounding boxes are [0,100]2). A uniform 101-by-101 grid was used with a constant time-step of 1/12. The standard ('lumped mass') Galerkin FEM was employed to solve the state equations and the adjoint equations, with zero flux boundary conditions. A 'steepest descent' algorithms was employed to update the control. |
Fish-nutrient-plankton dynamicsI am collaborating with a Postdoctoral Research Associate, Catalin Trenchea (Florida State University), on the optimal control of a system that is related to the predator-prey models discussed above (a nonlinear reaction-diffusion system). The predator-prey model is modified to account for the predation rate 'f' (the control variable) of fish on zooplankton 'v'. The kinetics are given by
where 'u' represents the phytoplankton and the parameters 'r','a','b', 'm', and 'g' are positive (see Figure 7). The positive control variable 'f' is a function of space and time and represents the biomass of zooplankton consumed at each small volume of water per unit time (average fish predation rate x density of fish). The model was originally formulated as an ordinary differential system by Scheffer (OIKOS Vol. 62, pp.271-282), and has since been generalised to include diffusion. The basic aim here is to enhance zooplankton, by reducing planktivorous fish, thereby, reducing algal biomass. In practice, planktivorous fish can be reduced by fish removal, or by piscivore stocking. This manipulation of the food-web is called biomanipulation, and is an important approach for improving water quality in eutrophic lakes. From a mathematical point of view it is important to choose a cost functional that can be analysed, but is still relevant to the field situation. We consider the problem of manipulating the densities of plankton by minimizing the quadratic functional 
where the desired densities of phytoplankton and zooplankton are denoted with a 'bar' over u and v, and Q represents the space - time domain of interest. In Figure 8 we show some numerical results for the optimal control problem that illustrates the ability of the control to drive the system from a chaotic regime to an ordered one (a rotating spiral wave). Due to quota constraints I cannot show an animation for t=0 to t=100. |
Last updated: May 08, 2006 by Marcus R Garvie