Figure 7: Interactions incorporated in a fish-nutrients-plankton model
Arrows indicate positive effects, circles indicate negative effects (redrawn
and modified from OIKOS Vol. 62, pp.271-282, 1991).
Figure 8: Snapshot at T = 100
Approximate 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.
I am collaborating with a Postdoctoral Research
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.