Figure 1: Snapshot at T = 150
Numerical solution of spiral waves for phytoplankton on [0,400]^{2} in a
phytoplanktonzooplankton reactiondiffusion model.
The standard Galerkin FEM using continuous piecewise linear basis
functions was employed with zero flux boundary conditions. Solved on a uniform
401by401 grid with a fixed timestep of 1/384.
Figure 2: Snapshot at T = 1000
Numerical solution of spatiotemporal 'chaos' for phytoplankton on [0,400]^{2}
in a phytoplanktonzooplankton reactiondiffusion model. The standard Galerkin FEM
using continuous piecewise linear basis functions was employed with zero flux boundary conditions.
Solved on a uniform 401by401 grid with a fixed timestep of 1/384.

Spatiallyextended predatorprey models
My recent work involved the mathematical and numerical study of computational algorithms (finite
element/difference) for nonlinear reactiondiffusion systems modeling
predatorprey interactions. I have been focusing on a welldocumented class of predatorprey models
with the Holling Type II functional response, which in the absense of predators has
logistic growth of the prey.
We have proved for the first time the rigorous wellposedness
of these nonlinear systems and rigorously proved stability and convergence results
for two fullypractical finite element approximations of these models.
For further details see my papers and for some free Matlab software
that I have written see my webpage below:
Our results cover an important example recently reviewed
in the context of marine plankton dynamics (SIAM Review, Vol.44, No.3, p.311).
The model reactiondiffusion system has the form
where 'u(x,t)' represents the phytoplankton density, 'v(x,t)' is the zooplankton
density, and the parameters 'alpha', 'beta', and 'gamma' are positive. In our
simulations we use the zeroflux boundary conditions, which reflects our
assumption that the plankton cannot leave the boundary of interest.
For various initial conditions, our computer simulations in 2D reveal that
the evolution of this system can lead to the formation of spiral patterns (see Figure 1),
followed by irregular 'patchy' structures in the whole domain, namely
spatiotemporal 'chaos' (see Figure 2). The animation of these dynamics for T=0 to T=1000 is
very interesting, but unfortunately due to quota constraints I cannot show them here. For the
same reason I've had to remove the link to my webpage for making independently playable movies
with Matlab.
The results of the simulations have important implications for
understanding the role of interspecific interactions in
the observed patchy distribution of plankton in marine
environments. In terrestrial environments spatiotemporal patterns
resembling periodic traveling waves have been observed recently in several natural
populations, for example, field voles and red grouse. The reactiondiffusion systems
are useful models for investigating possible mechanisms for this behavior (see Figure 3).
