Figure 1: Snapshot at T = 150
Numerical solution of spiral waves for phytoplankton on [0,400]2 in a
phytoplankton-zooplankton reaction-diffusion model.
The standard Galerkin FEM using continuous piecewise linear basis
functions was employed with zero flux boundary conditions. Solved on a uniform
401-by-401 grid with a fixed time-step of 1/384.
Figure 2: Snapshot at T = 1000
Numerical solution of spatiotemporal 'chaos' for phytoplankton on [0,400]2
in a phytoplankton-zooplankton reaction-diffusion model. The standard Galerkin FEM
using continuous piecewise linear basis functions was employed with zero flux boundary conditions.
Solved on a uniform 401-by-401 grid with a fixed time-step of 1/384.
Spatially-extended predator-prey models
My recent work involved the mathematical and numerical study of computational algorithms (finite
element/difference) for nonlinear reaction-diffusion systems modeling
predator-prey interactions. I have been focusing on a well-documented class of predator-prey 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 well-posedness
of these nonlinear systems and rigorously proved stability and convergence results
for two fully-practical finite element approximations of these models.
For further details see my papers and for some free Matlab software
that I have written see my web-page 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 reaction-diffusion 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 zero-flux boundary conditions, which reflects our
assumption that the plankton cannot leave the boundary of interest.
For various initial conditions, our computer simulations in 2-D 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
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 reaction-diffusion systems
are useful models for investigating possible mechanisms for this behavior (see Figure 3).