Dr. Marcus R Garvie

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 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 reaction-diffusion systems are useful models for investigating possible mechanisms for this behavior (see Figure 3).

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Last updated: Nov 16, 2006 by Marcus R Garvie