In collaboration with Simon A. Levin.

It has been shown that individual-based predator-prey systems can display at an intermediate spatial scale of aggregation, a dynamic regime in which oscillations differ from random fluctuations around a global average. This intermediate scale has been proposed as a natural size at which to aggregate individuals into densities. It can be identified with an approach based on a determinism test from nonlinear data analysis. I illustrate some of these ideas with an individual-based predator-prey model that is spatial, stochastic, and nonlinear. I then move to the open problem of deriving an appropiate approximation for the dynamics of densities at the selected intermediate scale. I show that the two simplest candidate models-the mean-field system with a limit cycle attractor and an extension of it that adds demographic noise-fail to provide a good approximation. Thus, space is nonnegligible. I end with a conjecture (and hopefully by the time of this talk, some evidence) on the type of model needed to approximate the aperiodic dynamics of densities.

References:

* Keeling, M.J., I. Mezic, R.J. Hendry, J. McGlade, and D.A. Rand. 1997. Characteristic length scales of spatial models in ecology via fluctuation analysis. Philosophical Transactions of the Royal Society of London Series B 352: 1589-1601.

* Pascual, M., and S.A. Levin. 1999. From individuals to population densities: searching for the intermediate scale of nontrivial determinism. Ecology in press.

* Rand, D.A., and H.B. Wilson. 1995. Using spatio-temporal chaos and intermediate-scale determinism to quantify spatially extended ecosystems. Proceedings of the Royal Society of London Series B 259: 111-117.

Back to Workshop Schedule

1998-1999 Mathematics in Biology