Biomathematics Seminar: Long-range dispersal, stochasticity and accelerating waves of of advance
10 February 2015 14:00 in CM105
The most common continuum model for the spread of disease or population is the well-known and much-used Fisher-Kolmogorov equation. This equation predicts that species invasion, for example, takes place through a wave of advance travelling at a constant speed related to the population diffusion and rate of population increase in a virgin environment. The underlying stochastic process involves a spatial dispersion kernel whose second moment is finite. It might be thought that an integral equation approach, which more specifically takes account of the dispersion kernel, would to be more accurate with respect to the long-time properties of the dispersion process, and more specifically the speed of the wave of advance. Such models have been widely studied and lead, for a variety of kernels decaying more slowly than exponentially, to accelerating waves of advance. Such “fat-tailed” kernels occur experimentally often in the ecological literature, and there presence might be thought to nullify the conclusions of Fisher and Kolmogorov. However, the integral (strictly integro-difference) equation models use a mean-field philosophy, and it is in fact they which can give rise to the misleading answers. It turns out that rare long distance dispersal events have a disproportionate impact on the spread of invasive species.
This talk presents a numerical study of a stochastic non-linear one-dimensional Ising-like lattice model in which demographic stochasticity and the dispersal regime can be systematically varied. Stochasticity has a profound effect on model behaviour, and usually breaks acceleration for sufficiently fat-tailed kernels. Exceptions are seen for some power law kernels, K(l) ∝ |l|−β with β < 3, for which acceleration persists despite stochasticity. Such kernels lack a second moment and are important in 'accelerating' phenomena such as Lévy flights. Furthermore, for long-range kernels the approach to the continuum limit behaviour as stochasticity is reduced is generally slow. Given that real-world populations are finite, stochastic models may give better predictive power when long-range dispersal is important. Thus insights from mean-field models should be applied with caution in such circumstances.
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