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Talk Abstract
Population-level Advection-diffusion Equations for Biased Random Walks by Individuals and Groups

Danny Grunbaum
Department of Zoology
University of Washington

The dynamics of many ecological systems are profoundly influenced by spatial heterogeneity in consumers and resources. Many consumers cope with environmental heterogeneity by using biased random walks, i.e., behaviors that are stochastic but have statistical biases that may lead to up-gradient movement over the long term. In particular, these organisms have internal state dynamics, in which internal state is affected by the environment and behavior is in turn determined by internal state. Population distributions that result from individual-level biased random walks can often approximated by advection-diffusion equations (ADEs), in which the diffusion and advection coefficients implicitly represent the sensory, physiological, or cognitive responses underlying the behavior. I will present some examples of biased random walks from marine and terrestrial systems (including some preliminary experimental results on zooplankton) and derive ADEs that may approximate the corresponding population fluxes. I will also present an extension of this theory that combines group fission/fusion dynamics with spatial random walks to obtain ADEs with density-dependent coefficients that describe how grouping behavior may alter population movements.

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1998-1999 Mathematics in Biology

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