Chapter

Spatial processes

Eric Renshaw

in Stochastic Population Processes

Published in print February 2011 | ISBN: 9780199575312
Published online September 2011 | e-ISBN: 9780191728778 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199575312.003.0009
Spatial processes

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All the previous analyses are based on the assumption that populations develop at a single site where individuals mix homogeneously. Whilst this is mathematically ideal, in that it facilitates theoretical development, in reality there are many situations in which it may be violated. For not only may a population be spatially distributed across several interlinked sites, but even within a specific site the chance of two individuals meeting and interacting may well depend on the distance between them. Although this fact was realized early on in the development of theoretical population dynamics, the high degree of mathematical intractability which rides along with it has meant that little analytic progress has been made relative to non-spatial scenarios. This chapter exposes the underlying theoretical difficulties, highlights directions in which some degree of progress can be made, and shows that the introduction of space generates a whole new concept of a stochastic dynamic. In this latter construct, single-site processes, which on their own result in early extinction, can generate long-term persistence when linked together.

Keywords: population dynamics; stochastic processes; space; single-site processes

Chapter.  58434 words.  Illustrated.

Subjects: Applied Mathematics

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