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Extinction in population dynamics

C. Escudero1, J. Buceta2, F. J. de la Rubia1, and Katja Lindenberg2

  • 1Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia, C/ Senda del Rey 9, 28040 Madrid, Spain
  • 2Department of Chemistry and Biochemistry, and Institute for Nonlinear Science, University of California San Diego, 9500 Gilman Dr., La Jolla, California 92093-0340, USA

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Phys. Rev. E 69, 021908 – Published 26 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.021908

Abstract

We study a generic reaction-diffusion model for single-species population dynamics that includes reproduction, death, and competition. The population is assumed to be confined in a refuge beyond which conditions are so harsh that they lead to certain extinction. Standard continuum mean field models in one dimension yield a critical refuge length Lc such that a population in a refuge larger than this is assured survival. Herein we extend the model to take into account the discreteness and finiteness of the population, which leads us to a stochastic description. We present a particular critical criterion for likely extinction, namely, that the standard deviation of the population be equal to the mean. According to this criterion, we find that while survival can no longer be guaranteed for any refuge size, for sufficiently weak competition one can make the refuge large enough (certainly larger than Lc) to cause extinction to be unlikely. However, beyond a certain value of the competition rate parameter it is no longer possible to escape a likelihood of extinction even in an infinite refuge. These unavoidable fluctuations therefore have a severe impact on refuge design issues.

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Phys. Rev. E 69, 021908– Published 26 February, 2004
  • Received 25 July 2003

© 2004 American Physical Society

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