Journal of Mathematical Biology

, Volume 19, Issue 2, pp 169–200

Asymptotic behavior of some stochastic difference equation population models

  • S. Ellner

DOI: 10.1007/BF00277745

Cite this article as:
Ellner, S. J. Math. Biology (1984) 19: 169. doi:10.1007/BF00277745


We consider a general class of Markov population models formulated as stochastic difference equations. The population density is shown to converge either to 0, to +∞, or to a unique stationary distribution concentrated on (0, +∞), depending on the signs of the mean log growth rates near 0 and +∞. These results are applied to the Watkinson-MacDonald “bottleneck” model of annual plants with a seedbank, extended to allow for random environmental fluctuations and competition among co-occurring species. We obtain criteria for long-term persistence of single-species populations, and for coexistence of two competing species, and the biological significance of the criteria is discussed. The lamentably few applications to the problem at hand of classical limit-theory for Markov chains are surveyed.

Key words

stochastic difference equations population dynamics Markov chains stationary distribution non-equilibrium coexistence seedbanks ergodic theory 

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • S. Ellner
    • 1
  1. 1.Department of Mathematics and Program in EcologyThe University of TennesseeKnoxvilleUSA

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