Advertisement

Beyond Diffusion: Conditional Dispersal in Ecological Models

  • Chris Cosner
Chapter
Part of the Fields Institute Communications book series (FIC, volume 64)

Abstract

Reaction-diffusion models have been widely used to describe the dynamics of dispersing populations. However, many organisms disperse in ways that depend on environmental conditions or the densities of other populations. Those can include advection along environmental gradients and nonlinear diffusion, among other possibilities. In this paper I will give a survey of some models involving conditional dispersal and discuss its effects and evolution. The presence of conditional dispersal can strongly influence the equilibria of population models, for example by causing the population to concentrate at local maxima of resource density. The analysis of the evolutionary aspects of dispersal is typically based on a study of models for two competing populations that are ecologically identical except for their dispersal strategies. The models consist of Lotka-Volterra competition systems with some spatially varying coefficients and with diffusion, nonlinear diffusion, and/or advection terms that reflect the dispersal strategies of the competing populations. The evolutionary stability of dispersal strategies can be determined by analyzing the stability of single-species equilibria in such models. In the case of simple diffusion in spatially varying environments it has been known for some time that the slower diffuser will exclude the faster diffuser, but conditional dispersal can change that. In some cases a population whose dispersal strategy involves advection along environmental gradients has the advantage or can coexist with a population that simply diffuses. As is often the case in reaction-diffusion theory, many of the results depend on the analysis of eigenvalue problems for linearized models.

Notes

Acknowledgements

Research partially supported by NSF grants DMS-0514839 and DMS-0816068.

Received 9/10/2009; Accepted 3/12/2012

References

  1. [1].
    H. Amann, Dynamic theory of quasilinear parabolic systems III. Global existence. Math. Z. 202, 219–250 (1989)MathSciNetCrossRefGoogle Scholar
  2. [2].
    H. Amann, Dynamic theory of quasilinear parabolic equations II: Reaction diffusion systems. Differ. Integr. Equat. 3, 13–75 (1990)Google Scholar
  3. [3].
    F. Belgacem, C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment. Canadian Appl. Math. Quart. 3, 379–397 (1995)Google Scholar
  4. [4].
    R.S. Cantrell, C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology (Wiley, Chichester, 2003)Google Scholar
  5. [5].
    R.S. Cantrell, C. Cosner, D. DeAngelis, V. Padron, The ideal free distribution as an evolutionarily stable strategy. J. Biol. Dynam. 1, 249–271 (2007)Google Scholar
  6. [6].
    R.S. Cantrell, C. Cosner, Y. Lou, Movement towards better environments and the evolution of rapid diffusion. Math. Biosci. 204, 199–214 (2006)MathSciNetCrossRefGoogle Scholar
  7. [7].
    R.S. Cantrell, C. Cosner, Y. Lou, Advection mediated coexistence of competing species. Proc. Roy. Soc. Edinb. 137A, 497–518 (2007)MathSciNetCrossRefGoogle Scholar
  8. [8].
    R.S. Cantrell, C. Cosner, Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations. J. Differ. Equat. 245, 3687–3703 (2008)MathSciNetCrossRefGoogle Scholar
  9. [9].
    X.F. Chen, Y. Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model. Indiana Univ. Math. J. 57, 627–658 (2008)MathSciNetCrossRefGoogle Scholar
  10. [10].
    X.F. Chen, R. Hambrock, Y. Lou, Evolution of conditional dispersal: a reaction-diffusion-advection model. J. Math Biol. 57, 361–386 (2008)MathSciNetCrossRefGoogle Scholar
  11. [11].
    C. Cosner, A dynamic model for the ideal free distribution as a partial differential equation. Theor. Pop. Biol. 67, 101–108 (2005)CrossRefGoogle Scholar
  12. [12].
    C. Cosner, Y. Lou, Does movement toward better environments always benefit a population? J. Math. Anal. App. 277, 489–503 (2003)MathSciNetCrossRefGoogle Scholar
  13. [13].
    J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dispersal rates: a reaction-diffusion model. J. Math. Biol. 37, 61–83 (1998)MathSciNetCrossRefGoogle Scholar
  14. [14].
    A. Hastings, Can spatial variation alone lead to selection for dispersal? Theor. Pop. Biol. 24, 244–251 (1983)MathSciNetCrossRefGoogle Scholar
  15. [15].
    M.A. McPeek, R.D. Holt, The evolution of dispersal in spatially and temporally varying environments. Am. Nat. 140, 1010–1027 (1992)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MiamiCoral GablesUSA

Personalised recommendations