Critical Patch-Size

  • Frithjof Lutscher
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 49)


One of the basic questions in spatial ecology is: how much space does a population need to persist? The critical patch-size is the size of the suitable habitat where population gain through reproduction balances population loss through dispersal. The question of how large a certain habitat has to be to support a given population has important applications in conservation biology, e.g., when designing a protected area to ensure the survival of an endangered population. The analysis in this chapter is based on linearization, thereby implicitly assuming that the population growth function has no Allee effect. We explicitly compute the critical size when dispersal is described by a Laplace kernel. We then compare how different dispersal patterns affect this critical size. At the end of the chapter, we consider the class of separable kernels and introduce an approximation method.


  1. Bramburger, J., & Lutscher, F. (2019) Analysis of integrodifference equations with a separable dispersal kernel. Acta Applicandae Mathematicae, 161(1), 127–151.MathSciNetCrossRefGoogle Scholar
  2. Cantrell, R. S., & Cosner, C. (2003). Spatial ecology via reaction-diffusion equations. Mathematical and computational biology. London: Wiley.zbMATHGoogle Scholar
  3. Caswell, H. (2001). Matrix population models. Sunderland: Sinauer Associates.Google Scholar
  4. Du, Y. (2006). Order structure and topological methods in nonlinear partial differential equations. Maximum principles and applications. Singapore: World Scientific.zbMATHGoogle Scholar
  5. Hardin, D., Takáč, P., & Webb, G. (1990). Dispersion population models discrete in time and continuous in space. Journal of Mathematical Biology, 28, 1–20.MathSciNetCrossRefGoogle Scholar
  6. Hutson, V., Martinez, S., Mischaikow, K., & Vickers, G. (2003). The evolution of dispersal. Journal of Mathematical Biology, 46, 483–517.MathSciNetCrossRefGoogle Scholar
  7. Iooss, G. (1979). Bifurcation of maps and applications. Mathematical studies (vol. 36). Amsterdam: North-Holland.zbMATHGoogle Scholar
  8. Keener, J. (2000). Principles of applied mathematics. Boulder: Westview.zbMATHGoogle Scholar
  9. Kierstead, H., & Slobodkin, L. B. (1953). The size of water masses containing plankton blooms. Journal of Marine Research, 12, 141–147.Google Scholar
  10. Kot, M., & Phillips, A. (2015). Bounds for the critical speed of climate-driven moving-habitat models. Mathematical Biosciences, 262, 65–72.MathSciNetCrossRefGoogle Scholar
  11. Kot, M., & Schaffer, W. (1986). Discrete-time growth-dispersal models. Mathematical Biosciences, 80, 109–136.MathSciNetCrossRefGoogle Scholar
  12. Krasnosel’skii, M. A. (1964). Positive solutions of operator equations. Groningen: Noordhoff LTD.zbMATHGoogle Scholar
  13. Krasnosel’skii, M. A., & Zabreiko, P. P. (1984). Geometrical methods of nonlinear analysis. Berlin: Springer.CrossRefGoogle Scholar
  14. Latore, J., Gould, P., & Mortimer, A. (1998). Spatial dynamics and critical patch size of annual plant populations. Journal of Theoretical Biology, 190, 277–285.CrossRefGoogle Scholar
  15. Lockwood, D., Hastings, A., & Botsford, L. (2002). The effects of dispersal patterns on marine reserve: Does the tail wag the dog? Theoretical Population Biology, 61, 297–309.CrossRefGoogle Scholar
  16. Lutscher, F., & Lewis, M. (2004). Spatially-explicit matrix models. A mathematical analysis of stage-structured integrodifference equations. Journal of Mathematical Biology, 48, 293–324.MathSciNetCrossRefGoogle Scholar
  17. Musgrave, J., & Lutscher, F. (2014a). Integrodifference equations in patchy landscapes I: Dispersal kernels. Journal of Mathematical Biology, 69(3), 583–615.MathSciNetCrossRefGoogle Scholar
  18. Robertson, S., & Cushing, J. (2011). Spatial segregation in stage-structured populations with an application to Tribolium. Journal of Biological Dynamics, 5(5), 398–409.MathSciNetCrossRefGoogle Scholar
  19. Skellam, J. G. (1951). Random dispersal in theoretical populations. Biometrika, 38, 196–218.MathSciNetCrossRefGoogle Scholar
  20. Van Kirk, R., & Lewis, M. (1997). Integrodifference models for persistence in fragmented habitats. Bulletin of Mathematical Biology, 59(1), 107–137.CrossRefGoogle Scholar
  21. Zhou, Y., & Kot, M. (2013). Life on the move: Modeling the effects of climate-driven range shifts with integrodifference equations. In Dispersal, individual movement and spatial ecology (pp. 263–292). Berlin: Springer.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Frithjof Lutscher
    • 1
  1. 1.Mathematics and StatisticsUniversity of OttawaOttawaCanada

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