Bulletin of Mathematical Biology

, Volume 68, Issue 2, pp 451–466

Density-dependent migration and synchronism in metapopulations

Original Article
  • 111 Downloads

Abstract

A spatially explicit metapopulation model with density-dependent dispersal is proposed in order to study the stability of synchronous dynamics. A stability criterion is obtained based on the computation of the transversal Liapunov number of attractors on the synchronous invariant manifold. We examine in detail a special case of density-dependent dispersal rule where migration does not occur if the patch density is below a certain critical density, while the fraction of individuals that migrate to other patches is kept constant if the patch density is above the threshold level. Comparisons with density-independent migration models indicate that this simple density-dependent dispersal mechanism reduces the stability of synchronous dynamics. We were able to quantify exactly this loss of stability through the frequency that synchronous trajectories are above the critical density.

Keywords

metapopulation density-dependent dispersal synchronism 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allen, J.C., Schaffer, W.M., Rosko, D., 1993. Chaos reduces species extinction by aplifying local population noise. Nature 364, 229–232.CrossRefPubMedGoogle Scholar
  2. Amarasekare, P., 2004. The role of density-dependent dispersal in source-sink dynamics. J. Theor. Biol. 226, 159–168.CrossRefPubMedMathSciNetGoogle Scholar
  3. Blasius, B., Huppert, A., Stone, L., 1999. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 353–359.CrossRefGoogle Scholar
  4. Caswell, H., 1989. Matrix Population Models, Sinauer, Sunderland, MA.Google Scholar
  5. Cazelles, B., 2001. Dynamics with riddled basins of attraction in models of interacting populations. Chaos, Solitons Fractals 12, 301–311.CrossRefMathSciNetMATHGoogle Scholar
  6. Cazelles, B., Bottani, S., Stone, L., 2001. Unexpected coherence and conservation. Proc. R. Soc. Lond. B268, 2595–2602.CrossRefGoogle Scholar
  7. De Castro, M.L., Silva, J.A.L., Justo, D.R., 2005. Stability in a age-structured metapopulaton model. J. Math. Biol. (Online).Google Scholar
  8. Gonzalez, A., Lawton, J.H., Gilbert, F.S., Blackburn, T.M., Evans-Freke, I., 1998. Metapopulation dynamics, abundance, and distribution in a microecosystem. Science 281, 2045–2047.Google Scholar
  9. Denno, R.F., Peterson, M.A., 1995. Density-dependent dispersal and its consequences for population dynamics. In: Capuccino, N., Price, P.W. (Eds.), Population Dynamics New Approaches and Synthesis. Academic Press, London, pp. 113–130.Google Scholar
  10. Doebeli, M., Ruxton, G.D., 1998. Stabilization through spatial pattern formation in metapopulations with long-range dispersal. Proc. R. Soc. Lond. B265, 1325–1332.CrossRefGoogle Scholar
  11. Earn, D.J.D., Levin, S.A., Rohani, P., 2000. Coherence and conservation. Science 290, 1360–1364.CrossRefPubMedGoogle Scholar
  12. Eckmann, J.P., Ruelle, D., 1985. Ergodic theory of chaos and strange attractors. Am. Phys. Soc. 57, 617–656.MathSciNetGoogle Scholar
  13. Hanski, I., Gilpin, M.E., 1997. Metapopulation Biology: Ecology Genetics and Evolution, Academic Press, London.MATHGoogle Scholar
  14. Harrison, M.A., Lai, Y.C., Holt, R.D., 2001. Dynamical mechanism for coexistence of dispersing species. J. Theor. Biol. 213, 53–72.CrossRefPubMedMathSciNetGoogle Scholar
  15. Heagy, J.F., Platt, N., Hammel, S.M., 1994. Characterization of on-off intermittency. Phys. Rev. E 49, 1140–1150.CrossRefGoogle Scholar
  16. Heino, M., Kaitala, V., Ranta, E., Lindström, J., 1997. Synchronous dynamics and rates of extiction in spatially structured populations. Proc. R. Soc. Lond. B264, 481–486.CrossRefGoogle Scholar
  17. Huang, Y., Diekmann, O., 2003. Interspecific influence on mobility and Turing instability. Bull. Math. Biol. 65, 143–156.CrossRefPubMedGoogle Scholar
  18. Huang, Y., Diekmann, O., 2004. Double-jump migration and diffusive instability. Bull. Math. Biol. 66, 487–504.CrossRefPubMedMathSciNetGoogle Scholar
  19. Jang, S.R.J., Mitra, A.K., 2000. Equilibrium stability of single—species metapopulations. Bull. Math. Biol. 62, 155–161.CrossRefPubMedGoogle Scholar
  20. Jansen, V.A.A., Lloyd, A., 2000. Local stability analysis of spatially homogeneous solutions of multi-patch systems. J. Math. Biol. 41, 232–252.CrossRefPubMedMathSciNetMATHGoogle Scholar
  21. Lancaster, P., Tismenetsky, M., 1985. The Theory of Matrices, Academic Press, London.MATHGoogle Scholar
  22. Lloyd, A.L., Jansen, V.A.A., 1994. Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models. Math. Biosci. 188, 1–16.CrossRefMathSciNetGoogle Scholar
  23. Matthysen, E., 2005. Density-dependent dispersal in birds and mammals. Ecography 28, 403–416.CrossRefGoogle Scholar
  24. May, R.M., 1976. Simple mathematical models with very complicated dynamics. Nature 261, 459–467.CrossRefGoogle Scholar
  25. Nagai, Y., Lai, Y.-C., 1997. Periodic-orbit theory of the blowout bifurcation. Phys. Rev. E56(4), 4031–4041.Google Scholar
  26. Ott, E., 1994. Chaos in Dynamical Systems. Cambridge University Press, New York.Google Scholar
  27. Ott, E., Sommerer, J.C., 1994. Blowout bifurcations: The occurence of riddled basins and on-off intermittency. Physics Letters A 188, 39–47.CrossRefGoogle Scholar
  28. Platt, N., Spiegel, E.A., Tresser, C., 1993. On-off intermittency: A mechanism for bursting. Phys. Rev. Lett. 70, 279–282.CrossRefPubMedGoogle Scholar
  29. Rohani, P.R., May, R.M., Hassel, M.P., 1996. Metapopulations and equilibrium stability: The effects of spatial structure. J. Theor. Biol. 181, 97–109.CrossRefPubMedGoogle Scholar
  30. Rohani, P., Ruxton, G.D., 1999. Dispersal-induced instabilities in host-parasitoid metapopulations. Theor. Popu. Biol. 55, 23–36.CrossRefMATHGoogle Scholar
  31. Rosenblum, M.G., Plkovsky, A.S., Kurths, J., 1996. Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807.CrossRefPubMedGoogle Scholar
  32. Ruxton, G.D., 1996. Density-dependent migration and stability in a system of linked populations. Bull. Math. Biol. 58, 643–660.CrossRefMATHGoogle Scholar
  33. Ruxton, G.D., Rohani, P., 1998. Fitness-dependent dispersal in metapopulations and its consequence for persistence and synchrony. J. Anim. Ecol. 67, 530–539.Google Scholar
  34. Scheuring, I., Jánosi, I.M., 1996. When two and two make four: A structured population without chaos. J. Theor. Biol. 178, 89–97.CrossRefGoogle Scholar
  35. Silva, J.A.L., De Castro, M.L., Justo, D.A.R., 2000. Synchronism in a metapopulation model. Bull. Math. Biol. 62, 337–349.CrossRefPubMedGoogle Scholar
  36. Silva, J.A.L., De Castro, M.L., Justo, D.A.R., 2001. Stability in a metapopulation model with density-dependent dispersal. Bull. Math. Biol. 63, 485–506.CrossRefPubMedGoogle Scholar
  37. Simberloff, D., Farr, J.A., Cox, J., Mehlman, D.W., 1992. Movement Corridors: conservaton bargains or poor investments? Conserv. Biol. 6, 493–504.Google Scholar
  38. Solé, R.V., Gamarra, J.P.G., 1998. Chaos, dispersal and extinction in coupled ecosystems. J. Theor. Biol. 193, 539–541.CrossRefPubMedGoogle Scholar
  39. Tilman, D., Kareiva, P., 1997. Spatial Ecology: The Role of Space in Population Dynamics and Iterspecific Interactions. Princeton University Press, Princeton.Google Scholar
  40. Ylikarjula, J., Alaja, S., Laakso, J., Tesar, D., 2000. Effects of patch number and dispersal patterns on population dynamics and synchrony. J. Theor. Biol. 207, 377–387.CrossRefPubMedGoogle Scholar

Copyright information

© Society for Mathematical Biology 2006

Authors and Affiliations

  1. 1.Departamento de Matemática Pura e Aplicada-IM-UFRGSPorto Alegre-RS, BrasilBrazil

Personalised recommendations