Bulletin of Mathematical Biology

, Volume 59, Issue 3, pp 497–515 | Cite as

Controlling spatial chaos in metapopulations with long-range dispersal

  • Michael Doebeli
  • Graeme D. Ruxton


We propose two methods to control spatial chaos in an ecological metapopulation model with long-range dispersal. The metapopulation model consists of local populations living in a patchily distributed habitat. The habitat patches are arranged in a one-dimensional array. In each generation, density-dependent reproduction occurs first in each patch. Then individuals disperse according to a Gaussian distribution. The model corresponds to a chain of coupled oscillators with long-range interactions. It exhibits chaos for a broad range of parameters. The proposed control methods are based on the method described by Güémez and Matías for single difference equations. The methods work by adjusting the local population sizes in a selected subset of all patches. In the first method (pulse control), the adjustments are made periodically at regular time intervals, and consist of always removing (or adding) a fixed proportion of the local populations. In the second method (wave control), the adjustments are made in every generation, but the proportion of the local population that is affected by the control changes sinusoidally. As long as dispersal distances are not too low, these perturbations can drive chaotic metapopulations to cyclic orbits whose period is a multiple of the control period. we discuss the influence of the magnitude of the pulses and wave amplitudes, and of the number and the distribution of controlled patches on the effectiveness of control. When the controls start to break down, interesting dynamic phenomena such as intermittent chaos can be observed.


Local Population Pulse Control Fixed Proportion Metapopulation Model Spatiotemporal Chaos 
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  1. Aranson, I., H. Levine and L. Tsimring. 1994. Controlling spatiotemporal chaos.Phys. Rev. Lett. 72, 2561–2564.CrossRefGoogle Scholar
  2. Astakhov, V. V., V. S. Ansihchenko and A. V. Shabunin. 1995. Controlling spatiotemporal chaos in a chain of coupled logistic maps.IEEE Trans. Circuits Syst. I 42, 352–357.CrossRefGoogle Scholar
  3. Bellows, T. S., Jr. 1981. The descriptive properties of some models for density dependence.J. Anim. Ecol. 50, 139–156.MathSciNetCrossRefGoogle Scholar
  4. Brayman, Y., J. F. Lindner and W. L. Ditto. 1995. Taming spatiotemporal chaos with disorder.Nature 378, 465–467.CrossRefGoogle Scholar
  5. Chakravarti, S., M. Marek and W. H. Ray. 1995. Reaction-diffusion system with Brusselar kinetics—control of a quasiperiodic route to chaos.Phys. Rev. E 52, 2407–2423.CrossRefGoogle Scholar
  6. Chow, S. N. and J. Mallet-Paret. 1995. Pattern-formation and spatial chaos in lattice dynamical systems,IEEE Trans. Circuits Syst. I 42, 746–751.MathSciNetCrossRefGoogle Scholar
  7. Doebeli, M. 1993. The evolutionary advantage of controlled chaos.Proc. Roy. Soc. London B 254, 281–286.Google Scholar
  8. Doebeli, M. 1994. Intermittent chaos in population dynamics.J. Theor. Biol. 166, 325–330.CrossRefGoogle Scholar
  9. Doebeli, M. 1995a. Dispersal and dynamics.Theor. Pop. Biol. 47, 82–106.MATHCrossRefGoogle Scholar
  10. Doebeli, M. 1995b. Updating Gillespie with controlled chaos.Amer. Nat. 146, 479–487.CrossRefGoogle Scholar
  11. Gavrilets, S. and A. Hastings. 1995. Intermittency and transient chaos from simple frequency-dependent selection.Proc. Roy. Soc. London B 261, 233–238.Google Scholar
  12. Gilpin, M. and I. Hanski (Eds). 1991.Metapopulation Dynamics: Empirical and Theoretical Investigations. London: Academic Press.Google Scholar
  13. Güémez, J. and M. A. Matías. 1993. Control of chaos in unidimensional maps.Phys. Lett. A 181, 29–32.MathSciNetCrossRefGoogle Scholar
  14. Gyllenberg, M., G. Söderbacka and S. Ericsson. 1993. Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model.Math. Biosci. 118, 25–49.MATHMathSciNetCrossRefGoogle Scholar
  15. Hassell, M. P. 1975. Density-dependence in single-species models.J. Anim. Ecol. 44, 283–296.CrossRefGoogle Scholar
  16. Hassell, M. P., H. N. Comins and R. M. May. 1991. Spatial structure and chaos in insect population dynamics.Nature 353, 255–258.CrossRefGoogle Scholar
  17. Hastings, A. 1993. Complex interactions between dispersal and dynamics: lessons from coupled logistic equations.Ecology 74, 1362–1372.CrossRefGoogle Scholar
  18. Hastings, A. and K. Higgins. 1994. Persistence of transients in spatially structured ecological models.Science 263, 1133–1136.Google Scholar
  19. Holt, R. D. and M. P. Hassell. 1993. Environmental heretogeneity and the stability to host-parasitoid interactions.J. Anim. Ecol. 62, 89–100.CrossRefGoogle Scholar
  20. Huffaker, C. B. 1958. Experimental studies on predation: dispersion factors and predator-prey oscillations.Hilgarida 27, 343–383.Google Scholar
  21. Levins, R. 1970. Extinction. InSome Mathematical Questions in Biology, M. Gerstenhaber (Ed), pp. 77–107. Providence, RI: American Mathematical Society.Google Scholar
  22. Lima, R. and M. Pettini. 1993. Suppression of chaos by resonant parametric perturbations.Phys. Rev. A 41, 726–733.MathSciNetCrossRefGoogle Scholar
  23. Lloyd, A. L. 1995. The coupled Logistic map—a simple model for the effects of spatial heterogeneity on population dynamics.J. Theor. Biol. 173, 217–230.CrossRefGoogle Scholar
  24. May, R. M. and G. F. Oster 1976. Bifurcations and dynamic complexity in simple ecological models.Amer. Nat. 110, 573–599.CrossRefGoogle Scholar
  25. Maynard Smith, J. and M. Slatkin. 1973. The stability of predator-prey systems.Ecology 54, 384–391.CrossRefGoogle Scholar
  26. McCallum, H. I. 1992. Effects of immigration on chaotic population dynamics.J. Theor. Biol. 154, 277–284.Google Scholar
  27. Ott, E., C. Gregobi and J. A. Yorke. 1990. Controlling chaos.Phys. Rev. Lett. 64, 1196–1199.MATHMathSciNetCrossRefGoogle Scholar
  28. Pomeau, Y. and P. Manneville, 1980. Intermittent transition to turbulence in dissipative dynamical systems.Physica A 74, 189–197.MathSciNetGoogle Scholar
  29. Ruxton, G. D. 1994. Low levels of immigration between chaotic populations can reduce system extinctions by inducing asynchronous regular cycles.Proc. Roy. Soc. London B 256, 189–193.Google Scholar
  30. Schoener, T. 1976. Alternatives to Lotka-Volterra competition: models of intermediate complexity.Theor. Pop. Biol. 10, 309–333.MATHMathSciNetCrossRefGoogle Scholar
  31. Sepulchre, J. A. and A. Baboyantz. 1993. Controlling chaos in a network of oscillators.Phys. Rev. E 48, 945–950.CrossRefGoogle Scholar
  32. Shinbrot, T., C. Grebogi, E. Ott and J. A. Yorke. 1993. Using small perturbations to control chaos.Nature 363, 411–417.CrossRefGoogle Scholar
  33. Solé, R. V. and L. Menéndez de la Prida. 1995. Controlling chaos in discrete neural networks.Phys. Lett. A 199, 65–69.CrossRefGoogle Scholar
  34. Stone, L. 1993. Period-doubling reversals and chaos in simple ecological models.Nature 365, 617–620.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1997

Authors and Affiliations

  • Michael Doebeli
    • 1
  • Graeme D. Ruxton
    • 2
  1. 1.Zoology InstituteUniversity of BaselBaselSwitzerland
  2. 2.Division of Environmental and Evolutionary BiologyUniversity of GlasgowGlasgowUK

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