Bulletin of Mathematical Biology

, Volume 71, Issue 1, pp 130–144 | Cite as

Small Heterogeneity Has Large Effects on Synchronization of Ecological Oscillators

Original Article


Heterogeneity in habitat plays a crucial role in the dynamics of spatially extended populations and is often ignored by both empiricists and theoreticians. A common assumption made is that spatially homogeneous systems and those with slight heterogeneity will behave similarly and, therefore, the results and data from studies of the former can be applied to the latter. Here, we test this assumption by deriving a phase model from two weakly coupled predator-prey oscillators and analyze the effect of spatial heterogeneity on the phase dynamics of this system. We find that even small heterogeneity between the two patches causes substantial changes in the phase dynamics of the system which can have dramatic effects on both population dynamics and persistence. Additionally, if the prey and predator time scales are similar, the effect of heterogeneity is much greater.


Heterogeneity Synchrony Dispersal Population persistence Predator-prey dynamics Phase response curve Moran effect 


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  1. Blasius, B., Stone, L., 2000. Chaos and phase synchronization in ecological systems. Int. J. Bifurc. Chaos 10, 2361–2380. Google Scholar
  2. Blasius, B., Huppert, A., Stone, L., 1999. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359. CrossRefGoogle Scholar
  3. Earn, D., Rohani, P., Grenfell, B., 1998. Persistence, chaos and synchrony in ecology and epidemiology. Proc. R. Soc. Lond. B 265, 7–10. CrossRefGoogle Scholar
  4. Ermentrout, G., 1981. n:m phase-locking of weakly coupled oscillators. J. Math. Biol. 12, 327–342. MATHCrossRefMathSciNetGoogle Scholar
  5. Ermentrout, G., 2002. Simulating, Analyzing, and Animating Dynamical Systems. SIAM, Philadelphia. MATHGoogle Scholar
  6. Goldwyn, E., Hastings, A., 2008. When can dispersal synchronize populations? Theor. Popul. Biol. 73, 395–402. CrossRefGoogle Scholar
  7. Grenfell, B., Bjørnstad, O., Kappey, J., 2001. Travelling waves and spatial hierarchies in measles epidemics. Nature 414, 716–723. CrossRefGoogle Scholar
  8. Hassell, M., May, R., 1974. Aggregation in predators and insect parasites and its effect on stability. J. Anim. Ecol. 43, 567–594. CrossRefGoogle Scholar
  9. Hastings, A., 1997. Population Biology: Concepts and Models. Springer, New York. Google Scholar
  10. Hastings, A., 2001. Transient dynamics and persistence of ecological systems. Ecol. Lett. 4, 215–220. CrossRefGoogle Scholar
  11. Holyoak, M., 2000. Habitat patch arrangement and metapopulation persistence of predators and prey. Am. Nat. 156, 378–389. CrossRefGoogle Scholar
  12. Hugueny, B., 2006. Spatial synchrony in population fluctuations: extending the Moran theorem to cope with spatially heterogeneous dynamics. Oikos 115, 3–14. CrossRefGoogle Scholar
  13. Izhikevich, E., 2000. Phase equations for relaxation oscillators. SIAM J. Appl. Math. 60, 1789–1805. MATHCrossRefMathSciNetGoogle Scholar
  14. Jansen, V., 2001. The dynamics of two diffusively coupled predator-prey populations. Theor. Popul. Biol. 59, 119–131. MATHCrossRefGoogle Scholar
  15. Kendall, B., Fox, G., 1998. Spatial synchrony, environmental heterogeneity, and population dynamics: analysis of the coupled logistic map. Theor. Popul. Biol. 54, 11–37. MATHCrossRefGoogle Scholar
  16. Kuramoto, Y., 1984. Chemical Oscillations, Waves, and Turbulence. Springer, Berlin. MATHGoogle Scholar
  17. Liebhold, A., Koenig, W., Bjørnstad, O., 2004. Spatial synchrony in population dynamics. Annu. Rev. Ecol. Evol. Syst. 35, 467–490. CrossRefGoogle Scholar
  18. Liebhold, A., Johnson, D., Bjørnstad, O., 2006. Geographic variation in density-dependent dynamics impacts the synchronizing effect of dispersal and regional stochasticity. Popul. Ecol. 48, 131–138. CrossRefGoogle Scholar
  19. Malkin, I., 1949. Methods of Poincare and Liapunov in Theory of Non-Linear Oscillations. Gostexizdat, Moscow. Google Scholar
  20. Malkin, I., 1956. Some Problems in Nonlinear Oscillation Theory. Gostexizdat, Moscow. Google Scholar
  21. Moran, P., 1953. The statistical analysis of the Canadian lynx cycle. ii. Synchronization and meteorology. Aust. J. Zool. 1, 291–298. CrossRefGoogle Scholar
  22. Peltonen, M., Liebhold, A., Bjørnstad, O., Williams, D.W., 2002. Spatial synchrony in forest insect outbreaks: roles of regional stochasticity and dispersal. Ecology 83, 3120–3129. CrossRefGoogle Scholar
  23. Ranta, E., Kaitala, V., Lindström, J., Lindén, H., 1995. Synchrony in population dynamics. Proc. R. Soc. Lond. B 262, 113–118. CrossRefGoogle Scholar
  24. Rosenzweig, M., MacArthur, R., 1963. Graphical representation and stability conditions of predator-prey interactions. Am. Nat. 97, 209–223. CrossRefGoogle Scholar
  25. Strogatz, S., 1994. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Perseus Books, Cambridge. Google Scholar
  26. Winfree, A., 1967. Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15. CrossRefGoogle Scholar
  27. Winfree, A., 2001. The Geometry of Biological Time. Springer, New York. MATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  1. 1.University of CaliforniaDavisUSA

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