Theoretical Ecology

, Volume 6, Issue 4, pp 405–418 | Cite as

Synchronization in ecological systems by weak dispersal coupling with time delay

  • Emily WallEmail author
  • Frederic Guichard
  • Antony R. Humphries
Original Paper


One of the most salient spatiotemporal patterns in population ecology is the synchronization of fluctuating local populations across vast spatial extent. Synchronization of abundance has been widely observed across a range of spatial scales in relation to the rate of dispersal among discrete populations. However, the dependence of synchrony on patterns of among-patch movement across heterogeneous landscapes has been largely ignored. Here, we consider the duration of movement between two predator–prey communities connected by weak dispersal and its effect on population synchrony. More specifically, we introduce time-delayed dispersal to incorporate the finite transmission time between discrete populations across a continuous landscape. Reducing the system to a phase model using weakly connected network theory, it is found that the time delay is an important factor determining the nature and stability of phase-locked states. Our analysis predicts enhanced convergence to stable synchronous fluctuations in general and a decreased ability of systems to produce in-phase synchronization dynamics in the presence of delayed dispersal. These results introduce delayed dispersal as a tool for understanding the importance of dispersal time across a landscape matrix in affecting metacommunity dynamics. They further highlight the importance of landscape and dispersal patterns for predicting the onset of synchrony between weakly coupled populations.


Synchronization Phase model Dispersal Time delay 



E. Wall is grateful to the McGill University Biology Department for a Science Undergraduate Research Award. F. Guichard and A.R. Humphries thank the Natural Sciences and Engineering Research Council of Canada for funding through the Discovery Grants Program.


  1. Allen LJS (1983) Persistence and extinction in Lotka-Volterra reaction–diffusion equations. Math Biosci 65(1):1–12. doi: 10.1016/0025-5564(83)90068-8 CrossRefGoogle Scholar
  2. Beninca E, Johnk KD, Heerkloss R, Huisman J (2009) Coupled predator–prey oscillations in a chaotic food web. Ecol Lett 12:1367–1378. doi: 10.1111/j.1461-0248.2009.01391.x PubMedCrossRefGoogle Scholar
  3. Blasius B, Tönjes R (2007) Predator–prey oscillations, synchronization and pattern formation in ecological systems. In: Schimansky-Geier L, Fiedler B, Kurths J, Schöll E (eds) Analysis and control of complex nonlinear processes in physics, chemistry and biology. World Scientific, Singapore, pp 397–427CrossRefGoogle Scholar
  4. Brady MJ, McAlpine CA, Possingham HP, Miller CJ, Baxter GS (2011) Matrix is important for mammals in landscapes with small amounts of native forest habitat. Landsc Ecol 26(5):617–628. doi: 10.1007/s10980-011-9602-6 CrossRefGoogle Scholar
  5. Bresloff PC, Lai YM (2013) Dispersal and noise: various modes of synchrony in ecological oscillators. J Math Biol. doi:  10.1007/s00285-012-0607-9 Google Scholar
  6. Brown JH, Kodric–Brown A (1977) Turnover rates in insular biogeography: effect of immigration on extinction. Ecology 58:445–449CrossRefGoogle Scholar
  7. Byers JA (2001) Correlated random walk equations of animal dispersal resolved by simulation. Ecology 82(6):1680–1690. doi: 10.2307/2679810 CrossRefGoogle Scholar
  8. Callaghan T, Karlson R (2002) Summer dormancy as a refuge from mortality in the freshwater bryozoan plumatella emarginata. Oecologia 132(1):51–59CrossRefGoogle Scholar
  9. Campbell SA, Kobelevskiy I (2012) Phase models and oscillators with time delayed coupling. Discret Conting Dyn Syst 32(8):2653–2673CrossRefGoogle Scholar
  10. Chesson P (2000) Mechanisms of maintenance of species diversity. Annu Rev Ecol Syst 31:343CrossRefGoogle Scholar
  11. DhamalaM, Jirsa V, Ding M (2004) Enhancement of neural synchrony by time delay. Phys Rev Lett 92(7):74104CrossRefGoogle Scholar
  12. Durrett R, Levin S (1994) The importance of being discrete (and spatial). Theo Popul Biol 46:363–394. doi: 10.1006/tpbi.1994.1032 CrossRefGoogle Scholar
  13. Ermentrout GB (1994) In: Ventriglia F (ed) In: Neural modeling and neural networks. Pergamon, OxfordGoogle Scholar
  14. Ermentrout GB (2002) Stimulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM, PhiladelphiaCrossRefGoogle Scholar
  15. Goldwyn EE, Hastings A (2008) When can dispersal synchronize populations? Theor Popul Biol 73:395–402. doi:  10.1016/j.tpb.2007.11.012 PubMedCrossRefGoogle Scholar
  16. Goldwyn EE, Hastings A (2011) The roles of the Moran effect and dispersal in synchronizing oscillating populations. Theor Popul Biol 289:237–246. doi: 10.1016/j.jtbi.2011.08.033 CrossRefGoogle Scholar
  17. Grenfell BT, Bjrnstad ON, Kappey J (2001) Travelling waves and spatial hierarchies in measles epidemics. Nature 414:716–723. doi: 10.1038/414716a PubMedCrossRefGoogle Scholar
  18. Hanski I (1999) Metapopulation ecology. Oxford University Press, OxfordGoogle Scholar
  19. He D, Stone L (2003) Spatio-temporal synchronization of recurrent epidemics. Proc R Soc Lond B 270:1519–1526. doi: 10.1098/rspb.2003.2366 CrossRefGoogle Scholar
  20. Holyoak M (2000) Habitat patch arrangement and metapopulation persistence of predators and prey. Am Nat 156:378–389CrossRefGoogle Scholar
  21. Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. Springer, New YorkCrossRefGoogle Scholar
  22. Huffaker C (1958) Experimental studies on predation: dispersion factors and predatorprey oscillations. Hilgardia 27:343–383Google Scholar
  23. Izhikivech EM (2008) Phase models with explicit time delays. Phys Rev E 58:905–908. doi: 10.1103/PhysRevE.58.905 CrossRefGoogle Scholar
  24. Kobelevskiy I (2008) Bifurcation analysis of a system of Morris-Lecar neurons with time delayed gap junctional coupling. Masters Dissertation. University of Waterloo, WaterlooGoogle Scholar
  25. Koelle K, Vandermeer J (2005) Dispersal-induced desynchronization: from metapopulations to metacommunities. Ecol Lett 8:167–175. doi: 10.1111/j.1461-0248.2004.00703.x CrossRefGoogle Scholar
  26. Koh LP, Lee TM, Sodhi NS, Ghazoul J (2010) An overhaul of the species-area approach for predicting biodiversity loss: incorporating matrix and edge effects. J Appl Ecol 47(5):1063–1070. doi: 10.1111/j.1365-2664.2010.01860 CrossRefGoogle Scholar
  27. Kuramoto Y (1984) Chemical oscillations, waves and turbulence. Springer, BerlinCrossRefGoogle Scholar
  28. Levins R (1969) Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull Entomol Soc Am 15:237–240Google Scholar
  29. Liebhold A, Koenig W, Bjørnstad O (2004) Spatial synchrony in population dynamics. Annu Rev Ecol Evol Syst 35:467–490. doi: 10.1146/annurev.ecolsys.34.011802.132516 CrossRefGoogle Scholar
  30. The MathWorks Inc. (2012) MATLAB R2012a. The MathWorks Inc., NatickGoogle Scholar
  31. Moran PAP (1953) The statistical analysis of the Canadian lynx cycle. II. Synchronization and metereology. Aust J Zool 1:291–298CrossRefGoogle Scholar
  32. Othmer HG, Dunbar SR, Alt W (1988) Models of dispersal in biological systems. J Math Ecol 26(3):263–98. doi: 10.1007/BF00277392 Google Scholar
  33. Peltonen M, Liebhold A, Bjornstad O, Williams D (2002) Spatial synchrony in forest insect outbreaks: roles of regional stochasticity and dispersal. Ecology 83(11):3120–3129CrossRefGoogle Scholar
  34. Prasad A, Dana SK, Karnatak R, Kurths J, Blasius B, Ramaswamy R (2008) Universal occurrence of the phase-flip bifurcation in time-delay coupled systems. Chaos 18(2):023111. doi: 10.1063/1.2905146 PubMedCrossRefGoogle Scholar
  35. Ranta E, Kaitala V, Lundberg P (1998) Population variability in space and time: the dynamics of synchronous populations. Oikos 83:376–382. doi: 10.2307/3546852 CrossRefGoogle Scholar
  36. Rothhaupt K (2000) Plankton population dynamics: food web interactions and abiotic constraints. Freshwater Biol 45(2):105–109CrossRefGoogle Scholar
  37. Schuster H, Wagner P (1989) Mutual entrainment of 2 limit-cycle oscillators with time delayed coupling. Prog Theor Phys 81(5):939–945CrossRefGoogle Scholar
  38. Strogatz SH (2000) Nonlinear dynamics and chaos with applications to physics, biology, chemistry, and engineering. Westview, CambridgeGoogle Scholar
  39. Turchin P (1998) Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. Sinauer Associates, SunderlandGoogle Scholar
  40. Turner MG, Gardner RH, O’Neill RV (2001) Landscape ecology in theory and practice: pattern and process. Springer, New YorkGoogle Scholar
  41. Winfree AT (1980) The geometry of biological time. Springer, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Emily Wall
    • 1
    Email author
  • Frederic Guichard
    • 1
  • Antony R. Humphries
    • 2
  1. 1.Department of BiologyMcGill UniversityMontrealCanada
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

Personalised recommendations