Skip to main content
SpringerLink
Log in

Dispersal and noise: Various modes of synchrony in ecological oscillators

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We use the theory of noise-induced phase synchronization to analyze the effects of dispersal on the synchronization of a pair of predator–prey systems within a fluctuating environment (Moran effect). Assuming that each isolated local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive a Fokker–Planck equation describing the evolution of the probability density for pairwise phase differences between the oscillators. In the case of common environmental noise, the oscillators ultimately synchronize. However the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of partially correlated noise with dispersal can lead to a multistable steady-state probability density.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Bressloff PC, Lai YM (2011) Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise. J Math Neurosci 1:2

    Article  MathSciNet  Google Scholar 

  • Colombo A, Dercole F, Rinaldi S (2008) Remarks on metacommunity synchronization with application to prey–predator systems. Am Nat 171:430–442

    Article  Google Scholar 

  • Earn DJD, Rohani P, Grenfell BT (1998) Persistence, chaos and synchrony in ecology and epidemiology. Proc R Soc Lond Ser B Biol Sci 265:7–10

    Article  Google Scholar 

  • Ermentrout GB, Kopell N (1991) Multiple pulse interactions and averaging in coupled neural oscillators. J Math Biol 29:195–217

    Google Scholar 

  • Evans SN, Ralph PL, Schreiber SJ, Sen A (2012) Stochastic population growth in spatially heterogeneous environments. J Math Biol (published online)

  • Galan RF, Ermentrout GB, Urban NN (2006) Correlation-induced synchronization of oscillations in olfactory bulb neurons. J Neurosci 26:3646–3655

    Article  Google Scholar 

  • Galan RF, Ermentrout GB, Urban NN (2008) Optimal time scale for spike-time reliability: theory, simulations and expeiments. J Neurophysiol 99:277–283

    Article  Google Scholar 

  • Gardiner CW (2009) Stochastic methods: a handbook for the natural and social sciences, 4th edn. Springer, New York

    Google Scholar 

  • Ghandi A, Levin S, Orszag S (1998) ‘Critical slowing down’ in time-to-extinction: an example of critical phenomena in ecology. J Theor Biol 192:363–376

    Article  Google Scholar 

  • Ghandi A, Levin S, Orszag S (2000) Moment expansions in spatial ecological models and moment closure through Gaussian approximation. Bull Math Biol 62:595–632

    Article  Google Scholar 

  • Goldobin DS, Pikovsky A (2005) Synchronization and desynchronization of self-sustained oscillators by common noise. Phys Rev E 71:045201

    Article  MathSciNet  Google Scholar 

  • Goldwyn EE, Hastings A (2008) When can dispersal synchronize populations? Theor Popul Biol 73:395–402

    Article  MATH  Google Scholar 

  • Goldwyn EE, Hastings A (2009) Small heterogeneity has large effects on synchronisation of ecological oscillators. Bull Math Biol 71:130–144

    Article  MathSciNet  MATH  Google Scholar 

  • Goldwyn EE, Hastings A (2011) The roles of the Moran effect and dispersal in synchronizing oscillating populations. J Theor Biol 289:237–246

    Article  MathSciNet  Google Scholar 

  • Hanski I (1998) Metapopulation dynamics. Nature 396:41–49

    Article  Google Scholar 

  • Hastings A (1997) Population biology: concepts and models. Springer, New York

    Book  Google Scholar 

  • Hudson PJ, Cattadori IM (1999) The Moran effect: a cause of population synchrony. Trends Ecol Evol 14:1–2

    Article  Google Scholar 

  • Jansen VAA, Lloyd AL (2000) Local stability analysis of spatially homogeneous solutions of multi-patch systems. J Math Biol 41:232–252

    Article  MathSciNet  MATH  Google Scholar 

  • Klausmeier CA (2010) Successional state dynamics: a novel approach to modeling nonequilibrium foodweb dynamics. J Theor Biol f262:585–595

    MathSciNet  Google Scholar 

  • Kuramoto Y (1984) Chemical oscillations, waves and turbulence. Springer, Berlin

    Book  MATH  Google Scholar 

  • Lai YM, Newby J, Bressloff PC (2011) Effects of demographic noise on the synchronization of metacommunities by a fluctuating environment. Phys Rev Lett 107:118102

    Article  Google Scholar 

  • Liebhold A, Koenig WD, Bjornstad ON (2004) Spatial synchrony in population dynamics. Annu Rev Ecol Evol Syst 35:467–490

    Article  Google Scholar 

  • Ly C, Ermentrout GB (2009) Synchronization of two coupled neural oscillators receiving shared and unshared noisy stimuli. J Comput Neurosci 26:425–443

    Article  MathSciNet  Google Scholar 

  • Mainen ZF, Sejnowski TJ (1995) Reliability of spike timing in neocortical neurons. Science 268:1503–1506

    Article  Google Scholar 

  • Marella S, Ermentrout GB (2008) Class-II neurons display a higher degree of stochastic synchronization than class-I neurons. Phys Rev E 77:41918

    Article  MathSciNet  Google Scholar 

  • McCauley E, Wilson WG, Roos AM (1993) Dynamics of age-structured predator–prey interactions: individual based models and population level formulations. Am Nat 142:412–442

    Article  Google Scholar 

  • McKane AJ, Newman TJ (2004) Stochastic models in population biology and their deterministic analogs. Phys Rev E 70:041902

    Article  MathSciNet  Google Scholar 

  • McKane AJ, Newman TJ (2005) Predator–prey cycles from resonant amplification of demographic stochasticity. Phys Rev Lett 94:218102

    Article  Google Scholar 

  • Moran PAP (1953) The statistical analysis of the Canadian lynx cycle. II. Synchronisation and meteorology. Aust J Zool 1:291–298

    Article  Google Scholar 

  • Nakao H, Arai K, Kawamura Y (2007) Noise-induced synchronization and clustering in ensembles of uncoupled limit cycle oscillators. Phys Rev Lett 98:184101

    Article  Google Scholar 

  • Oksendal B (2000) Stochastic differential equations: an introduction with applications. Springer, Berlin

    Google Scholar 

  • Pecora LM, Carroll TL (1998) Master stability functions for synchronized coupled systems. Phys Rev Lett 80:2109–2112

    Article  Google Scholar 

  • Pikovsky AS (1984) Synchronization and stochastization of an ensemble of autogenerators by external noise. Radiophysics 27:576–581

    Google Scholar 

  • Renshaw E (1993) Modelling biological populations in space and time. Cambridge University Press, Cambridge

  • Rohani P, Earn DJD, Grenfell BT (1999) Opposite patterns of synchrony in sympatric disease metapopulations. Science 286:968–971

    Article  Google Scholar 

  • Roos AM, McCauley E, Wilson WG (1991) Mobility versus density-limited predator–prey dynamics on different spatial scales. Proc R Soc Lond B 246:117–122

    Article  Google Scholar 

  • Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97:209–223

    Article  Google Scholar 

  • Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20

    Article  MathSciNet  MATH  Google Scholar 

  • Teramae JN, Tanaka D (2004) Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. Phys Rev Lett 93:204103

    Article  Google Scholar 

  • Teramae JN, Nakao H, Ermentrout GB (2009) Stochastic phase reduction for a general class of noisy limit cycle oscillators. Phys Rev Lett 102:194102

    Article  Google Scholar 

  • Turelli M (1978) Random environments and stochastic calculus. Theor Popul Biol 12:140–178

    Article  MathSciNet  Google Scholar 

  • Van Kampen NG (1992) Stochastic processes in physics and chemistry. North-Holland, Amsterdam

    Google Scholar 

  • Vasseur DA, Fox JW (2009) Phase-locking and environmental fluctuations generate synchrony in a predator–prey community. Nature 460:1007–1010

    Article  Google Scholar 

  • Yoshimura K, Arai K (2008) Phase reduction of stochastic limit cycle oscillators. Phys Rev Lett 101:154101

    Article  Google Scholar 

Download references

Acknowledgments

This publication was based on work supported in part by the National Science Foundation (DMS-1120327) and the King Abdullah University of Science and Technology Award No. KUK-C1-013-04.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Paul C. Bressloff

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Yi Ming Lai

Authors
  1. Paul C. Bressloff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Ming Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul C. Bressloff.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bressloff, P.C., Lai, Y.M. Dispersal and noise: Various modes of synchrony in ecological oscillators. J. Math. Biol. 67, 1669–1690 (2013). https://doi.org/10.1007/s00285-012-0607-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-012-0607-9

Keywords

Mathematics Subject Classification

Search

Navigation