Population Ecology

, Volume 45, Issue 3, pp 227–237

Neighbourhood size, dispersal distance and the complex dynamics of the spatial Ricker model

Original Article

Abstract

Amongst the most frequently made assumptions in simple population models are that individuals interact equally with every other individual and that dispersal occurs with equal likelihood to any location. This is especially true for models of a single population (as opposed to a patchy population or metapopulation). For many species of animals and probably for all plant species these assumptions are unlikely to hold true. Here one much-studied population model—the Ricker model—is reformulated such that interactions occur only between individuals located within a certain distance of each other and dispersal distance is finite. Two alternative reformulations are presented. Results demonstrate that both limiting the interaction neighbourhood and reducing dispersal distance tend to stabilise the global population dynamics, although the extent to which this occurs depends upon the reformulation used. Spatial pattern formation is a feature of the simulated population. At lower intrinsic rates of growth (r) these patterns tend to be static, while for higher r, they are dynamic. Both the stabilisation of global dynamics and spatial pattern formation are well-described features of metapopulation models. Here, similar effects are shown to occur on a single contiguous patch of habitat.

Keywords

Complex dynamics Chaos Continuous space Discrete time Population model Zone of influence 

References

  1. Berec L (2002) Techniques of spatially explicit individual-based models: construction, simulation and mean-field analysis. Ecol Model 150:55–81CrossRefGoogle Scholar
  2. Biging GS, Dobbertin W (1992) A comparison of distance-dependent competition measures for height and basal growth of individual conifer trees. For Sci 38:695–720Google Scholar
  3. Bolker BM, Pacala SW (1997) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor Popul Biol 52179–197Google Scholar
  4. Bolker BM, Pacala SW (1999) Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am Nat 157:217–230Google Scholar
  5. Bolker BM, Pacala SW, Levin SA (2000) Moment methods for ecological processes in continuous space. In: Dieckmann U, Law R, Metz JAJ (eds) The Geometry of ecological interactions. Cambridge University Press, Cambridge, pp 388–411Google Scholar
  6. Costantino RF, Desharnais RA, Cushing JM, Dennis B (1997) Chaotic dynamics in an insect population. Science 275:389–341CrossRefPubMedGoogle Scholar
  7. Czaran T (1998) Spatiotemporal models of population and community dynamics. Chapman and Hall, LondonGoogle Scholar
  8. Dennis B, Desharnais RA, Cushing JM, Costantino RF (1995) Nonlinear demographic dynamics: mathematical models, statistical methods and biological experiments. Ecol Monogr 65:261–268Google Scholar
  9. Ellner S, Turchin P (1995) Chaos in a noisy world: new methods and evidence from time series analysis. Am Nat 145:343–375CrossRefGoogle Scholar
  10. Godfray HCJ, Grenfell BT (1993) The continuing quest for chaos. Trends Ecol Evol 8:43–44CrossRefGoogle Scholar
  11. Hanski I (1999) Metapopulation ecology. Oxford University Press, OxfordGoogle Scholar
  12. Hassell MP, Lawton JH, May RM (1976) Patterns of dynamical behaviour in single species populations. J Anim Ecol 42:471–486Google Scholar
  13. Hassell MP, Comins HN, May RM (1991) Spatial structure and chaos in insect population dynamics. Nature 353:255–258CrossRefGoogle Scholar
  14. Hassell MP, Comins HN, May RM (1994) Species co-existence and self-organizing spatial dynamics. Nature 370:290–294CrossRefGoogle Scholar
  15. Hastings A, Hom CL, Ellner S, Turchin P, Godfray HCJ (1993) Chaos in ecology: Is mother nature a strange attractor? Annu Rev Ecol Syst 24:1–33Google Scholar
  16. Jaggi S, Joshi A (2001) Incorporating spatial variation in density enhances the stability of simple population dynamics models. J Theor Biol 209:249–255CrossRefPubMedGoogle Scholar
  17. Law R, Dieckmann U (2000) A dynamical system for neighbourhoods in plant communities. Ecology 81:2137–2148Google Scholar
  18. Law R, Murrell DJ, Dieckmann U (2002) On population growth in space and time: the spatial logistic equation. Ecology 84:252–262Google Scholar
  19. Matsuda HN, Ogita A, Sasaki A, Sato K (1992) Statistical mechanics of population: the lattice Lotka-Volterra model. Prog Theor Phys 88:1035–1049Google Scholar
  20. May RM (1974) Biological populations with nonoverlapping generations: stable points, stable cycles and chaos. Science 186:645–647PubMedGoogle Scholar
  21. Neubert MG, Kot M, Lewis MA (1995) Dispersal and pattern-formation in a discrete-time predator-prey model. Theor Popul Biol 48:7–43CrossRefGoogle Scholar
  22. Pacala SW (1986) Neighborhood models of plant population dynamics. 2. Multi-species models of annuals. Theor Popul Biol 29:262–292Google Scholar
  23. Pacala SW (1987) Neighborhood models of plant population dynamics. III. Models with spatial heterogeneity in the physical environment. Theor Popul Biol 31:359–392Google Scholar
  24. Pacala SW, Silander JA Jr (1985) Neighborhood models of plant population dynamics. I. Single-species models of annuals. Am Nat 125:385–411CrossRefGoogle Scholar
  25. Ricker WE (1958) Handbook of computations for biological statistics of fish populations. Fisheries Research Board of Canada, VancouverGoogle Scholar
  26. Rohde K, Rohde PP (2001) Fuzzy chaos: reduced chaos in the combined dynamics of several independently chaotic populations. Am Nat 158:553–556CrossRefGoogle Scholar
  27. Soares P, Tome M (1999) Distance-dependent competition measures for eucalyptus plantations in Portugal. Ann For Sci 56:307–319Google Scholar
  28. Stenseth NC, Bjǿrnstad ON, Saitoh T (1996) A gradient from stable to cyclic populations of Clethrionomys rufocanus in Hokkaido, Japan. Proc R Soc London B 263:1117–1126Google Scholar
  29. Thomas WR, Pomerantz MJ, Gilpin ME (1980). Chaos, asymmetric growth and group selection for dynamical stability. Ecology 61:1312–1320Google Scholar
  30. Travis JMJ, Dytham C (1999) Habitat persistence, habitat availability and the evolution of dispersal. Proc R Soc London B 266:1837–1842CrossRefGoogle Scholar
  31. Turchin P, Taylor AD (1992) Complex dynamics in ecological time series. Ecology 73:289–305Google Scholar
  32. Wang MH, Kot M, Neubert MG (2002) Integrodifference equations, Allee effects, and invasions. J Math Biol 44:150–168CrossRefPubMedGoogle Scholar
  33. Wilson WG, Harrison SP, Hastings A, McCann K (1999) Exploring stable pattern formation in models of tussock moth populations. J Anim Ecol 68:94–107CrossRefGoogle Scholar

Copyright information

© The Society of Population Ecology and Springer-Verlag Tokyo 2004

Authors and Affiliations

  1. 1.Centre for Conservation Science, The ObservatoryUniversity of St AndrewsSt AndrewsUK

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