Abstract
A central goal of population ecology is to understand and predict fluctuations in population numbers. Until recently, much of the debate focused on the issue of population regulation by density-dependent factors. In this paper, I describe an approach to nonlinear modeling of time-series data that is designed to go beyond this question by investigating the possibility of complex population dynamics, characterized by lags in regulation and periodic or chaotic oscillations. The questions motivating this approach are: what are relative contributions of endogenousvs. exogenous components of dynamics? Is the irregular component in fluctuations entirely due to exogenous noise, or do nonlinearities contribute to it, too? I describe the philosophy and the technical details of the nonlinear modeling approach, and then apply it to a collection of time-series data on vole population fluctuations in northern Europe. The results suggest that population dynamics of European voles undergo a latitudinal shift from stability to chaos. Dynamics in northern Fennoscandia are characterized by positive Lyapunov exponent estimates, and a high degree of short-term (one year ahead) predictability, suggesting a strong endogenous component. In more southerly populations estimated Lyapunov exponents are negative, and there is no one-step ahead predictability, suggesting that fluctuations are driven by exogenous factors.
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Andrewartha, H. G. and L. C. Birch (1954)The distribution and abundance of animals. University of Chicago Press, Chicago.
Argoul, F., A. Arneodo, P. Richetti, J. C. Roux and H. L. Swinnery (1987) Chemical chaos: from hints to confirmation.Accounts of Chemical Research 20: 436–442.
Ayala, F. J., M. E. Gilpin and J. G. Ehrenfeld (1973) Competition between species: theoretical models and experimental tests.Theoretical Population Biology 4: 331–356.
Berryman, A. A. (1978) Population cycles of the douglas-fir tussock moth (Lepidoptera: Lymantriidae): the time-delay hypothesis.Canadian Entomologist 110: 513–518.
Berryman, A. A. (1986) On the dynamics of blackheaded budworm populations.Canadian Entomologist 118: 775–779.
Berryman, A. A. (1991) Chaos in ecology and resource management: what causes it and how to avoid it. pp. 23–38.In J. A. Logan and F. Hain (eds.)Chaos and insect ecology. Virginia Experiment Station Information Series 91-3. Blacksburg, VA.
Berryman, A. A. and J. A. Millstein (1990)Population analysis system. POPSYS series 1, single species analysis. Ecological Systems Analysis, Pullman, WA.
Box, G. E. P. and D. R. Cox (1964) An analysis of transformations.Journal of Royal Statistical Society B26: 211–252.
Box, G. E. P. and N. R. Draper (1987)Empirical model-building and response surfaces. John Wiley & Sons, New York.
Bulmer, M. G. (1974) A statistical analysis of the 10-year cycle in Canada.Journal of Animal Ecology 43: 701–718.
Casdagli, M. (1989) Nonlinear prediction of chaotic time series.Physica D 35: 335–356.
Casdagli, M., S. Eubank, J. D. Farmer and J. Gibson (1991) State space reconstruction in the presence of noise.Physica D 51: 52–98.
Costantino, R. F., J. M. Cushing, B. Dennis and R. A. Desharnais (1995) Experimentally induced transitions in the dynamics behaviour of insect populations.Nature 375: 227–230.
Ellner, S. (1991) Detecting low-dimensional chaos in population dynamics data: a critical review.In F. Hain and J. Logan (eds.)Proceedings of the symposium “Does chaos exist in ecological systems” XIX IUFRO World Congress.
Ellner, S., A. R. Gallant, D. McCaffrey and D. Nychka (1991) Convergence rate and data requirements for Jacobian-based estimates of Lyapunov exponents from data.Physics Letters A 153: 357–363.
Ellner, S., D. W. Nychka and A. R. Gallant (1992)LENNS, a program to estimate the dominant Lyapunov exponent of noisy nonlinear systems from time series data. North Carolina State University, Raleigh, NC.
Ellner, S. and P. Turchin (1995) Chaos in a noisy world: new methods and evidence from time series modeling.American Naturalist 145: 343–375.
Ellner, S., B. Bailey, G. Bobashev, A. R. Gallant, B. Grenfell and D. W. Nychka (1996) Noise and determinism in measles epidemic dynamics: estimates from nonlinear forecasting. (in press)
Finerty, J. P. (1980)The population ecology of cycles in small mammals. Yale University Press, New Haven.
Godfray, H. C. J. and S. P. Blythe (1990) Complex dynamics in multispecies communities.Philosophical Transactions of the Royal Society London B 330: 221–233.
Hanski, I., L. Hansson and H. Henttonen (1991) Specialist predators, generalist predators, and the microtine rodent cycle.Journal of Animal Ecology 60: 353–367.
Hanski, I., H. Henttonen and L. Hansson (1994) Temporal variability and geographic patterns in the population density of microtine rodents: a reply to Xia and Boonstra.American Naturalist 144: 329–342.
Hanski, I. and E. Korpimaki (1995) Microtine rodent dynamics in northern Europe: parameterized models for the predator-prey interaction.Ecology 76: 840–850.
Hanski, I., P. Turchin, E. Korpimaki and H. Henttonen (1993) Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos.Nature 364: 232–235.
Hansson, L. and H. Henttonen (1985) Gradients in density variations of small rodents: the importance of latitude and snow cover.Oecologia 67: 394–402.
Hansson, L. and H. Henttonen (1988) Rodent dynamics as a community processes.Trends in Ecology and Evolution 3: 195–200.
Hassell, M. P., J. H. Lawton and R. M. May (1976) Patterns of dynamical behavior in single species populations.Journal of Animal Ecology 45: 471–486.
Hastings, A., C. L. Hom, S. Ellner, P. Turchin and H. C. J. Godfray (1993) Chaos in ecology; is mother nature a strange attractor?Annual Review of Ecology and Systematics 24: 1–33.
Henttonen, H., T. Oksanen, A. Jortikka and V. Haukisalmi (1987) How much do weasels shape microtine cycles in the northern Fennoscandian taiga?Oikos 50: 353–365.
Hörnfeldt, B. (1994) Delayed density dependence as a determinant of vole cycles.Ecology 75: 791–806.
Ivankina, E. V. (1987)Numerical dynamics and population structure of bank vole near Moscow. Ph. D. Thesis. Moscow University, Moscow.
Ivanter, E. V. (1981) Dinamika chislennosti. pp. 245–267.In N. V. Bashenina (ed.)Evropeyskaya ryzhaya polevka. Nauka, Moscow. (in Russian)
Korpimäki, E. (1994) Rapid or delayed tracking of multi-annual vole cycles by avian predators?Journal of Animal Ecology 63: 619–628.
Korpimäki, E. and K. Norrdahl (1991) Numerical and functional responses of kestrels, short-eared owls, and long-eared owls to vole densities.Ecology,72: 814–826.
Koshkina, T. V. (1966) On the periodical changes in the numbers of voles (as exemplified by the Kola Peninsula).Bulletin of the Moscow Society of Naturalists, Biological Section 71: 14–26. (in Russian).
Laine, K. and H. Henttonen (1983) The role of plant production in microtine cycles in northern Fennoscandia.Oikos 40: 407–415.
Leigh, E. (1968) The ecological role of Volterra's equations. pp. 1–61.In M. Gerstenhaber (ed.)Some mathematical problems in biology. American Mathematical Society, Providence.
Lewontin, R. C. (1966) On the measurement of relative variability.Systematic Zoology 57: 15: 141–142.
Lindström, E. R., H. Andrén, P. Angelstam, G. Cederlund, B. Hörnfeldt, L. Jäderberg, P. A. Lemnell, B. Martinsson, K. Sköld and J. E. Swenson (1994) Disease reveals the predator: sarcoptic mange, red fox predation, and prey populations.Ecology 75: 1042–1049.
Logan, J. and J. C. Allen (1992) Nonlinear dynamics and chaos in insect populations.Annual Review of Entomology 37: 455–477.
May, R. M. (1974) Biological populations with nonoverlapping populations: stable points, stable cycles, and chaos.Science 186: 645–647.
May, R. M. (1976) Simple mathematical models with very complicated dynamics.Nature 261: 459–467.
May, R. M. (1987) Chaos and the dynamics of biological populations.Proceedings of Royal Society, London A 413: 27–44.
Millstein, J. A. and P. Turchin (1994)RAMAS/time, ecological time series modeling and forecasting. Applied Biomathematics, Setauket, New York.
Moran, P. A. P. (1953a) The statistical analysis of the Canadian Lynx cycle. I. Structure and prediction.Australian Journal of Zoology 1: 163–173.
Moran, P. A. P. (1953b) The statistical analysis of the Canadian Lynx cycle. II. Synchronization and meteorology.Australian Journal of Zoology 1: 291–298.
Morris, W. F. (1990) Problems in detecting chaotic behavior in natural populations by fitting simple discrete models.Ecology 71: 1849–1862.
Mueller, L. D. and F. J. Ayala (1981) Dynamics of single-species population growth: stability of chaos?Ecology 62: 1148–1154.
Nicholson, A. J. (1954) An outline of the dynamics of animal populations.Australian Journal of Zoology 2: 9–65.
Nychka, D., S. Ellner, D. McCaffrey and A. R. Gallant (1992) Finding chaos in noisy systems.Journal of Royal Statistical Society B 54: 399–426.
Olsen, L. F. and W. M. Schaffer (1990) Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics.Science 249: 499–504.
Perry, J. N., I. P. Woiwod and I. Hanski (1993) Using responsesurface methodology to detect chaos in ecological time series.Oikos 68: 329–339.
Pollard, E., K. H. Lakhani and P. Rothery (1987) The detection of density-dependence from a series of annual censuses.Ecology 68: 2046–2055.
Pucek, Z., W. Jedrzejewski, B. Jedrzejewska and M. Pucek (1993) Rodent population dynamics in a primeveal deciduous forest (Bialowieza National Park) in relation to weather, seed crop, and predation.Acta Theriologica 38: 199–232.
Rand, D. A. and H. B. Wilson (1991) Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.Proceedings of the Royal Society London, Series B 246: 179–184.
Royama, T. (1977) Population persistence and density dependence.Ecological Monographs 47: 1–35.
Royama, T. (1981) Fundamental concepts and methodology for the analysis of animal population dynamics, with particular reference to univoltine species.Ecological Monographs 51: 473–493.
Royama, T. (1992)Analytical population dynamics. Chapman & Hall, London.
Ruppert, D. (1989) Fitting mathematical models to data: a review of recent developments.In C. Castillo-Chavez, S. A. Levin and C. A. Shoemaker (eds.)Mathematical approaches to problems in resource management and epidemiology. Lecture Notes in Biomathematics 81: 274–284.
Schaffer, W. M. (1985) Order and chaos in ecological systems.Ecology 66: 93–106.
Schaffer, W. M. and M. Kot (1985a) Nearly one-dimensional dynamics in an epidemic.Journal of Theoretical Biology 1112: 403–427.
Schaffer, W. M. and M. Kot (1985b) Do strange attractors govern ecological systems?Bioscience 35: 342–350.
Schaffer, W. M. and M. Kot (1986) Chaos in ecological systems: the coals that Newcastle forgot.Trends in Ecology and Evolution 1: 58–63.
Sokal, R. R. and F. J. Rohlf (1981)Biometry. Freeman, New York.
Southern, H. N. (1979) The stability and instability of small mammal populations. pp. 103–134.In D. M. Stoddard (ed.)Ecology of small mammals. Chapman & Hall, London.
Stenseth, N. C. and E. Framstad (1980) Reproductive effort and optimal reproductive rates in small rodents.Oikos 34: 23–34.
Sugihara, G. and R. M. May (1990) Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series.Nature 344: 734–741.
Turchin, P. (1990) Rarity of density dependence or population regulation with lags?Nature 344: 660–663.
Turchin, P. (1991a) Reconstructing endogenous dynamics of a laboratoryDrosophila population.Journal of Animal Ecology 60: 1091–1098.
Turchin, P. (1991b) Nonlinear modeling of time series data: limit cycles and chaos in forest insects, voles, and epidemics. pp. 39–62.In J. A. Logan and F. P. Hain (eds.)Chaos and insect ecology. Virginia Experiment Station Information Series 91-3. Blacksburg, VA.
Turchin, P. (1993) Chaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis.Oikos 68: 167–172.
Turchin, P. (1995) Population regulation: old arguments and a new synthesis. pp. 19–40.In N. Cappuccino and P. Price (eds.)Population dynamics. Academic Press, New York.
Turchin, P. and I. Hanski (1997) An empirically-based model for the latitudinal gradient in vole population dynamics.American Naturalist (in press)
Turchin, P. and J. A. Millstein (1994) Theoretical background. pp. 1–50.In J. A. Millstein and P. Turchin.RAMAS/time, ecological time series modeling and forecasting. Applied Biomathematics, Setauket, New York.
Turchin, P. and A. Taylor (1992) Complex dynamics in ecological time series.Ecology 73: 289–305.
Wolf, A., J. B. Swift, H. L. Swinney and J. A. Vastano (1985) Determining Lyapunov exponents from a time series.Physica D 16: 285–317.
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Turchin, P. Nonlinear time-series modeling of vole population fluctuations. Res Popul Ecol 38, 121–132 (1996). https://doi.org/10.1007/BF02515720
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DOI: https://doi.org/10.1007/BF02515720