Abstract
Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ’saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.
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References
Beniot, H. P., E. McCauley and J. R. Post (1998). Testing the demographic consequences of cannibalism in Tribolium confusum. Ecology 79, 2839–2851.
Costantino, R. F., J. M. Cushing, B. Dennis and R. A. Desharnais (1995). Experimentally induced transitions in the dynamic behavior of insect populations. Nature 375, 227–230.
Costantino, R. F., J. M. Cushing, B. Dennis, R. A. Desharnais and S. M. Henson (1998). Resonant population cycles in alternating habitats. Bull. Math. Biol. 60, 247–273.
Costantino, R. F., R. A. Desharnais, J. M. Cushing and B. Dennis (1997). Chaotic dynamics in an insect population. Science 275, 389–391.
Cushing, J. M., B. Dennis, R. A. Desharnais and R. F. Costantino (1996). An interdisciplinary approach to understanding non-linear ecological dynamics. Ecol. Model. 92, 111–119.
Cushing, J. M., B. Dennis, R. A. Desharnais and R. F. Costantino (1998). Moving toward an unstable equilibrium: saddle nodes in population systems. J. Anim. Ecol. 67, 298–306.
Dennis, B., R. A. Desharnais, J. M. Cushing and R. F. Costantino (1995). Non-linear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecol. Monogr. 65, 261–281.
Dennis, B., R. A. Desharnais, J. M. Cushing and R. F. Costantino (1997). Transitions in population dynamics: equilibria to periodic cycles to aperiodic cycles. J. Anim. Ecol. 66, 704–729.
Desharnais, R. A. and R. F. Costantino (1980). Genetic analysis of a population of Tribolium. VII. Stability: response to genetic and demographic perturbations. Can. J. Genetics Cytol. 22, 577–589.
Desharnais, R. A., R. F. Costantino, J. M. Cushing and B. Dennis (1997). Estimating chaos in an insect population. Science 276, 1881–1882.
Henson, S. M. (1999). Multiple attractors and resonance in periodically-forced population models, (Submitted.)
Henson, S. M. and J. M. Cushing (1997). The effect of periodic habitat fluctuations on a non-linear insect population model. J. Math. Biol. 36, 201–226.
Henson, S. M., J. M. Cushing, R. F. Costantino, B. Dennis and R. A. Desharnais (1998). Phase switching in population cycles. Proc. R. Soc. Lond. B 265, 2229–2234.
Jillson, D. (1980). Insect populations respond to fluctuating environments. Nature 288, 699–700.
May, R. M. (1974). Biological populations with non-overlapping generations: stable points, stable cycles, and chaos. Science 186, 645–647.
May, R. M. (1977). Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269, 471–477.
Moffa, A. M. and R. F. Costantino (1977). Genetic analysis of a population of Tribolium. VI. Polymorphism and demographic equilibrium. Genetics 87, 785–805.
Neubert, M. G. (1997). A simple population model with qualitatively uncertain dynamics. J. Theor. Biol. 189, 399–411.
Nisbet, R. M. and W. S. C. Gurney (1981). Modelling Fluctuating Populations, New York: Wiley and Sons.
Nusse, H. E. and J. A. Yorke (1996). Basins of attraction. Science 271, 1376–1380.
Oster, G. and Y. Takahashi (1974). Models for age-specific interactions in a periodic environment. Ecol. Monogr. 44, 483–501.
Park, T., M. Nathanson, J. R. Ziegler and D. B. Mertz (1970). Cannibalism of pupae by mixed-species populations of adult Tribolium. Physiol. Zool. 43, 166–184.
Petraitis, P. S. and R. E. Latham (1999). The importance of scale in testing the origins of alternative community states. Ecology 80, 429–442.
Renshaw, E. (1991). Modelling Biological Populations in Space and Time, Cambridge: Cambridge University Press.
Shaffer, M. L. (1981). Minimum population sizes for species conservation. Bioscience 31, 131–134.
Slobodkin, L. B. (1961). Growth and Regulation of Animal Populations, New York: Holt, Rinehart and Winston.
Tong, H. (1990). Non-linear Time Series: a Dynamical Approach, Oxford, England: Oxford University Press.
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Henson, S.M., Costantino, R.F., Cushing, J.M. et al. Multiple attractors, saddles, and population dynamics in periodic habitats. Bull. Math. Biol. 61, 1121–1149 (1999). https://doi.org/10.1006/bulm.1999.0136
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DOI: https://doi.org/10.1006/bulm.1999.0136