Phase switching in population cycles
22 Nov 1998-Proceedings of The Royal Society B: Biological Sciences (The Royal Society)-Vol. 265, Iss: 1411, pp 2229-2234
TL;DR: In this paper, phase shifts correspond to stochastic jumps between basins of attraction in an appropriate phase space which associates the different phases of a periodic cycle with distinct attractors, which accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium.
Abstract: Oscillatory populations may exhibit a phase change in which, for example, a high–low periodic pattern switches to a low–high pattern. We propose that phase shifts correspond to stochastic jumps between basins of attraction in an appropriate phase space which associates the different phases of a periodic cycle with distinct attractors. This mechanism accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium .
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TL;DR: In this paper, the authors used the TSIR model to predict the long-term dynamics of measles and the balance between noise and determinism, as a function of population size.
Abstract: Two key linked questions in population dynamics are the relative importance of noise vs. density-dependent nonlinearities and the limits on temporal predictability of population abundance. We propose that childhood microparasitic infections, notably mea- sles, provide an unusually suitable empirical and theoretical test bed for addressing these issues. We base our analysis on a new mechanistic time series model for measles, the TSIR model, which captures the mechanistic essence of epidemic dynamics. The model, and parameter estimates based on short-term fits to prevaccination measles time series for 60 towns and cities in England and Wales, is introduced in a companion paper. Here, we explore how well the model predicts the long-term dynamics of measles and the balance between noise and determinism, as a function of population size. The TSIR model captures the basic dynamical features of the long-term pattern of measles epidemics in large cities remarkably well (based on time and frequency domain analyses). In particular, the model illustrates the impact of secular increases in birth rates, which cause a transition from biennial to annual dynamics. The model also captures the observed increase in epidemic irregularity with decreasing population size and the onset of local extinction below a critical community size. Decreased host population size is shown to be associated with an increased impact of demographic stochasticity. The interaction between nonlinearity and noise is explored using local Lyapunov exponents (LLE). These testify to the high level of stability of the biennial attractor in large cities. Irregularities are due to the limit cycle evolving with changing human birth rates and not due to complex dynamics. The geometry of the dynamics (sign and magnitude of the LLEs across phase space) is similar in the cities and the smaller urban areas. The qualitative difference in dynamics between small and large host communities is that demographic and extinction-recolonization stochasticities are much more influential in the former. The regional dynamics can therefore only be understood in terms of a core-satellite metapopulation structure for this host-enemy system. We also make a preliminary exploration of the model's ability to predict the dynamic consequences of measles vaccination.
250 citations
Cites background from "Phase switching in population cycle..."
...In contrast, the biennial regime exhibits two coexisting attractors that differ only with respect to whether the major peak falls in the odd or the even year (see Henson et al. 1998 for a related discussion)....
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TL;DR: The long-term history of Zeiraphera diniana Gn.
Abstract: The long-term history of Zeiraphera diniana Gn. (the larch budmoth, LBM) outbreaks was reconstructed from tree rings of host subalpine larch in the European Alps. This record was derived from 47513...
203 citations
Cites background from "Phase switching in population cycle..."
...While such transient dynamics have been observed in microcosm experiments (Henson et al. 1998), less evidence exists in natural populations....
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TL;DR: In this article, a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium.
Abstract: A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, non- linear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attri- buting the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic vari- ability and obtained conditional least-squares estimates for the model parameters. We gen- erated 2000 ''bootstrapped data sets'' with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distri- butions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, in- cluding treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.
192 citations
Cites background from "Phase switching in population cycle..."
...For example, noise may cause a cycle to shift phase (Henson et al. 1998) or, in a regime with multiple deterministic attractors, may cause an orbit to jump from one cyclic attractor to another (Henson et al. 1999)....
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TL;DR: This work used 1/ƒ β models to test cycles in the wavelet spectrum against a null hypothesis that takes into account the highly autocorrelated nature of ecological time series and used the maximum covariance analysis to compare the time-frequency patterns of numerous time series.
Abstract: In nature, non-stationarity is rather typical, but the number of statistical tools allowing for non-stationarity remains rather limited. Wavelet analysis is such a tool allowing for non- stationarity but the lack of an appropriate test for statistical inference as well as the difficulty to deal with multiple time series are 2 important shortcomings that limits its use in ecology. We present 2 approaches to deal with these shortcomings. First, we used 1/ƒ β models to test cycles in the wavelet spectrum against a null hypothesis that takes into account the highly autocorrelated nature of ecological time series. To illustrate the approach, we investigated the fluctuations in bluefin tuna trap catches with a set of different null models. The 1/ƒ β models approach proved to be the most consistent to discriminate significant cycles. Second, we used the maximum covariance analysis to compare, in a quantitative way, the time-frequency patterns (i.e. the wavelet spectra) of numerous time series. This approach built cluster trees that grouped the wavelet spectra according to their time-frequency patterns. Controlled signals and time series of sea surface temperature (SST) in the Mediterranean Sea were used to test the ability and power of this approach. The results were satisfactory and clusters on the SST time series displayed a hierarchical division of the Mediterranean into a few homogeneous areas that are known to display different hydrological and oceanic patterns. We discuss the limits and potentialities of these methods to study the associations between ecological and environmental fluctuations.
112 citations
Cites methods from "Phase switching in population cycle..."
...…to compare time series by using their raw properties, the fitted parameters of autoregressive moving average (ARMA) models or their 20 Rouyer et al.: Wavelet analysis of multiple time series rhythmic properties (e.g. Henson et al. 1998, Keogh & Pazzani 1998, Xiong & Yeung 2002, Cazelles 2004)....
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01 Jan 2004
TL;DR: Two main topics are explored: the analogies between the disciplines of ecology and epidemiology at the metapopulation level, and how the metAPopulation theory can help understand the epidemiological dynamics.
Abstract: Publisher Summary
This chapter uses extensive data sets and realistic dynamic models to discuss the metapopulation dynamics of infectious disease. Because of the dual spatial scales of regulation, an extended metapopulation paradigm is central to infectious disease dynamics in two important ways. First, the metapopulation approach can help researchers understand disease dynamics at different spatial scales. Second, there are important conceptual insights into the eradication of infections by vaccination to be gained from studies of the persistence of metapopulations. This chapter therefore explores two main topics: the analogies between the disciplines of ecology and epidemiology at the metapopulation level, and how the metapopulation theory can help understand the epidemiological dynamics. It discusses these issues by using a set of detailed models and high-resolution space-time data of disease incidence.
92 citations
References
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TL;DR: In this article, the authors identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model.
Abstract: 1. We identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model.
2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not.
3. Using two quantitative indices, we examine time series from a Tribolium experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment.
92 citations
"Phase switching in population cycle..." refers background in this paper
...This ¢xed point is stable in some directions and unstable in other directions (i.e. it is a saddle point; see Cushing et al. (1998))....
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TL;DR: A discrete stage-structured mathematical model is derived that explains the phenomenon of animal numbers being larger in cultures grown in a periodically fluctuating volume of medium than in cultures growing in a constant volume of the same average size.
Abstract: Experiments with the flour beetle Tribolium have revealed that animal numbers were larger in cultures grown in a periodically fluctuating volume of medium than in cultures grown in a constant volume of the same average size. In this paper we derive and analyze a discrete stage-structured mathematical model that explains this phenomenon as a kind of resonance effect. Habitat volume is incorporated into the model by the assumption that all rates of cannibalism (larvae on eggs, adults on eggs and pupae) are inversely proportional to the volume of the culture medium. We tested this modeling assumption by conducting and statistically analyzing laboratory experiments. For parameter estimates derived from experimental data, our model indeed predicts, under certain circumstances, a larger (cycle-average) total population abundance when the habitat volume periodically fluctuates than when the habitat volume is held constant at the average volume. The model also correctly predicts certain phase relationships and transient dynamics observed in data. The analyses involve a thorough integration of mathematics, statistical methods, biological details and experimental data.
79 citations
"Phase switching in population cycle..." refers background in this paper
...…including the transitions between dynamic regimes (such as equilibria, two-cycles, three-cycles, invariant loops, and chaos), multiple attractors, and saddle in£uences (Costantino et al. 1995, 1997, 1998; Cushing 1996, 1998; Dennis et al. 1995, 1997; Desharnais et al. 1997; Henson et al. 1999)....
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57 citations
Additional excerpts
...2 in Desharnais & Liu (1987)....
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TL;DR: A research program which covers a spectrum of activities essential to testing nonlinear population theory, from the translation of the biology into the formal language of mathematics, to the analysis of mathematical models, and the development and application of statistical techniques.
Abstract: We describe a research program which covers a spectrum of activities essential to testing nonlinear population theory: from the translation of the biology into the formal language of mathematics, to the analysis of mathematical models, to the development and application of statistical techniques for the analysis of data, to the design and implementation of biological experiments. The statistical analyses, mathematics, and biology are thoroughly integrated. We review several aspects of our current research effort that demonstrate this integration.
51 citations
TL;DR: 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.
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.
48 citations
"Phase switching in population cycle..." refers background in this paper
...…including the transitions between dynamic regimes (such as equilibria, two-cycles, three-cycles, invariant loops, and chaos), multiple attractors, and saddle in£uences (Costantino et al. 1995, 1997, 1998; Cushing 1996, 1998; Dennis et al. 1995, 1997; Desharnais et al. 1997; Henson et al. 1999)....
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...(1995) has successfully explained and predicted nonlinear phenomena in a variety of contexts, including the transitions between dynamic regimes (such as equilibria, two-cycles, three-cycles, invariant loops, and chaos), multiple attractors, and saddle in£uences (Costantino et al. 1995, 1997, 1998; Cushing 1996, 1998; Dennis et al. 1995, 1997; Desharnais et al. 1997; Henson et al. 1999)....
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