Phase switching in population cycles

doi:10.1098/RSPB.1998.0564

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Phase switching in population cycles

Journal Article•DOI•

Phase switching in population cycles

Shandelle M. Henson, Jim M. Cushing¹, R. F. Costantino², Brian Dennis³, Robert A. Desharnais⁴ - Show less +1 more•Institutions (4)

University of Arizona¹, University of Rhode Island², University of Idaho³, California State University, Los Angeles⁴

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.

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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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Journal Article•DOI•

Dynamics of measles epidemics: scaling noise, determinism, and predictability with the tsir model

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Bryan T. Grenfell¹, Ottar N. Bjørnstad², Bärbel Finkenstädt³•Institutions (3)

University of Cambridge¹, Pennsylvania State University², University of Warwick³

01 May 2002-Ecological Monographs

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.

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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.

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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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Journal Article•DOI•

1200 years of regular outbreaks in alpine insects

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Jan Esper, Ulf Büntgen, David Frank, Daniel Nievergelt, Andrew M. Liebhold¹ - Show less +1 more•Institutions (1)

United States Forest Service¹

07 Mar 2007-Proceedings of The Royal Society B: Biological Sciences

TL;DR: The long-term history of Zeiraphera diniana Gn.

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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...

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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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Journal Article•DOI•

Estimating chaos and complex dynamics in an insect population

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Brian Dennis¹, Robert A. Desharnais², Jim M. Cushing³, Shandelle M. Henson⁴, R. F. Costantino⁵ - Show less +1 more•Institutions (5)

University of Idaho¹, California State University, Los Angeles², University of Arizona³, College of William & Mary⁴, University of Rhode Island⁵

01 May 2001-Ecological Monographs

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.

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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.

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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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Journal Article•DOI•

Analysing multiple time series and extending significance testing in wavelet analysis

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Tristan Rouyer, Jean-Marc Fromentin, Nils Christian Stenseth¹, Bernard Cazelles•Institutions (1)

University of Oslo¹

05 May 2008-Marine Ecology Progress Series

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.

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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.

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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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Book Chapter•DOI•

Metapopulation Dynamics of Infectious Diseases

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Matthew James Keeling¹, Ottar N. Bjørnstad, Bryan T. Grenfell²•Institutions (2)

University of Warwick¹, University of Cambridge²

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.

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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.

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92 citations

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Journal Article•DOI•

Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments'

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Brian Dennis¹, Robert A. Desharnais¹, Jim M. Cushing, R. F. Costantino¹•Institutions (1)

University of Idaho¹

01 Feb 1995-Ecological Monographs

TL;DR: This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

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Abstract: Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

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229 citations

"Phase switching in population cycle..." refers background or methods in this paper

...In Dennis et al. (1995), the maximum likely parameters estimated from the control replicates reported in Desharnais & Costantino (1980) were b 11.6772, l 0.5129, cpa 0.0178, cea 0.0110, cel 0.0093, and a 0.1108, with S 0:2771 0:0279 0:0098 0:0279 0:4284 ÿ0:0081 0:0098 ÿ0:0081 0:0111 0@ 1A: At these…...
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...(a) The LPA model The discrete stage-structured `LPA' Tribolium model of Dennis et al. (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…...
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...The cultures were shown to be oscillating with period two (Dennis et al. 1995)....
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...…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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...We now use the LPA model to explain the chicken-steps observed in Desharnais & Costantino (1980) and Dennis et al. (1995)....
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Journal Article•DOI•

Experimentally induced transitions in the dynamic behaviour of insect populations

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R. F. Costantino¹, Jim M. Cushing², Brian Dennis³, Robert A. Desharnais⁴•Institutions (4)

University of Rhode Island¹, University of Arizona², University of Idaho³, California State University, Los Angeles⁴

18 May 1995-Nature

TL;DR: A simple model describing population growth in the flour beetle Tribolium was developed and it was predicted that changes in adult mortality would produce substantial shifts in population dynamic behaviour, and changes in the dynamics from stable fixed points to periodic cycles to aperiodic oscillations were observed.

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Abstract: SIMPLE nonlinear models can generate fixed points, periodic cycles and aperiodic oscillations in population abundance without any external environmental variation Another familiar theoretical result is that shifts in demographic parameters (such as survival or fecundity) can move a population from one of these behaviours to another1–4 Unfortunately, empirical evidence to support these theoretical possibilities is scarce5–15 We report here a joint theoretical and experimental study to test the hypothesis that changes in demographic parameters cause predictable changes in the nature of population fluctuations Specifically, we developed a simple model describing population growth in the flour beetle Tribolium16 We then predicted, using standard mathematical techniques to analyse the model, that changes in adult mortality would produce substantial shifts in population dynamic behaviour Finally, by experimentally manipulating the adult mortality rate we observed changes in the dynamics from stable fixed points to periodic cycles to aperiodic oscillations that corresponded to the transitions forecast by the mathematical model

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205 citations

"Phase switching in population cycle..." refers background or methods in this paper

...Figure 1b plots larval numbers for two replicate time-series reported in Costantino et al. (1995)....
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...(b) Tribolium castaneum two-cycle data from Costantino et al. (1995). Replicate A (squares) does not...
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...Figure 1b plots larval numbers for two replicate time-series reported in Costantino et al. (1995). These cultures were also shown to be in a two-cycle regime (Dennis et al. 1997). One of the replicates appearing in ¢gure 1b shifts phase, and at that particular time the replicates become asynchronous. An example with a di¡erent animal can be seen in the data recorded by Gause (1964; tab. 3). Figure 1c presents time-series data for two replicate Paramecium caudatum cultures grown separately. Both replicates show two-cycle oscillations followed by several phase shifts. A ¢nal example illustrates the phenomenon of phase switching in data cycling with period three.Three replicate time-series forTribolium castaneum, as reported in Costantino et al. (1997), are shown in ¢gure 1d....
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...Figure 1b plots larval numbers for two replicate time-series reported in Costantino et al. (1995). These cultures were also shown to be in a two-cycle regime (Dennis et al....
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...…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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Journal Article•DOI•

Transitions in population dynamics: Equilibria to periodic cycles to aperiodic cycles

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Brian Dennis¹, Robert A. Desharnais², Jim M. Cushing, R. F. Costantino•Institutions (2)

University of Idaho¹, California State University, Los Angeles²

01 Sep 1997-Journal of Animal Ecology

TL;DR: The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.

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Abstract: 1. We experimentally set adult mortality rates, μ a , in laboratory cultures of the flour beetle Tribolium at values predicted by a biologically based, nonlinear mathematical model to place the cultures in regions of different asymptotic dynamics. 2. Analyses of time-series residuals indicated that the stochastic stage-structured model described the data quite well. Using the model and maximum-likelihood parameter estimates, stability boundaries and bifurcation diagrams were calculated for two genetic strains. 3. The predicted transitions in dynamics were observed in the experimental cultures. The parameter estimates placed the control and μ a = 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At μ a = 0.27 and 0.50 the populations were located in the two-cycle region. With μ a = 0.73 one genetic strain was close to a two-cycle boundary while the other strain underwent another transition and was in a region of equilibrium. In the μ a = 0.96 treatment both strains were close to the boundary at which a bifurcation to aperiodicities occurs; one strain was just outside this boundary, the other just inside the boundary. 4. The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.

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123 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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...These cultures were also shown to be in a two-cycle regime (Dennis et al. 1997)....
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Journal Article•DOI•

Population Fluctuation, an Experimental and Theoretical Approach

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Syunro Utida

01 Jan 1957-Cold Spring Harbor Symposia on Quantitative Biology

109 citations

"Phase switching in population cycle..." refers background in this paper

...Examples include paramecia (Gause 1964), blow £ies (Nicholson 1957), bean weevils (Utida 1957), and £our beetles....
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Journal Article•DOI•

The effect of periodic habitat fluctuations on a nonlinear insect population model

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Shandelle M. Henson¹, Jim M. Cushing¹•Institutions (1)

University of Arizona¹

27 Jun 1997-Journal of Mathematical Biology

TL;DR: These integrated theoretical and experimental results provide the first convincing example illustrating the possibility of increased population numbers in a periodically fluctuating environment.

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Abstract: Laboratory data show that populations of flour beetles (Tribolium), when grown in a periodically fluctuating volume of flour, can exhibit significant increases in numbers above those attained when grown in a constant volume (of the same average). To analyze and explain this phenomenon a discrete stage-structured model of Tribolium dynamics with periodic environmental forcing is introduced and studied. This model is an appropriately modified version of an experimentally validated model for flour beetle populations growing in a constant volume of flour, in which cannibalism rates are assumed inversely proportional to flour volume. This modeling assumption has been confirmed by laboratory experiments. Theorems implying the existence and stability of periodic solutions of the periodically forced model are proved. The time averages of periodic solutions of the forced model are compared with the equilibrium levels of the unforced model (with the same average flour volume). Parameter constraints are determined for which the average population numbers in the periodic environment are greater than (or less than) the equilibrium population numbers in the associated constant environment. Sample parameter estimates taken from the literature show that these constraints are fulfilled. These theoretical results provide an explanation for the experimentally observed increase in flour beetle numbers as a result of periodically fluctuating flour volumes. More generally, these integrated theoretical and experimental results provide the first convincing example illustrating the possibility of increased population numbers in a periodically fluctuating environment.

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96 citations

"Phase switching in population cycle..." refers background in this paper

...Relevant mathematical theorems concerning properties of the LPA model appear in Henson & Cushing (1997) ....
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...Relevant mathematical theorems concerning properties of the LPA model appear in Henson & Cushing (1997)....
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