# Chaotic Dynamics in an Insect Population

17 Jan 1997-Science (American Association for the Advancement of Science)-Vol. 275, Iss: 5298, pp 389-391

TL;DR: A nonlinear demographic model was used to predict the population dynamics of the flour beetle Tribolium under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior.

Abstract: A nonlinear demographic model was used to predict the population dynamics of the flour beetle Tribolium under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior. With the adult mortality rate experimentally set high, the dynamics of animal abundance changed from equilibrium to quasiperiodic cycles to chaos as adult-stage recruitment rates were experimentally manipulated. These transitions in dynamics corresponded to those predicted by the mathematical model. Phase-space graphs of the data together with the deterministic model attractors provide convincing evidence of transitions to chaos.

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TL;DR: This work has shown that measles is a natural ecological system that exhibits different dynamical transitions at different times and places, yet all of these transitions can be predicted as bifurcations of a single nonlinear model.

Abstract: Dramatic changes in patterns of epidemics have been observed throughout this century. For childhood infectious diseases such as measles, the major transitions are between regular cycles and irregular, possibly chaotic epidemics, and from regionally synchronized oscillations to complex, spatially incoherent epidemics. A simple model can explain both kinds of transitions as the consequences of changes in birth and vaccination rates. Measles is a natural ecological system that exhibits different dynamical transitions at different times and places, yet all of these transitions can be predicted as bifurcations of a single nonlinear model.

665 citations

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TL;DR: This work discusses recent advances in understanding ecological dynamics and testing theory using long-term data and reviews how dynamical forces interact to generate some central field and laboratory time series.

Abstract: Both biotic interactions and abiotic random forcing are crucial influences on population dynamics. This frequently leads to roughly equal importance of deterministic and stochastic forces. The resulting tension between noise and determinism makes ecological dynamics unique, with conceptual and methodological challenges distinctive from those in other dynamical systems. The theory for stochastic, nonlinear ecological dynamics has been developed alongside methods to test models. A range of dynamical components has been considered-density dependence, environmental and demographic stochasticity, and climatic forcing-as well as their often complex interactions. We discuss recent advances in understanding ecological dynamics and testing theory using long-term data and review how dynamical forces interact to generate some central field and laboratory time series.

587 citations

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TL;DR: In this article, the authors derived three properties of stochastic multispecies communities that measure different characteristics associated with community stability using first-order multivariate autoregressive (MAR(1)) models.

Abstract: Natural ecological communities are continuously buffeted by a varying environment, often making it difficult to measure the stability of communities using concepts requiring the existence of an equilibrium point. Instead of an equilibrium point, the equilibrial state of communities subject to environmental stochasticity is a stationary distribution, which is characterized by means, variances, and other statistical moments. Here, we derive three properties of stochastic multispecies communities that measure different characteristics associated with community stability. These properties can be estimated from multispecies time-series data using first-order multivariate autoregressive (MAR(1)) models. We demonstrate how to estimate the parameters of MAR(1) models and obtain confidence intervals for both parameters and the measures of stability. We also address the problem of estimation when there is observation (measurement) error. To illustrate these methods, we compare the stability of the planktonic commun...

478 citations

### Cites background from "Chaotic Dynamics in an Insect Popul..."

...Since many of these mechanisms involve nonlinearities, they demand nonlinear models (May and Oster 1976, Hastings et al. 1993, Carpenter et al. 1994a, Costantino et al. 1997, Ives et al. 2000a)....

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TL;DR: The numerically integrated state-space (NISS) method as mentioned in this paper was proposed to fit models to time series of population abun- dances that incorporate both process noise and observation error in a likelihood framework.

Abstract: We evaluate a method for fitting models to time series of population abun- dances that incorporates both process noise and observation error in a likelihood framework. The method follows the probability logic of the Kalman filter, but whereas the Kalman filter applies to linear, Gaussian systems, we implement the full probability calculations numerically so that any nonlinear, non-Gaussian model can be used. We refer to the method as the "numerically integrated state-space (NISS) method" and compare it to two common methods used to analyze nonlinear time series in ecology: least squares with only process noise (LSPN) and least squares with only observation error (LSOE). We compare all three methods by fitting Beverton-Holt and Ricker models to many replicate model-generated time series of length 20 with several parameter choices. For the Ricker model we chose parameters for which the deterministic part of the model produces a stable equilibrium, a two-cycle, or a four-cycle. For each set of parameters we used three process-noise and observation-error scenarios: large standard deviation (0.2) for both, and large for one but small (0.05) for the other. The NISS method had lower estimator bias and variance than the other methods in nearly all cases. The only exceptions were for the Ricker model with stable-equilibrium parameters, in which case the LSPN and LSOE methods has lower bias when noise variances most closely met their assumptions. For the Beverton-Holt model, the NISS method was much less biased and more precise than the other methods. We also evaluated the utility of each method for model selection by fitting simulated data to both models and using information criteria for selection. The NISS and LSOE methods showed a strong bias toward selecting the Ricker over the Beverton-Holt, even when data were generated with the Beverton-Holt. It remains unclear whether the LSPN method is generally superior for model selection or has fortuitously better biases in this particular case. These results suggest that information criteria are best used with caution for nonlinear population models with short time series. Finally we evaluated the convergence of likelihood ratios to theoretical asymptotic distributions. Agreement with asymptotic distributions was very good for stable-point Rick- er parameters, less accurate for two-cycle and four-cycle Ricker parameters, and least accurate for the Beverton-Holt model. The numerically integrated state-space method has a number of advantages over least squares methods and offers a useful tool for connecting models and data and ecology.

438 citations

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University of Amsterdam

^{1}, University of Rostock^{2}, Leibniz Association^{3}, Cornell University^{4}TL;DR: The first experimental demonstration of chaos in a long-term experiment with a complex food web, isolated from the Baltic Sea, demonstrates that species interactions in food webs can generate chaos, and implies that stability is not required for the persistence of complex food webs, and that the long- term prediction of species abundances can be fundamentally impossible.

Abstract: Mathematical models predict that species interactions such as competition and predation can generate chaos. However, experimental demonstrations of chaos in ecology are scarce, and have been limited to simple laboratory systems with a short duration and artificial species combinations. Here, we present the first experimental demonstration of chaos in a long-term experiment with a complex food web. Our food web was isolated from the Baltic Sea, and consisted of bacteria, several phytoplankton species, herbivorous and predatory zooplankton species, and detritivores. The food web was cultured in a laboratory mesocosm, and sampled twice a week for more than 2,300 days. Despite constant external conditions, the species abundances showed striking fluctuations over several orders of magnitude. These fluctuations displayed a variety of different periodicities, which could be attributed to different species interactions in the food web. The population dynamics were characterized by positive Lyapunov exponents of similar magnitude for each species. Predictability was limited to a time horizon of 15-30 days, only slightly longer than the local weather forecast. Hence, our results demonstrate that species interactions in food webs can generate chaos. This implies that stability is not required for the persistence of complex food webs, and that the long-term prediction of species abundances can be fundamentally impossible.

382 citations

##### References

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01 Jan 1990

TL;DR: Non-linear least-squares prediction based on non-linear models and case studies and an introduction to dynamical systems.

Abstract: Preface Acknowlegement Introduction 1. An introduction to dynamical systems 2. Some non-linear time series models 3. Probability structure 4. Statistical aspects 5. Non-linear least-squares prediction based on non-linear models 6. Case studies

2,209 citations

### "Chaotic Dynamics in an Insect Popul..." refers methods in this paper

...These assumptions were found acceptable for many previous data sets (12, 13) by standard diagnostic analyses of time-series residuals (14)....

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TL;DR: This paper presents a dynamical regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur.

Abstract: Some of the simplest nonlinear difference equations describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between 2 population points, to stable cycles with 4, 8, 16, . . . points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized analyses; its existence in such fully deterministic nonlinear difference equations is a fact of considerable mathematical and ecological interest.

1,456 citations

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TL;DR: It is shown that as a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes".

Abstract: Many biological populations breed seasonally and have nonoverlapping generations, so that their dynamics are described by first-order difference equations, Nt+1 = F (Nt). In many cases, F(N) as a function of N will have a hump. We show, very generally, that as such a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes." We give a detailed account of the underlying mathematics of this process and review other situations (in two- and higher dimensional systems, or in differential equation systems) where apparently random dynamics can arise from bifurcation processes. This complicated behavior, in simple deterministic models, can have disturbing implications for the analysis and interpretation of biological data.

1,119 citations