The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A

We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheep-blowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a well-defined mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking natural behavior.

In a classical paper, Hutchinson (1961) argued that the large number of species in most plankton communities is remarkable in view of the competitive exclusion principle, which suggests that in homogeneous, well-mixed environments species that compete for the same resources cannot coexist. Few ideas in aquatic ecology have evoked more research than this `paradox of the plankton'. This review is an effort to put the main solutions to the paradox that have been proposed over the years into perspective. Hutchinson himself already suggested that the explanation could be that plankton communities are not in equilibrium at all due to weather-driven fluctuations. Subsequent research confirmed that such externally imposed variability can allow many species to coexist. Another important point is that in practice the homogeneous well-mixed conditions assumed in the competitive exclusion principle hardly exist. Even the open ocean, for instance, has a spatial complexity resulting from meso-scale vortices and fronts that can facilitate coexistence of species. Perhaps most excitingly, theoretical work on species interactions has given a counter-intuitive new dimension to the understanding of diversity. Various competition and predation models suggest that even in homogeneous and constant environments plankton will never settle to equilibrium. Instead, interactions between multiple species may give rise to oscillations and chaos, with a continuous wax and wane of species within the community. Long-term laboratory experiments support this view. This chaotic behavior implies among other things that plankton dynamics are intrinsically unpredictable in the long run when viewed in detail. Nonetheless, on a higher aggregation level, indicators such as total algal biomass may show quite reg...

reas for two consecutive breeding seasons. This confirmed a strong spatial structuring within grouse populations, which prevented immigration from neighbouring higherdensity areas. In the second autumn, testosterone was not implanted but the recruitment rate remained significantly lower and cock density continued to decline more on the experimental than on the control areas. 5. The results suggest that cocks continued to be aggressive and to maintain large territories for at least a year after aggressiveness was increased experimentally, and therefore that autumn aggressiveness is influenced by previous territorial contests. 6. The experiment validates key assumptions of the `territorial behaviour' hypothesis for red grouse cycles. Population models in a subsequent paper demonstrate how changes in aggressiveness can cause population cycles. Key-words:experiment, testosterone, territorial behaviour, unstable population dynamics. Journal of Animal Ecology (2003) 72, 1073 -- 1

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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 47 513 maximum latewood density measurements, and highlights the impact of contemporary climate change on ecological disturbance regimes. With over 1000 generations represented, this is the longest annually resolved record of herbivore population dynamics, and our analysis demonstrates that remarkably regular LBM fluctuations persisted over the past 1173 years with population peaks averaging every 9.3 years. These regular abundance oscillations recurred until 1981, with the absence of peak events during recent decades. Comparison with an annually resolved, millennium-long temperature reconstruction representative for the European Alps (rZ0.72, correlation with instrumental data) demonstrates that regular insect population cycles continued despite major climatic changes related to warming during medieval times and cooling during the Little Ice Age. The late twentieth century absence of LBM outbreaks, however, corresponds to a period of regional warmth that is exceptional with respect to the last 1000C years, suggesting vulnerability of an otherwise stable ecological system in a warming environment.

Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory complements the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific models of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood function of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right. 1

Time series analysis for nonlinear dynamical systems with applications to modeling of infectious diseases by

Xia and Tong have made a novel contribution to the debate on whether and how to carry out some sort of feature matching in preference to a statistically efficient alternative such as the maximum likelihood estimate (MLE). They show that an estimation criterion emphasizing long-term predictions has some advantages over the MLE on some misspecified time series models. However, emphasizing long-term predictions must lead to a down-weighting of higher-frequency information in the data. In particular, Xia and Tong’s catch-all approach does not typically share the statistical efficiency of MLE when the model fits the data adequately. Further, it is necessarily the case (whatever fitting method is used) that some scientific inferences one might wish to conclude from fitting a misspecified model are statistically invalid. Scientific interpretation of fitted parameter values and predictions using a model that is a statistically poor match to the data therefore requires considerable care. One seeks models that are simultaneously scientifically relevant and provide an adequate statistical description of the data, and then statistical efficiency becomes an important consideration for drawing scientific conclusions from limited data. Flexible modern inference methods facilitate the development and statistical analysis of such models. I will discuss these issues in the context of Xia and Tong’s analysis of Nicholson’s blowfly data. Similar considerations arise in their measles example, and have been investigated by He, Ionides and King (2010). Xia and Tong’s APE(≤1) estimate is equivalent to the MLE only for a specific choice of stochastic model. From their equation (3.12), we see that APE(≤1) corresponds to the MLE for additive, Gaussian, constantvariance process noise with no measurement error. For Xia and Tong’s blowfly model, the log-likelihood at the APE(≤1) point estimate is −1568.5 whereas the log-likelihood at the APE(≤T) point estimate is −1569.5. A chi-squared approximation indicates that

www.elsevier.com/locate/tpb Stochastic analogues of deterministic single-species population models