A. Kleczkowski

Seasonality and Extinction in Chaotic Metapopulations

A body of recent work has used coupled logistic maps to show that these model metapopulations show a decrease in global extinction rate in the chaotic region of model behaviour. In fact, many of the main ecological candidates for low-dimensional chaos are continuous-time host-parasite and predator-prey systems, driven by strong seasonal ‘forcing’ of one or more population parameters. This paper, therefore, explores the relation between seasonal forcing and metapopulation extinction for such systems. We base the analysis on extensive simulations of a stochastic metapopulation model for measles, based on a standard compartmental model, tracking the density of susceptible, exposed, infectious and recovered individuals (the SEIR model). The results show that, by contrast with coupled logistic maps, the increased seasonality which causes chaos maintains or increases levels of global extinction of infection, by increasing the synchrony of sub-population epidemics. The model also illustrates that the population interaction (here between susceptible and infective hosts) has a significant effect on patterns of synchrony and extinction.

124,000-year periodicity in terrestrial vegetation change during the late Pliocene epoch

The late Pliocene (∼3–2.6 million years ago) is an interval of exceptional interest for understanding the Earths climate system. It was a time of progressive global cooling, resulting in the growth of large terrestrial ice sheets and the initiation of extensive Northern Hemisphere glaciation,. The build up of the ice sheets was cyclical and apparently paced by the orbitally driven oscillations in incoming solar radiation (Milankovitch cycles) at periods of approximately 41 kyr (obliquity) and 23–19 kyr (precession). Here we present a high-resolution continental record of late Pliocene climate change, detailing the response of terrestrial vegetation to this interval of dramatic global environmental change. The annually laminated sequence of lake sediments from Pula maar, in Hungary, represents approximately 320 kyr of accumulation between ∼3.0 and 2.6 million years ago. Spectral analyses of the record indicate terrestrial responses to incoming solar radiation at obliquity and precession periodicities, but the strongest response appears at a period of ∼124 kyr. Calculations indicate that variations in insolation forcing at this periodicity were negligible at this time. The Pula record thus demonstrates that internally driven nonlinear responses of the climate system, at a period of ∼124 kyr, were at least as important as external forcing at the orbital frequencies of precession and obliquity in driving late Pliocene large-scale environmental change.

Mean-field-type equations for spread of epidemics: the ‘small world’ model

In the paper we study a cellular automata (CA) model of epidemic dynamics. The effects of local spatial correlations on a temporal (aggregated) spread of single epidemics are studied, as a function of increasing proportion of global contacts (‘small world’ model). We conjecture that even in the presence of high local correlations, the aggregated (mean-field-type) models can be quite successful, if the contact rate is treated as a free parameter. The dependence of the (estimated) contact rate on the mixing parameter can be understood in terms of a simple probabilistic model. The contact rate reflects not only a microscopic and epidemiological situation, but also a complicated social pattern, including short- and long-range contacts as well as a possibly hierarchical structure of human society.

Dynamically Generated Variability in Plant-Pathogen Systems with Biological Control

Using a combination of replicated microcosm experiments, simple nonlinear modelling and model fitting we show that unexpected levels of variability can be detected and described in the dynamics of plant disease. Temporal development of damping-off disease of radish seedlings caused by an economically important plant pathogen, Rhizoctonia solani, is quantified, with and without the addition of an antagonistic fungus, Trichoderma viride. The biological control agent reduces the average amount of disease but also greatly enhances the variability among replicates. The results are shown to be consistent with predictions from a nonlinear model that exhibits dynamically generated variability in which small differences in the initiation of infection associated with the antagonist are later amplified as the pathogen spreads from plant to plant. The effect of dynamically generated variability is mediated by the interruption of transient disease progress curves for separate replicates by an exponential decrease in susceptibility of the host over time. The decay term essentially ‘freezes’ the dynamics of the transient behaviour so that the solutions for different replicates settle on asymptotes that depend on initial conditions and parameter values. The effect is further magnified by nonlinear terms in the infection force in the models. A generalization of the Lyapunov exponent is introduced to quantify the amplification. The observed behaviour has profound consequences for the design and interpretation of ecological experiments, and can also account for the notorious failure of many biological control strategies through the creation of ‘hot spots’, created by the amplification of plant to plant infection, where the control by the antagonist is locally unsuccessful.

Measles as a case study in nonlinear forecasting and chaos

This paper uses measles incidence in developed countries as the basis of a case study in nonlinear forecasting and chaos. It uses a combination of epidemiological modelling and nonlinear forecasting to explore a range of issues relating to the predictability of measles before and after the advent of mass vaccination. A comparison of the pre-vaccination self-predictability of measles in England and Wales indicates relatively high predictability of these predominantly biennial epidemic series, compared to New York City, which shows mixtures of one-, twoand three-year epidemics. This analysis also indicates the importance of choosing correct embeddings to avoid bias in prediction. Forecasting for English cities indicates significant spatial heterogeneity in predictability before vaccination and an overall drop in predictability during the vaccination era. The interpretation of predictions of observed measles series by epidemiological models is explored and areas for refinement of current models discussed.

Scaling and spatial dynamics in plant–pathogen systems: from individuals to populations

Components of transmission for primary infection from soil–borne inoculum and secondary (plant to plant) infection are estimated from experiments involving single plants. The results from these individual–based experiments are used in a probabilistic spatial contact process (cellular automaton) to predict the progress of an epidemic. The model accounts for spatial correlations between infected and susceptible plants due to inhomogeneous mixing caused by restricted movement of the pathogen in soil. It also integrates nonlinearities in infection, including small stochastic differences in primary infection that become amplified by secondary infection. The model predicts both the mean and the variance of the infection dynamics of R. solani when compared with replicated epidemics in populations of plants grown in microcosms. The broader consequences of the combination of experimental and modelling approaches for scaling–up from individual to population behaviour are discussed.

Modelling control of epidemics spreading by long-range interactions

We have studied the spread of epidemics characterized by a mixture of local and non-local interactions. The infection spreads on a two-dimensional lattice with the fixed nearest neighbour connections. In addition, long-range dynamical links are formed by moving agents (vectors). Vectors perform random walks, with step length distributed according to a thick-tail distribution. Two distributions are considered in this paper, an α-stable distribution describing self-similar vector movement, yet characterized by an infinite variance and an exponential power characterized by a large but finite variance. Such long-range interactions are hard to track and make control of epidemics very difficult. We also allowed for cryptic infection, whereby an infected individual on the lattice can be infectious prior to showing any symptoms of infection or disease. To account for such cryptic spread, we considered a control strategy in which not only detected, i.e. symptomatic, individuals but also all individuals within a certain control neighbourhood are treated upon the detection of disease. We show that it is possible to eradicate the disease by using such purely local control measures, even in the presence of long-range jumps. In particular, we show that the success of local control and the choice of the optimal strategy depend in a non-trivial way on the dispersal patterns of the vectors. By characterizing these patterns using the stability index of the α-stable distribution to change the power-law behaviour or the exponent characterizing the decay of an exponential power distribution, we show that infection can be successfully contained using relatively small control neighbourhoods for two limiting cases for long-distance dispersal and for vectors that are much more limited in their dispersal range.

Quenched disorder and long-tail distributions

A model of overdamped and externally stimulated oscillators is discussed. It is shown analytically that in the uncoupled case a wide class of random distributions of parameters of individual oscillators leads to a long-tail distribution of resting points. Interactions between the individual oscillators destroy these long tails partially (nearest-neighbours interaction) or completely (mean field interactions). As the levels of a local coupling increase, domains of similarly acting oscillators are formed. The collective behaviour becomes important for large local coupling at which the long tails are destroyed. In this case, the observed pattern of resting states is a reflection of both the quenched disorder and interactions between the oscillators.