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We use two modelling approaches to forecast synthetic Ebola epidemics in the context of the RAPIDD Ebola Forecasting Challenge. The first approach is a standard stochastic compartmental model that aims to forecast incidence, hospitalization and deaths among both the general population and health care workers. The second is a model based on the renewal equation with latent variables that forecasts incidence in the whole population only. We describe fitting and forecasting procedures for each model and discuss their advantages and drawbacks. We did not find that one model was consistently better in forecasting than the other. Keywords: Forecasting, Epidemics, Renewal equation, Compartmental model

Background West Nile virus (WNV) is a mosquito-transmitted disease of birds that has caused bird population declines and can spill over into human populations. Previous research has identified bird species that infect a large fraction of the total pool of infected mosquitoes and correlate with human infection risk; however, these analyses cover small spatial regions and cannot be used to predict transmission in bird communities in which these species are rare or absent. Here we present a mechanistic model for WNV transmission that predicts WNV spread (R0) in any bird community in North America by scaling up from the physiological responses of individual birds to transmission at the level of the community. We predict unmeasured bird species’ responses to infection using phylogenetic imputation, based on these species’ phylogenetic relationships with bird species with measured responses. Results We focused our analysis on Texas, USA, because it is among the states with the highest total incidence of WNV in humans and is well sampled by birders in the eBird database. Spatio-temporal patterns: WNV transmission is primarily driven by temperature variation across time and space, and secondarily by bird community composition. In Texas, we predicted WNV R0 to be highest in the spring and fall when temperatures maximize the product of mosquito transmission and survival probabilities. In the most favorable months for WNV transmission (April, May, September and October), we predicted R0 to be highest in the “Piney Woods” and “Oak Woods & Prairies” ecoregions of Texas, and lowest in the “High Plains” and “South Texas Brush County” ecoregions. Dilution effect: More abundant bird species are more competent hosts for WNV, and predicted WNV R0 decreases with increasing species richness. Keystone species: We predicted that northern cardinals (Cardinalis cardinalis) are the most important hosts for amplifying WNV and that mourning doves (Zenaida macroura) are the most important sinks of infection across Texas. Conclusions Despite some data limitations, we demonstrate the power of phylogenetic imputation in predicting disease transmission in heterogeneous host communities. Our mechanistic modeling framework shows promise both for assisting future analyses on transmission and spillover in heterogeneous multispecies pathogen systems and for improving model transparency by clarifying assumptions, choices and shortcomings in complex ecological analyses. Electronic supplementary material The online version of this article (10.1186/s13071-019-3656-8) contains supplementary material, which is available to authorized users.

Sexual reproduction persists in nature despite its large cost. The Red Queen Hypothesis postulates that parasite pressure maintains sexual reproduction in the host population by selecting for the ability to produce rare genotypes that are resistant to infection. Mathematical models have been used to lay theoretical foundations for the hypothesis; empirical studies have confirmed these predictions. For example, Lively used a simple host-parasite model to predict that the frequency of sexual hosts should be positively correlated with the prevalence of infection. Lively et al. later confirmed the prediction through numerous field studies of snail-trematode systems in New Zealand. In this study, we fit a simple metapopulation host-parasite coevolution model to three data sets, each representing a different snail-trematode system, by matching the observed prevalence of sexual reproduction and trematode infection among hosts. Using the estimated parameters, we perform a power analysis to test the feasibility of observing the positive correlation predicted by Lively. We discuss anomalies in the data that are poorly explained by the model and provide practical guidance to both modelers and empiricists. Overall, our study suggests that a simple Red Queen model can only partially explain the observed relationships between parasite infection and the maintenance of sexual reproduction.

Summary Integral projection models (IPMs) allow us to describe quantitatively the dynamics of a population structured by a continuous variable. They rely on information gathered at the individual level by recording survival, reproduction and changes in some structuring variable over time. This requires the ability to track individuals over the course of their entire life cycle. When this is not feasible, we would like to use alternative information to infer a population's dynamics. Time series of population-level data are an option. An inverse modelling approach allows inferring the vital rates of a population when only population-level data, in the form of a time series of the size of a population and the distribution of its individuals along a structuring variable, are available. The approach also allows incorporating estimates obtained through individual-level data. Here, we explore how inverse modelling performs with simulated data and a relatively simple demographic model. We explore scenarios of data availability in terms of time-series length, per-year sample size and availability of independent vital-rate estimates. We also test model performance in a real system using a 15-year long data set from a chamaephyte plant, Cryptantha flava. We show that an inverse model can provide accurate reconstructions of the vital rates in a scenario where no individual-level information is available. Better results can be obtained if independent estimates on any vital rate are provided, as was the case for C. flava where high interannual variation is present. Parameter estimation becomes more difficult with shorter time series, but per-year sample size can be greatly reduced without significantly affecting parameter accuracy. Inverse modelling of IPMs allow for the estimation of unobserved vital rates, which is important for systems where any or all of the vital rates are hard to quantify. It also helps to determine whether a forward IPM is capturing the population dynamics: if the inverse version produces incorrect reconstructions of the vital rates, the forward IPM can be considered as inadequately describing the system.

Post-establishment dynamics of invasive species have been under-studied. However, understanding these dynamics is particularly important for the management of invasive species known to impact native communities. Following the invasion of a highly invasive species, the round goby (Neogobius melanostomus), we document long-term population changes after establishment and address how population dynamics of a successful invader change through persistence and integration. Round goby present a threat to the areas they invade by out-competing native species for resources. Furthermore, as a pollution tolerant species, round goby present a second threat by acting as a possible vector for contaminant transfer to higher trophic levels in invaded ecosystems with areas of contamination. We sampled round goby for 11 years (2002–2012) at four low contamination sites and two high contamination sites within Hamilton Harbour ON, Canada, an International Joint Commission Area of Concern. Across sampling years, we show that round goby abundance has declined at low contamination sites, while remaining stable at high contamination sites. Moreover, we show that average body size decreased and reproductive investment increased both across sampling years and between sites of low and high contamination. Our results document population demographic shifts in a persisting invasive species, and underscore the importance of management practices for this species in contaminated environments.

The description of a species’ mating patterns is often based on observations from a single exemplar population; however, environmental variation can lead to variation in mating patterns and to differences in the strength of sexual selection among populations. In this study, we explored how resource distribution across a species’ range affects competition and the strength of sexual selection in a northern and southern population of plainfin midshipman (Porichthys notatus), a species with 2 male reproductive tactics. Male plainfin midshipman can be guarders that compete for nest sites and court females, or sneakers that attempt to steal fertilizations from the guarder males during spawning. Males from the north population grow larger, suggesting that there might be more competition among males in the north. However, we found that the variance in body size and in nest availability were similar between populations, suggesting instead a similar degree of male-male competition. We found no significant population differences in reproductive success (north: 517 ± 50 eggs/nest ± SE; south: 412 ± 68 eggs/nest ± SE), paternity (north: 52%; south: 58% for the guarding male), or tactic frequencies (north: 88% guarders; south: 91% guarders). There was a marginally steeper Bateman gradient in the south populat ion but no difference at 8 other measures of the strength of sexual selection between the 2 populations. Thus, despite a wide geographic distance, our results show remarkable conservation of mating patterns between the north and south populations of this benthic toadfish.

Genetic introgression has been suggested as a management tool for mitigating detrimental effects of inbreeding depression, but the role of introgression in species conservation has been controversial, partly because population-level impacts of genetic introgressions are not well understood. Concerns about potential inbreeding depression in the endangered Florida panther (Puma con- color coryi) led to the release of eight female Texas pumas (P. c. stanleyana) into the Florida panther population in 1995. We used long-term reproductive data (1995-2008) collected from 61 female Florida panthers to estimate and model reproduction probability (probability of producing a litter) and litter size, and to investigate the influence of intentional genetic introgression on these parameters. Overall, 6-month probability of reproduction (±1SE) was 0.232 ± 0.021 and average litter size was 2.60 ± 0.09. Although F1 admixed females had a lower reproduction probability than females with other ancestries, this was most likely because kittens born to F1 females survive better; consequently, these females are unavailable for breeding until kittens are independent. There was no evi- dence for the effect of ancestry on litter size or of hetero- zygosity on probability of reproduction or litter size. In contrast, earlier studies have shown that genetic intro- gression positively affected Florida panther survival. Our results, along with those of earlier studies, clearly suggest that genetic introgression can have differential effects on components of fitness and highlight the importance of examining multiple demographic parameters when evalu- ating the effects of management actions.

Animals face trade-offs between predation risk and foraging success depending on their location in the landscape; for example, individuals that remain near a common shelter may be safe from predation but incur stronger competition for resources. Despite a long tradition of theoretical exploration of the relationships among foraging success, conspecific competition, predation risk, and population distribution in a heterogeneous environment, the scenario we describe here has not been explored theoretically. We construct a model of habitat use rules to predict the distribution of a local population (prey sharing a common shelter and foraging across surrounding habitats). Our model describes realized habitat quality as a ratio of density- and location-dependent mortality to density-dependent growth. We explore how the prey distribution around a shelter is expected to change as the parameters governing the strength of density dependence, landscape characteristics, and local abundance vary. Within the range of parameters where prey spend some time away from shelter but remain site-attached, the prey density decreases away from shelter. As the distance at which prey react to predators increases, the population range generally increases. At intermediate reaction distances, however, increases in the reaction distance lead to decreases in the maximum foraging distance because of increased evenness in the population distribution. As total abundance increases, the population range increases, average population density increases, and realized quality decreases. The magnitude of these changes differs in, for example, ‘high-’ and ‘low-visibility’ landscapes where prey can detect predators at different distances.

Upper respiratory tract disease (URTD) caused by Mycoplasma agassizii has been hypothesized to contribute to the decline of some wild populations of gopher tortoises (Gopherus polyphemus). However, the force of infection (FOI) and the effect of URTD on survival in free-ranging tortoise populations remain unknown. Using four years (2003-2006) of mark-recapture and epidemiological data collected from 10 populations of gopher tortoises in central Florida, USA, we estimated the FOI (probability per year of a susceptible tortoise becoming infected) and the effect of URTD (i.e., seropositivity to M. agassizii) on apparent survival rates. Sites with high (or = 25%) seroprevalence had substantially higher FOI (0.22 +/- 0.03; mean +/- SE) than low (25%) seroprevalence sites (0.04 +/- 0.01). Our results provide the first quantitative evidence that the rate of transmission of M. agassizii is directly related to the seroprevalence of the population. Seropositive tortoises had higher apparent survival (0.99 +/- 0.0001) than seronegatives (0.88 +/- 0.03), possibly because seropositive tortoises represent individuals that survived the initial infection, developed chronic disease, and experienced lower mortality during the four-year span of our study. However, two lines of evidence suggested possible effects of mycoplasmal URTD on tortoise survival. First, one plausible model suggested that susceptible (seronegative) tortoises in high seroprevalence sites had lower apparent survival rates than did susceptible tortoises in low seroprevalence sites, indicating a possible acute effect of infection. Second, the number of dead tortoise remains detected during annual site surveys increased significantly with increasing site seroprevalence, from approximately 1 to approximately 5 shell remains per 100 individuals. If (as our results suggest) URTD in fact reduces adult survival, it could adversely influence the population dynamics and persistence of this late- maturing, long-lived species.

Traditional mixed stock analyses use morphological, chemical, or genetic markers measured in several source populations and in a single mixed population to estimate the proportional contribution of each source to the mixed population. In many systems, however, different individuals from a particular source population may go to a variety of mixed populations. Now that data are becoming available from (meta)populations with multiple mixed stocks, the need arises to estimate contributions in this 'many-to-many' scenario. We suggest a Bayesian hierarchical approach, an extension of previous Bayesian mixed stock analysis algorithms, that can estimate contributions in this case. Applying the method to mitochondrial DNA data from green turtles (Chelonia mydas) in the Atlantic gives results that are largely consistent with previous results but makes some novel points, e.g. that the Florida, Bahamas and Corisco Bay foraging grounds have greater contributions than previously thought from distant foraging grounds. More generally, the 'many-to-many' approach gives a more complete understanding of the spatial ecology of organisms, which is especially important in species such as the green turtle that exhibit weak migratory connectivity (several distinct subpopulations at one end of the migration that mix in unknown ways at the other end).

We consider the evolution of mutation rate in a seasonally forced, deterministic, compartmental epidemiological model with a transmission-virulence trade-off. We model virulence as a quantitative genetic trait in a haploid population and mutation as continuous diffusion in the trait space. There is a mutation rate threshold above which the pathogen cannot invade a wholly susceptible population. The evolutionarily stable (ESS) mutation rate is the one which drives the lowest average density, over the course of one forcing period, of susceptible individuals at steady state. In contrast with earlier eco-evolutionary models in which higher mutation rates allow for better evolutionary tracking of a dynamic environment, numerical calculations suggest that in our model the minimum average susceptible population, and hence the ESS, is achieved by a pathogen strain with zero mutation. We discuss how this result arises within our model and how the model might be modified to obtain a nonzero optimum.

The role of density-dependent processes in sea turtle populations remains largely unknown. This study quantified density-dependent and density-independent processes that underlie hatchling production in the green turtle nesting population at Tortuguero, Costa Rica, and estimated current mean hatchling output and potential carrying capacity of Tortuguero beach for hatchlings and nesting females. Density-dependent effects of nest destruction by nesting females and coatis were evaluated in the 2000 nesting season along the 28.8 km nesting beach. To quantify factors affecting hatchling production, the fates of nests were monitored in twelve 50 m long study plots. Density-dependent factors included nest destruction by nesting females and predation by coatis, whereas density-independent factors included beach erosion, beach flooding, and below-beach- surface destruction by crabs, ants, microbes, and plant roots. Calculations indicated that between 5 and 6 million hatchlings are currently produced, while a simulation model suggested that at carrying capacity, 6 to 10 times as many hatchlings could be produced by more than 600 000 nesting females. The current mean number of females nesting at Tortuguero is between 3 and 4% of the population that is estimated to nest at carrying capacity. This estimate is consistent with previous estimates that modern day populations of Caribbean green turtles represent only 3 to 7% of pre-exploitation levels. The hatchling production model is applicable to other beaches and sea turtle species, and provides a framework to evaluate recovery goals for sea turtles.

Studies of host–parasite dynamics usually consider one, or at most two, host species, neglecting the possible effects of other species on the focal hosts and vice versa. To explore the interaction of community structure with host–parasite dynamics, we model the invasion of stable communities of varying size by a parasite. The communities are generated with random interaction coefficients and connectance 0.5. Each community is invaded by parasites with different values of virulence (disease-induced host mortality rate), specificity and transmission rate. The result of each invasion is determined by numerically simulating the dynamics of the community. We classify the outcomes by whether the parasite successfully establishes in the focal host population(s), and, if so, by the proportion of host and non-host species that go extinct as a result of the parasite's introduction. We discuss how the structure of the community and the interaction between hosts and other species affect several important processes of disease ecology: the density threshold for parasite invasion, extinction cascades caused by the parasite, and the frequency of extinctions of hosts and non-hosts. In our simulated communities, non-host species went extinct more frequently than hosts, suggesting the importance of the community context of disease. In some cases, the parasite's invasion induced regular population cycles in the previously stable community.

Although spatial patterns of seed distribution are thought to vary greatly among plant species dispersed by different vectors, few studies have directly examined this assumption. We compared patterns of seed rain of nine species of trees disseminated by large birds, monkeys, and wind in a closed canopy forest in Cameroon. We used maximum-likelihood methods to fit seed rain data to four dispersal functions: inverse power, negative exponential, Gaussian, and Student t. We then tested for differences in dispersal characteristics (1) among individuals within species, and (2) among species dispersed by the same vector. In general, an inverse power function best described animal-dispersed species and the Gaussian and Student t functions best described wind-dispersed species. Animal-dispersed species had longer mean dispersal distances than wind-dispersed species, but lower fecundities. In addition to these distinct differences in average dispersal distance and functional form of the seed shadow between animal- and wind-dispersed species, seed shadows varied markedly within species and vector, with conspecifics and species within vector varying in their dispersal scale, fecundity, and clumping parameters. Dispersal vectors determine a significant amount of variation in seed distribution, but much variation remains to be explained. Finally, we demonstrate that most seeds, regardless of vector, fall directly under the parent canopy. Long-distance dispersal events (>60 m) account for a small proportion of the seed crop but may still be important in terms of the absolute numbers of dispersed seeds and effects on population and community dynamics.

For an epidemic to occur in a closed population, the transmission rate must be above a threshold level. In plant populations, the threshold depends not only on host density, but on the distribution of hosts in space. This paper presents an alternative analysis of a previously presented stochastic model for an epidemic in continuous space (Bolker, 1999, Bull. Math. Biol. 61, 849-874). A variety of moment closures are investigated to determine the dependence of the epidemic threshold on host spatial distribution and pathogen dispersal. Local correlations that arise during the early phase of the outbreak determine whether a true global epidemic will occur.

Analytically tractable models of dynamics in continuous space rarely incorporate both endogenous and exogenous spatial heterogeneity. We use spatial moment equations in combination with simulation models to analyze the combined effects of endogenous and exogenous variability on population viability in a simple single-population model where landscape heterogeneity and local population density both affect mortality rate. The equations partition the effects of heterogeneity into an effect of local crowding and an effect of habitat association caused by differential mortality. Exogenous heterogeneity in mortality rate increases population viability through habitat association and decreases it through increased crowding; the net effect of exogenous heterogeneity is generally to improve population viability. This result is contrary to some (but not all) conclusions in the literature, which usually focus on the effects of fragmentation rather than the benefits of refuges to short-dispersing individuals.

Fever is commonly attenuated with antipyretic medication as a means to treat unpleasant symptoms of infectious diseases. We highlight a potentially important negative effect of fever suppression that becomes evident at the population level: reducing fever may increase transmission of associated infections. A higher transmission rate implies that a larger proportion of the population will be infected, so widespread antipyretic drug use is likely to lead to more illness and death than would be expected in a population that was not exposed to antipyretic pharmacotherapies. We assembled the published data available for estimating the magnitudes of these individual effects for seasonal influenza. While the data are incomplete and heterogeneous, they suggest that, overall, fever suppression increases the expected number of influenza cases and deaths in the US: for pandemic influenza with reproduction number , the estimated increase is 1% (95% CI: 0.0–2.7%), whereas for seasonal influenza with , the estimated increase is 5% (95% CI: 0.2–12.1%).

Two processes are universally recognized for inducing spatial synchrony in abundance: dispersal and correlated environmental stochasticity. In the present study we seek the expected relationship between synchrony and distance in populations that are synchronized by density-independent dispersal. In the absence of dispersal, synchrony among populations with simple dynamics has been shown to echo the correlation in the environment. We ask what functional form we may expect between synchrony and distance when dispersal is the synchronizing agent. We formulate a continuous-space, continuous-time model that explicitly represents the time evolution of the spatial covariance as a function of spatial distance. Solving this model gives us two simple canonical functions for dispersal-induced covariance in spatially extended populations. If dispersal is rare relative to birth and death, then covariances between nearby points will follow the dispersal distance distribution. At long distances, however, the covariance tails off according to exponential or Bessel functions (depending on whether the population moves in one or two dimensions). If dispersal is common, then the covariances will follow the mixture distribution that is approximately Gaussian around the origin and with an exponential or Bessel tail. The latter mixture results regardless of the original dispersal distance distribution. There are hence two canonical functions for dispersal-induced synchrony

In most ecological studies, within-group variation is a nuisance that obscures patterns of interest and reduces statistical power. However, patterns of within-group variability often contain information about ecological processes. In particular, such patterns can be used to detect positive growth autocorrelation (consistent variation in growth rates among individuals in a cohort across time), even in samples of unmarked individuals. Previous methods for detecting autocorrelated growth required data from marked individuals. We propose a method that requires only estimates of within-cohort variance through time, using maximum likelihood methods to obtain point estimates and confidence intervals of the correlation parameter. We test our method on simulated data sets and determine the loss in statistical power due to the inability to identify individuals. We show how to accommodate nonlinear growth trajectories and test the effects of size-dependent mortality on our method's accuracy. The method can detect significant growth autocorrelation at moderate levels of autocorrelation with moderate-sized cohorts (for example, statistical power of 80% to detect growth autocorrelation ρ (2) = 0.5 in a cohort of 100 individuals measured on 16 occasions). We present a case study of growth in the red-eyed tree frog. Better quantification of the processes driving size variation will help ecologists improve predictions of population dynamics. This work will help researchers to detect growth autocorrelation in cases where marking is logistically infeasible or causes unacceptable decreases in the fitness of marked individuals.

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.

We aim to develop a simple model to explore how disturbance and propagule pressure determine conditions for successful invasion in systems where recruitment occurs only in disturbed sites. Disturbance is often thought to favour invaders as it allows recruitment; however, the effects of disturbance are more complicated when it results in mortality of the invader. When disturbance rates in both invader occupied and unoccupied sites are the same, recruitment and mortality effects are exactly balanced, and successful invasion is independent of the disturbance regime. Differences in the disturbance rates between invader occupied and unoccupied sites can occur through invader modification or management of disturbance. Under these conditions, we found a novel mechanism for the generation of an Allee effect, which occurs when the invader promotes disturbance in sites it already occupies. When Allee effects occur one-off, large-scale disturbances can result in permanent, dramatic shifts in invader abundance; and conversely, reducing the population below a critical threshold can cause extinction.

In this paper, we introduce a Levins-type metapopulation model with empty and occupied patches, and dispersing population. We structure the proportion of occupied patches according to the patch-occupancy age. We observe that patch-occupancy age may destabilize the metapopulation, leading to persistent oscillations. We also allow for the dispersal rate to vary with the proportion of empty patches in a monotone or unimodal way. The unimodal dependence leads to multiple non-trivial equilibria and bistability when the reproduction number of the metapopulation R1 but greater than a lower critical value R(*). We show that the metapopulation will persist independently of its initial status if R1.

Publisher Summary This chapter provides an overview of models for ecological dynamics in continuous space, while focusing on spatial rather than landscape ecology. In particular, it explores spatial moment equations, a relatively new framework for analyzing spatial dynamics in terms of mean population densities and spatial covariances. The chapter also gives a sample derivation of a set of spatial moment equations; summarizes various applications of moment equations for single-species and community dynamics; contrasts the strengths and weaknesses of the approach with other frameworks such as PDEs and IPSs; and discusses future directions and potential of spatial moment equations. Although this chapter concentrates on the effects of endogenous rather than exogenous variability, it also describes some strategies for incorporating both kinds of heterogeneities and bridging the gap between spatial and landscape ecology. Spatial moment equations are a powerful tool for this task; other advantages include preservation of the spatial and stochastic character of ecological systems; analytical tractability; and simple connections to field data on individual dispersal and performance and to well-established spatial statistical measures.

Introduction Spatial dynamics of populations have long been of interest to ecologists (Skellam 1951; Levin and Paine 1974; Andow et al . 1990), but recent advances in data collection and in computational power have put these concepts within the reach of many ecologists for the first time. Computational models (Wilson et al . 1993, 1995b; McCauley et al . 1993; Pacala and Deutschman 1995) suggest important and previously unexplored effects of space and discrete individuals on population dynamics. Analytic approaches that capture these effects are emerging, building on methods developed in other contexts. This chapter presents a general method for deriving approximate equations for spatial dynamics in continuous space and time that has advantages over classical and many modern approaches. We are interested in spatial pattern formation in plant communities and in the effects of pattern on plant competition. Our goal is to find general methods for exploring this problem that are analytically tractable, so that we can gain insight into qualitative behaviors of the system and analyze how they depend on the parameters; sufficiently general, so that some of the same tools can be applied to answer a range of different questions about spatial dynamics in ecology; and close enough to the characteristics of real populations that we can eventually fit the models to field data on individual behavior. We focus on spatial point processes (Diggle 1983; Gandhi et al . 1998), continuous-time dynamical systems for discrete individuals interacting in a continuous habitat.

Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean and spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by "moment closure," neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models. Copyright 1997 Academic Press. Copyright 1997 Academic Press

There is currently considerable interest in the role of nonlinear phenomena in the population dynamics of infectious diseases. Childhood diseases such as measles are particularly well documented dynamically, and have recently been the subject of analyses (of both models and notification data) to establish whether the pattern of epidemics is chaotic. Though the spatial dynamics of measles have also been extensively studied, spatial and nonlinear dynamics have only recently been brought together. The present review concentrates mainly on describing this synthesis. We begin with a general review of the nonlinear dynamics of measles models, in a spatially homogeneous environment. Simple compartmental models (specifically the SEIR model) can behave chaotically, under the influence of strong seasonal 'forcing' of infection rate associated with patterns of schooling. However, adding observed heterogeneities such as age structure can simplify the deterministic dynamics back to limit cycles. By contrast all current strongly seasonally forced stochastic models show large amplitude irregular fluctuations, with many more 'fadeouts' of infection that is observed in real communities of similar size. This indicates that (social and/ or geographical) spatial heterogeneity is needed in the models. We review the exploration of this problem with nonlinear spatiotemporal models. The few studies to date indicate that spatial heterogeneity can help to increase the realism of models. However, a review of nonlinear analyses of spatially subdivided measles data show that more refinements of the models (particularly in representing the impact of human demo graphic changes on infection dynamics) are required. We conclude with a discussion of the implication of these results for the dynamics of infectious diseases in general and, in particular, the possibilities of cross fertilization between human disease epidemiology and the study of plant and animal diseases.

Tree cover is a fundamental structural characteristic and driver of ecosystem processes in terrestrial ecosystems, and trees are a major global carbon (C) sink. Fire and herbivores have been hypothesized to play dominant roles in regulating trees in African savannas, but the evidence for this is conflicting. Moving up a trophic scale, the factors that regulate fire occurrence and herbivores, such as disease and predation, are poorly understood for any given ecosystem. We used a Bayesian state-space model to show that the wildebeest population irruption that followed disease (rinderpest) eradication in the Serengeti ecosystem of East Africa led to a widespread reduction in the extent of fire and an ongoing recovery of the tree population. This supports the hypothesis that disease has played a key role in the regulation of this ecosystem. We then link our state-space model with theoretical and empirical results quantifying the effects of grazing and fire on soil carbon to predict that this cascade may have led to important shifts in the size of pools of C stored in soil and biomass. Our results suggest that the dynamics of herbivores and fire are tightly coupled at landscape scales, that fire exerts clear top-down effects on tree density, and that disease outbreaks in dominant herbivores can lead to complex trophic cascades in savanna ecosystems. We propose that the long-term status of the Serengeti and other intensely grazed savannas as sources or sinks for C may be fundamentally linked to the control of disease outbreaks and poaching.

View Large Image | Download PowerPoint SlideRuss Lande, Steinar Engen and Bernt-Erik Saether have been using stochastic models to understand ecological communities for a combined total of nearly 60 years, individually and in collaboration. Their new book, Stochastic Population Dynamics in Ecology and Conservation, presents a broad sampling of these efforts, and is a smorgasbord that will be palatable to mathematically proficient readers, or to those willing to take some of the results on faith.The book covers a wide range of topics that are relevant to both basic and applied ecology: linear and nonlinear population growth, extinction dynamics, population viability analysis, sustainable harvesting, biodiversity metrics, and neutral theories in community ecology, among others. The authors prefer simple models, aiming for general conclusions rather than mathematical rigor. They rightly criticize overdependence on simulations, but use them where appropriate to extend the conclusions of analytical models. Lande et al. follow in the tradition of ecological modelers such as Nisbet and Gurney [1xNisbet, R.M and Gurney, W.S.C. See all References[1], emphasizing the power and applicability of linear approximations rather than the complexities of strongly nonlinear models. They also sensibly eschew discrete models, such as branching processes, which achieve realism by following the fates of discrete individuals but pay a high price in flexibility and tractability. The authors show that the continuous-population alternative, stochastic partial differential equations, are a powerful framework for answering a broad spectrum of ecological questions (they wisely sweep the difficult mathematical assumptions that underlie these models under the rug).The drawback of the tremendous breadth of the book is that it prevents it from being a solid introduction to any particular topic. I was hoping to find an introductory text on stochastic dynamics (complementary to Denny and Gaines' excellent Chance [2xDenny, M and Gaines, S. See all References[2]), but this book doesn't fill that niche. The authors do introduce the basics of stochastic dynamics in the first chapter, and generally present modeling frameworks, such as diffusion approximations, early and return them in later chapters. However, the style assumes a high mathematical comfort level, and the book sometimes glosses over the deeper context of the methods: the assurance of the Introduction that readers need only a working knowledge of elementary calculus and statistics is technically correct, but optimistic. For this reason, Stochastic Population Dynamics will work best as a sampler of applications for mathematically inclined ecologists who can then go back to the primary literature for more detail.The authors challenge their theories with empirical data, weighing in on a variety of current topics. For example, they point out that temporal autocorrelation in population dynamics, often taken as a proxy for density dependence, can also arise from lags implicit in the life history of an organism. They present a straightforward method, based on simulating population dynamics with bootstrapped parameters, to address recent concerns [3xWhen is it meaningful to estimate an extinction probability?. Fieberg, J and Ellner, S.P. Ecology. 2000; 81: 2040–2047CrossrefSee all References[3] that population viability analysis fails to incorporate uncertainty in parameters. (Although I like their approach, I was disappointed by their rote dismissal of Bayesian approaches. I also disagree with their recommendation to present only the upper confidence limit of the extinction probability, which leads to the conclusion that any poorly understood species is endangered. Isn't it more honest to say that the confidence limits on extinction probability are 1–99% than to say only that there is a chance that extinction rate could be as high as 99%?) The authors also showcase their recent work on the roles of dispersal and environmental variation in driving spatial synchrony in population fluctuations. Finally, they use a new approach to partitioning metrics of biodiversity, which I initially took to be of theoretical interest only, as a platform to test (and reject) Hubbell's neutral theory of community assembly [4xHubbell, S.P. See all References[4] for a community of tropical butterflies.Stochastic Population Dynamics provides a snapshot of research into stochastic population dynamics by some of the best workers in the field. It gives the flavor of the authors' accomplished and well balanced approach to stochastic dynamic modeling, but it is definitely a sampler rather than a practical introduction to the methods. It will appeal to ecologists who are up for a mathematical challenge and who want to see the variety of theoretical and applied questions that one can answer by clever application of stochastic models to ecological systems.