When mechanism matters: Bayesian forecasting using models of ecological diffusion

Ecological diffusion is a theory that can be used to understand and forecast spatio‐temporal processes such as dispersal, invasion, and the spread of disease. Hierarchical Bayesian modelling provides a framework to make statistical inference and probabilistic forecasts, using mechanistic ecological models. To illustrate, we show how hierarchical Bayesian models of ecological diffusion can be implemented for large data sets that are distributed densely across space and time. The hierarchical Bayesian approach is used to understand and forecast the growth and geographic spread in the prevalence of chronic wasting disease in white‐tailed deer (Odocoileus virginianus). We compare statistical inference and forecasts from our hierarchical Bayesian model to phenomenological regression‐based methods that are commonly used to analyse spatial occurrence data. The mechanistic statistical model based on ecological diffusion led to important ecological insights, obviated a commonly ignored type of collinearity, and was the most accurate method for forecasting.     

Current approaches using genetic distances produce poor estimates of landscape resistance to interindividual dispersal

Landscape resistance reflects how difficult it is for genes to move across an area with particular attributes (e.g. land cover, slope). An increasingly popular approach to estimate resistance uses Mantel and partial Mantel tests or causal modelling to relate observed genetic distances to effective distances under alternative sets of resistance parameters. Relatively few alternative sets of resistance parameters are tested, leading to relatively poor coverage of the parameter space. Although this approach does not explicitly model key stochastic processes of gene flow, including mating, dispersal, drift and inheritance, bias and precision of the resulting resistance parameters have not been assessed. We formally describe the most commonly used model as a set of equations and provide a formal approach for estimating resistance parameters. Our optimization finds the maximum Mantel r when an optimum exists and identifies the same resistance values as current approaches when the alternatives evaluated are near the optimum. Unfortunately, even where an optimum existed, estimates from the most commonly used model were imprecise and were typically much smaller than the simulated true resistance to dispersal. Causal modelling using Mantel significance tests also typically failed to support the true resistance to dispersal values. For a large range of scenarios, current approaches using a simple correlational model between genetic and effective distances do not yield accurate estimates of resistance to dispersal. We suggest that analysts consider the processes important to gene flow for their study species, model those processes explicitly and evaluate the quality of estimates resulting from their model.     

Modelling larval dispersal and behaviour of coral reef fishes

Coral reef fish spend their first few weeks developing in the open ocean, where eggs and larvae appear merciless to tides and currents, before attempting to leave the pelagic zone and settle on a suitable reef. This pelagic dispersal phase is the process that determines population connectivity and allows replenishment of harvested populations across multiple coral reef habitats. Until recently this pelagic larval dispersal phase has been poorly understood and has often been referred to as the ‘black-box’ in the life-history of coral reef fishes. In this perspective article we highlight three areas where mathematical and computational approaches have been used to aid our understanding of this important ecological process. We discuss models that provide insights into the evolution of the pelagic larval phase in coral reef fish, an unresolved question which lends itself well to a modelling approach due to the difficulty in obtaining empirical data on this life history strategy. We describe how studies of fish hearing and physical sound propagation models can be used to predict the detection distance of reefs for settling larval fish, and the potential impact of anthropogenic noise. We explain how random walk models can be used to explore individual- and group-level behaviour in larval fish during the dispersal and settlement stage of their life-history. Finally, we discuss the mutual benefits that mathematical and computational approaches have brought to and gained from the field of larval behaviour and dispersal of reef fishes.     

Spatially explicit modelling of transgenic maize pollen dispersal and cross-pollination

Modelling of pollen dispersal and cross-pollination is of great importance for the ongoing discussion on thresholds for the adventitious presence of genetically modified material in food and feed. Two different modelling approaches for pollen dispersal are used to simulate the cross-pollination rate of pollen emerged from an adjacent transgenic crop field. The models are applied to cross-pollination data from field experiments with transgenic maize (Zea mays). The data were generated by an experimental setup specifically designed to suit the demands of mathematical modelling. First a Gaussian plume model is used for the simulation of pollen transport in and from plant canopies. This is a semiempirical approach combining the atmospheric diffusion equation and Lagrangian methodology. The second model is derived from the localised near field (LNF) theory and based on the physical processes in the canopy. Both modelling approaches prove to be appropriate for the simulation of the cross-pollination rates at distances of about 7.5m and more from the transgene source. The simulation of the cross-pollination rate is less precise at the edge of the source plot especially with the LNF theory. However, the simulation results lie within the range of variability of the observations. Concluding can be pointed out that both models might be adapted to other pollen dispersal experiments of different crops and plot sizes.     

Development of a new version of the Liverpool Malaria Model. II. Calibration and validation for West Africa

Mathematical models have been used to provide an explicit framework for understanding malaria transmission dynamics in human population for over 100 years. With the disease still thriving and threatening to be a major source of death and disability due to changed environmental and socio-economic conditions, it is necessary to make a critical assessment of the existing models, and study their evolution and efficacy in describing the host-parasite biology. In this article, starting from the basic Ross model, the key mathematical models and their underlying features, based on their specific contributions in the understanding of spread and transmission of malaria have been discussed. The first aim of this article is to develop, starting from the basic models, a hierarchical structure of a range of deterministic models of different levels of complexity. The second objective is to elaborate, using some of the representative mathematical models, the evolution of modelling strategies to describe malaria incidence by including the critical features of host-vector-parasite interactions. Emphasis is more on the evolution of the deterministic differential equation based epidemiological compartment models with a brief discussion on data based statistical models. In this comprehensive survey, the approach has been to summarize the modelling activity in this area so that it helps reach a wider range of researchers working on epidemiology, transmission, and other aspects of malaria. This may facilitate the mathematicians to further develop suitable models in this direction relevant to the present scenario, and help the biologists and public health personnel to adopt better understanding of the modelling strategies to control the disease     

Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?

What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.     

Making statistics biologically relevant in fragmented landscapes

The biological impacts of habitat fragmentation are routinely assessed using standard statistical modelling techniques that are used across many ecological disciplines. However, to assess the biological relevance of fragmentation impacts, we must consider an extra, spatial dimension to the standard statistical model: the biological importance of a significant and well supported model with large effect sizes crucially depends on the configuration of habitat within the study area. We argue that mapping the outputs from statistical models across a study area generates biologically meaningful estimates of fragmentation impacts. Integrating traditional statistical approaches with geographic information systems will facilitate rigorous comparisons of fragmentation impacts between taxa, studies and ecosystems.     

Lessons from sea louse and salmon epidemiology

Effective disease management can benefit from mathematical models that identify drivers of epidemiological change and guide decision-making. This is well illustrated in the host–parasite system of sea lice and salmon, which has been modelled extensively due to the economic costs associated with sea louse infections on salmon farms and the conservation concerns associated with sea louse infections on wild salmon. Consequently, a rich modelling literature devoted to sea louse and salmon epidemiology has been developed. We provide a synthesis of the mathematical and statistical models that have been used to study the epidemiology of sea lice and salmon. These studies span both conceptual and tactical models to quantify the effects of infections on host populations and communities, describe and predict patterns of transmission and dispersal, and guide evidence-based management of wild and farmed salmon. As aquaculture production continues to increase, advances made in modelling sea louse and salmon epidemiology should inform the sustainable management of marine resources.     

Gene flow and functional connectivity in the natterjack toad

Functional connectivity is a key factor for the persistence of many specialist species in fragmented landscapes. However, connectivity estimates have rarely been validated by the observation of dispersal movements. In this study, we estimated functional connectivity of a real landscape by modelling dispersal for the endangered natterjack toad (Bufo calamita) using cost distance. Cost distance allows the evaluation of 'effective distances', which are distances corrected for the costs involved in moving between habitat patches in spatially explicit landscapes. We parameterized cost-distance models using the results of our previous experimental investigation of natterjack's movement behaviour. These model predictions (connectivity estimates from the GIS study) were then confronted to genetic-based dispersal rates between natterjack populations in the same landscape using Mantel tests. Dispersal rates between the populations were inferred from variation at six microsatellite loci. Based on these results, we conclude that matrix structure has a strong effect on dispersal rates. Moreover, we found that cost distances generated by habitat preferences explained dispersal rates better than did the Euclidian distances, or the connectivity estimate based on patch-specific resistances (patch viscosity). This study is a clear example of how landscape genetics can validate operational functional connectivity estimates.     

Estimation of the seed dispersal kernel from exact identification of source plants

The exact identification of individual seed sources through genetic analysis of seed tissue of maternal origin has recently brought the full analytical potential of parentage analysis to the study of seed dispersal. No specific statistical methodology has been described so far, however, for estimation of the dispersal kernel function from categorical maternity assignment. In this study, we introduce a maximum-likelihood procedure to estimate the seed dispersal kernel from exact identification of seed sources. Using numerical simulations, we show that the proposed method, unlike other approaches, is independent of seed fecundity variation, yielding accurate estimates of the shape and range of the seed dispersal kernel under varied sampling and dispersal conditions. We also demonstrate how an obvious estimator of the dispersal kernel, the maximum-likelihood fit of the observed distribution of dispersal distances to seed traps, can be strongly biased due to the spatial arrangement of seed traps relative to source plants. Finally, we illustrate the use of the proposed method with a previously published empirical example for the animal-dispersed tree species Prunus mahaleb.     

一种等分作物花粉扩散圆形源区的方法

【背景】转基因作物花粉在大气中扩散会引起基因漂流,从而可导致不可预知的环境风险,运用模型预测可评估其花粉扩散状况、定量确定可靠的安全扩散距离。为了应用高斯烟羽模型应用于模拟转基因作物花粉在大气中的扩散浓度,提出了一种如何将半径为R的圆形花粉源区划分成许多等面积小面元的方法。【方法】通过数学推导,首次提出了各等面积面元中心点坐标的计算公式。【结果】根据在中国东北地区吉林省公主岭玉米花粉扩散和基因漂流的试验观测资料,将该公式应用到高斯烟羽扩散模型中,模拟了玉米花粉扩散到源区外不同距离处的浓度,对比花粉扩散模拟值与实测值,两者具有较好的一致性,表明应用该公式模拟花粉扩散能取得令人满意的效果。【结论与意义】分析证明这种推导划分半径为R的圆形花粉源区各面元中心点坐标的计算公式是可靠的,该方法可为应用高斯烟羽模型计算花粉源区内不同位置点单个源强对源区外某一距离处的花粉浓度的贡献提供便利。     

Analyses of genetic ancestry enable key insights for molecular ecology

Gene flow and recombination in admixed populations produce genomes that are mosaic combinations of chromosome segments inherited from different source populations, that is, chromosome segments with different genetic ancestries. The statistical problem of estimating genetic ancestry from DNA sequence data has been widely studied, and analyses of genetic ancestry have facilitated research in molecular ecology and ecological genetics. In this review, we describe and compare different model‐based statistical methods used to infer genetic ancestry. We describe the conceptual and mathematical structure of these models and highlight some of their key differences and shared features. We then discuss recent empirical studies that use estimates of genetic ancestry to analyse population histories, the nature and genetic basis of species boundaries, and the genetic architecture of traits. These diverse studies demonstrate the breadth of applications that rely on genetic ancestry estimates and typify the genomics‐enabled research that is becoming increasingly common in molecular ecology. We conclude by identifying key research areas where future studies might further advance this field.     

A genetic evaluation of seed dispersal in the neotropical tree Jacaranda copaia (Bignoniaceae)

Seed dispersal is a critical but poorly understood life-history stage of plants. Here we use a genetic approach to describe seed dispersal patterns accurately in a natural population of the Neotropical tree species Jacaranda copaia (Bignoniaceae). We used microsatellite genotypes from maternally derived tissue on the diaspore to identify which individual of all possible adult trees in the population was the true source of a given seed collected after it dispersed. Wind-dispersed seeds were captured in two different years in a large array of seed traps in an 84-ha mapped area of tropical forest on Barro Colorado Island, Panama. We were particularly interested in the proportion of seeds that traveled long distances and whether there was evidence for direct dispersal into gaps, which are required for successful recruitment of this pioneer tree species. Maximum likelihood procedures were used to fit single- and multiple-component dispersal kernels to the distance data. Mixture models, with separate distributions near and far, best fit the observed dispersal distances, albeit with considerable uncertainty in the tail. We discuss the results in light of different mechanisms responsible for separate distributions near the adult source and in the tail of the curve.     

Can the cause of aggregation be inferred from species distributions?

Species distributions often show an aggregated pattern, which can be due to a number of endo- and exogenous factors. While autologistic models have been used for modelling such data with statistical rigour, little emphasis has been put on disentangling potential causes of aggregation. In this paper we ask whether it is possible to infer sources of aggregation in species distributions from a single set of occurrence data by comparing the performance of various autologistic models. We create simulated data sets, which show similar occupancy patterns, but differ in the process that causes the aggregation. We model the distribution of these data with various autologistic models, and show how the relative performance of the models is sensitive to the factor causing aggregation in the data. This information can be used when modelling real species data, where causes of aggregation are typically unknown. To illustrate, we use our approach to assess the potential causes of aggregation in data of seven bird species with contrasting statistical patterns. Our findings have important implications for conservation, as understanding the mechanisms that drive population fluctuations in space and time is critical for the development of effective management actions for long-term conservation.     

Observer-based techniques for the identification and analysis of avascular tumor growth

Cancer represents one of the most challenging issues for the biomedical research, due its large impact on the public health state. For this reason, many mathematical methods have been proposed to forecast the time evolution of cancer size and invasion. In this paper, we study how to apply the Gompertz’s model to describe the growth of an avascular tumor in a realistic setting. To this aim, we introduce mathematical techniques to discretize the model, an important requirement when discrete-time measurements are available. Additionally, we describe observed-based techniques, borrowed from the field of automation theory, as a tool to estimate the model unknown parameters. This identification approach is a promising alternative to traditional statistical methods, and it can be easily extended to other models of cancer growth as well as to the evaluation of not measurable variables, on the basis of the available measurements. We show an application of this method to the analysis of solid tumor growth and parameters estimation in presence of a chemotherapy agent.     

An open source simulation model for soil and sediment bioturbation

Bioturbation is one of the most widespread forms of ecological engineering and has significant implications for the structure and functioning of ecosystems, yet our understanding of the processes involved in biotic mixing remains incomplete. One reason is that, despite their value and utility, most mathematical models currently applied to bioturbation data tend to neglect aspects of the natural complexity of bioturbation in favour of mathematical simplicity. At the same time, the abstract nature of these approaches limits the application of such models to a limited range of users. Here, we contend that a movement towards process-based modelling can improve both the representation of the mechanistic basis of bioturbation and the intuitiveness of modelling approaches. In support of this initiative, we present an open source modelling framework that explicitly simulates particle displacement and a worked example to facilitate application and further development. The framework combines the advantages of rule-based lattice models with the application of parameterisable probability density functions to generate mixing on the lattice. Model parameters can be fitted by experimental data and describe particle displacement at the spatial and temporal scales at which bioturbation data is routinely collected. By using the same model structure across species, but generating species-specific parameters, a generic understanding of species-specific bioturbation behaviour can be achieved. An application to a case study and comparison with a commonly used model attest the predictive power of the approach.     

Using theoretical models to explore dispersal variation and fragmentation in urban environments

Cities are fundamentally changing the environment that plants inhabit, most notably through habitat fragmentation. Urban plant habitat patches are separated by impervious surfaces like buildings and roads and may vary from small, isolated green spaces to large green spaces like parks. Understanding the consequences of this urban fragmentation on seed dispersal is essential to both maintain urban biodiversity and mitigate the spread of unwanted weeds or invasive species but we currently lack enough empirical data to draw generalities. Theoretical reasoning (via both verbal and mathematical models) is well positioned to contribute to this knowledge gap in dispersal by providing useful predictions when empirical data are lacking. Variation in dispersal can easily be captured by models by incorporating different dispersal kernel shapes, and multiple habitat configurations can be examined. Urban environments are rarely considered by mathematical models, and our literature review indicates that most models that include dispersal variation via a dispersal kernel use only one or two shapes, suggesting a gap in the theoretical literature as well. We present a proof-of-concept model of fragmentation in an urban environment illustrating how varying habitat width can lead to different outcomes depending on the dispersal kernel. We also provide some thoughts for future directions on the application of mathematical models in urban areas.     

Spatial heterogeneity and critical patch size: Area effects via diffusion in closed environments

We describe a class of mathematical models for critical patch size in which the mechanisms inducing area effects are based on source-sink population dynamics arising from dispersal throughout a closed, finite, but spatially heterogeneous environment. Our models are reaction-diffusion equations, but unlike classical KISS models for area effects they do not assume that there is dispersal across the boundary of the environment into a hostile exterior. We observe that simple rescaling has the same effects in our models as in KISS models and hence predicts the same sort of area effects, but that other sorts of rescaling may not predict area effects. The models considered here provide an alternative to the KISS models used in our previous work on species-area relationships in island biogeography.     

Population dynamics can be more important than physiological limits for determining range shifts under climate change

Evidence is accumulating that species' responses to climate changes are best predicted by modelling the interaction of physiological limits, biotic processes and the effects of dispersal‐limitation. Using commercially harvested blacklip (Haliotis rubra) and greenlip abalone (Haliotis laevigata) as case studies, we determine the relative importance of accounting for interactions among physiology, metapopulation dynamics and exploitation in predictions of range (geographical occupancy) and abundance (spatially explicit density) under various climate change scenarios. Traditional correlative ecological niche models (ENM) predict that climate change will benefit the commercial exploitation of abalone by promoting increased abundances without any reduction in range size. However, models that account simultaneously for demographic processes and physiological responses to climate‐related factors result in future (and present) estimates of area of occupancy (AOO) and abundance that differ from those generated by ENMs alone. Range expansion and population growth are unlikely for blacklip abalone because of important interactions between climate‐dependent mortality and metapopulation processes; in contrast, greenlip abalone should increase in abundance despite a contraction in AOO. The strongly non‐linear relationship between abalone population size and AOO has important ramifications for the use of ENM predictions that rely on metrics describing change in habitat area as proxies for extinction risk. These results show that predicting species' responses to climate change often require physiological information to understand climatic range determinants, and a metapopulation model that can make full use of this data to more realistically account for processes such as local extirpation, demographic rescue, source‐sink dynamics and dispersal‐limitation.     

Computational and mathematical modeling of the effects of tailbeat frequency and flexural stiffness in swimming fish

In this paper we describe how we combine computational and mathematical models to form virtual fish to explore different hypotheses about the impact of centra. We show how we create simulation models using a combination of a mathematical model of a fish-like robot using caudal fin propulsion, a propulsion model, and an optimizer, to explore the impact of centra under various scenarios. The optimizer uses the mathematical model to construct valid configurations of the digital robot and uses the utility function and propulsion model to evaluate the performance of each configuration. The evaluations are used to explore the adaptive landscape and find high-performing configurations. Our results show that the high-performing configurations have both increased (flexural) stiffness of the tail and higher tailbeat frequencies.