Dynamic energy budget theory and population ecology: lessons from Daphnia

Dynamic energy budget (DEB) theory offers a perspective on population ecology whose starting point is energy utilization by, and homeostasis within, individual organisms. It is natural to ask what it adds to the existing large body of individual-based ecological theory. We approach this question pragmatically--through detailed study of the individual physiology and population dynamics of the zooplankter Daphnia and its algal food. Standard DEB theory uses several state variables to characterize the state of an individual organism, thereby making the transition to population dynamics technically challenging, while ecologists demand maximally simple models that can be used in multi-scale modelling. We demonstrate that simpler representations of individual bioenergetics with a single state variable (size), and two life stages (juveniles and adults), contain sufficient detail on mass and energy budgets to yield good fits to data on growth, maturation and reproduction of individual Daphnia in response to food availability. The same simple representations of bioenergetics describe some features of Daphnia mortality, including enhanced mortality at low food that is not explicitly incorporated in the standard DEB model. Size-structured, population models incorporating this additional mortality component resolve some long-standing questions on stability and population cycles in Daphnia. We conclude that a bioenergetic model serving solely as a 'regression' connecting organismal performance to the history of its environment can rest on simpler representations than those of standard DEB. But there are associated costs with such pragmatism, notably loss of connection to theory describing interspecific variation in physiological rates. The latter is an important issue, as the type of detailed study reported here can only be performed for a handful of species.     

Potential applications of mussel modelling

Mussels are extensively cultivated worldwide and are of growing economic importance. However, constraints on the exploitation of wild mussel resources have necessitated the need for tools to improve the management of mussel cultivation towards increased production. Ecological models are increasingly being used as a management tool, and therefore the existing approaches to modelling mussels have been reviewed with respect to their possible application to the improvement of shellfish management strategies. We suggest that dynamic energy budget (DEB) models have the greatest potential in this area, and discuss the mussel DEB models that have been developed to date in terms of their physiological complexity, accuracy of prediction of individual mussel growth and ability to predict mussel population production. Individual mussel production has been predicted; however, the focus of many of the models has been on the growth and reproduction of a single mussel and therefore population effects generally have not been included. Other models at the population level have included only limited population effects, and this has reduced the capacity of many of the models to accurately predict mussel production at the population level. Interactions at the population level (self-thinning and predation) are discussed and the models that describe the consequences of these processes are examined. In future DEB models will need to include the ability to parameterise population level processes if we are to have greater confidence in their application to shellfish management. Electronic Publication     

Extrapolating toxic effects on individuals to the population level: the role of dynamic energy budgets

The interest of environmental management is in the long-term health of populations and ecosystems. However, toxicity is usually assessed in short-term experiments with individuals. Modelling based on dynamic energy budget (DEB) theory aids the extraction of mechanistic information from the data, which in turn supports educated extrapolation to the population level. To illustrate the use of DEB models in this extrapolation, we analyse a dataset for life cycle toxicity of copper in the earthworm Dendrobaena octaedra. We compare four approaches for the analysis of the toxicity data: no model, a simple DEB model without reserves and maturation (the Kooijman-Metz formulation), a more complex one with static reserves and simplified maturation (as used in the DEBtox software) and a full-scale DEB model (DEB3) with explicit calculation of reserves and maturation. For the population prediction, we compare two simple demographic approaches (discrete time matrix model and continuous time Euler-Lotka equation). In our case, the difference between DEB approaches and population models turned out to be small. However, differences between DEB models increased when extrapolating to more field-relevant conditions. The DEB3 model allows for a completely consistent assessment of toxic effects and therefore greater confidence in extrapolating, but poses greater demands on the available data.     

Experimental testing of dynamic energy budget models

1. Dynamic energy budget (DEB) models describing the allocation of assimilate to the competing processes of growth, reproduction and maintenance in individual organisms have been applied to a variety of species with some success. There are two contrasting model formulations based on dynamic allocation rules that have been widely used (net production and net assimilation formulations). However, the predictions of these two classes of DEB models are not easily distinguished on the basis of simple growth and fecundity data.
2. It is shown that different assumptions incorporated in the rules determining allocation to growth and reproduction in two classes of commonly applied DEB models predict qualitatively distinct patterns for an easily measured variable, cumulative reproduction by the time an individual reaches an arbitrary size.
3. A comparison with experimental data from Daphnia pulex reveals that, in their simplest form, neither model predicts the observed qualitative pattern of reproduction, despite the fact that both formulations capture basic growth features.
4. An examination of more elaborate versions of the two models, in which the allocation rules are modified to account for brief periods of starvation experienced in the laboratory cultures, reveals that a version of the net production model can predict the qualitative pattern seen for cumulative eggs as a function of mass in D. pulex . The analysis leads to new predictions which can be easily tested with further laboratory experiments.     

Calibration of parameters in Dynamic Energy Budget models using Direct-Search methods

Dynamic Energy Budget (DEB) theory aims to capture the quantitative aspects of metabolism at the individual level, for all species. The parametrization of a DEB model is based on information obtained through the observation of natural populations and experimental research. Currently the DEB toolbox estimates these parameters using the Nelder–Mead Simplex method, a derivative-free direct-search method. However, this procedure presents some limitations regarding convergence and how to address constraints. Framed in the calibration of parameters in DEB theory, this work presents a numerical comparison between the Nelder–Mead Simplex method and the SID-PSM algorithm, a Directional Direct-Search method for which convergence can be established both for unconstrained and constrained problems. A hybrid version of the two methods, named as Simplex Directional Direct-Search, provides a robust and efficient algorithm, able to solve the constrained optimization problems resulting from the parametrization of the biological models.

     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

What is the status of metabolic theory one century after Pütter invented the von Bertalanffy growth curve?

Metabolic theory aims to tackle ecological and evolutionary problems by explicitly including physical principles of energy and mass exchange, thereby increasing generality and deductive power. Individual growth models (IGMs) are the fundamental basis of metabolic theory because they represent the organisational level at which energy and mass exchange processes are most tightly integrated and from which scaling patterns emerge. Unfortunately, IGMs remain a topic of great confusion and controversy about the origins of the ideas, their domain and breadth of application, their logical consistency and whether they can sufficiently capture reality. It is now 100 years since the first theoretical model of individual growth was put forward by Pütter. His insights were deep, but his model ended up being attributed to von Bertalanffy and his ideas largely forgotten. Here I review Pütter's ideas and trace their influence on existing theoretical models for growth and other aspects of metabolism, including those of von Bertalanffy, the Dynamic Energy Budget (DEB) theory, the Gill-Oxygen Limitation Theory (GOLT) and the Ontogenetic Growth Model (OGM). I show that the von Bertalanffy and GOLT models are minor modifications of Pütter's original model. I then synthesise, compare and critique the ideas of the two most-developed theories, DEB theory and the OGM, in relation to Pütter's original ideas. I formulate the Pütter, DEB and OGM models in the same structure and with the same notation to illustrate the major similarities and differences among them. I trace the confusion and controversy regarding these theories to the notions of anabolism, catabolism, assimilation and maintenance, the connections to respiration rate, and the number of parameters and state variables their models require. The OGM model has significant inconsistencies that stem from the interpretation of growth as the difference between anabolism and maintenance, and these issues seriously challenge its ability to incorporate development, reproduction and assimilation. The DEB theory is a direct extension of Pütter's ideas but with growth being the difference between assimilation and maintenance rather than anabolism and catabolism. The DEB theory makes the dynamics of Pütter's ‘nutritive material’ explicit as well as extending the scheme to include reproduction and development. I discuss how these three major theories for individual growth have been used to explain ‘macrometabolic’ patterns including the scaling of respiration, the temperature–size rule (first modelled by Pütter), and the connection to life history. Future research on the connections between theory and data in these macrometabolic topics have the greatest potential to advance the status of metabolic theory and its value for pure and applied problems in ecology and evolution.     

Quantitative aspects of metabolic organization: a discussion of concepts

Metabolic organization of individual organisms follows simple quantitative rules that can be understood from basic physical chemical principles. Dynamic energy budget (DEB) theory identifies these rules, which quantify how individuals acquire and use energy and nutrients. The theory provides constraints on the metabolic organization of subcellular processes. Together with rules for interaction between individuals, it also provides a basis to understand population and ecosystem dynamics. The theory, therefore, links various levels of biological organization. It applies to all species of organisms and offers explanations for body-size scaling relationships of natural history parameters that are otherwise difficult to understand. A considerable number of popular empirical models turn out to be special cases of the DEB model, or very close numerical approximations. Strong and weak homeostasis and the partitionability of reserve kinetics are cornerstones of the theory and essential for understanding the evolution of metabolic organization.     

Physiologically structured models – from versatile technique to ecological theory

A ubiquitous feature of natural communities is the variation in size that can be observed between organisms, a variation that to a substantial degree is intraspecific. Size variation within species by necessity implies that ecological interactions vary both in intensity and type over the life cycle of an individual. Physiologically structured population models (PSPMs) constitute a modelling approach especially designed to analyse these size‐dependent interactions as they explicitly link individual level processes such as consumption and growth to population dynamics. We discuss two cases where PSPMs have been used to analyse the dynamics of size‐structured populations. In the first case, a model of a size‐structured consumer population feeding on a non‐structured prey was successful in predicting both qualitative (mechanisms) and quantitative (individual growth, survival, cycle amplitude) aspects of the population dynamics of a planktivorous fish population. We conclude that single generation cycles as a result of intercohort competition is a general outcome of size‐structured consumer–resource interactions. In the second case, involving both cohort competition and cannibalism, we show that PSPMs may predict double asymptotic growth trajectories with individuals ending up as giants. These growth trajectories, which have also been observed in field data, could not be predicted from individual level information, but are emergent properties of the population feedback on individual processes. In contrast to the size‐structured consumer–resource model, the dynamics in this case cannot be reduced to simpler lumped stage‐based models, but can only be analysed within the domain of PSPMs. Parameter values used in PSPMs adhere to the individual level and are derived independently from the system at focus, whereas model predictions involve both population level processes and individual level processes under conditions of population feedback. This leads to an increased ability to test model predictions but also to a larger set of variables that is predicted at both the individual and population level. The results turn out to be relatively robust to specific model assumptions and thus render a higher degree of generality than purely individual‐based models. At the same time, PSPMs offer a much higher degree of realism, precision and testing ability than lumped stage‐based or non‐structured models. The results of our analyses so far suggest that also in more complex species configurations only a limited set of mechanisms determines the dynamics of PSPMs. We therefore conclude that there is a high potential for developing an individual‐based, size‐dependent community theory using PSPMs.     

昆虫种群动态模拟模型

昆虫是动物界中最大的类群,与人类有着密切的利害关系。对昆虫的数量预测与符合经济和生态规律的管理,一直都被国内外列入重点研究课题。种群动态模拟是害虫管理中重要的基础工作。近十年来,关于昆虫种群动态模型的理论和实验研究进展迅速。现分别从单种种群和多种种群两个方面对国内外近些年来昆虫种群动态模拟模型的研究进展进行了概括和总结。单种种群从两个方面阐述:一是最基本的种群动态模拟模型Log istic方程的研究成果,包括方程的修正、参数的拟合与最优捕获策略等;另一个方面是对种群动态模拟常用的矩阵模型的概述,主要介绍不等期年龄组、矩阵维数的变化、矩阵维数与历期的关系、个体之间的发育差异以及发育速率差异等等对昆虫种群动态模型的影响。多种群主要从建模和模型应用两个部分对国内外研究成果进行综述。最后,对种群动态模拟模型研究的发展方向做了深入地讨论,即在原有的数据采集工作的基础上,使用面向对象程序设计语言,把各种要素包括各种物种及各种环境条件抽象成类,用消息传递来表示昆虫种群内个体与个体、昆虫种群与环境之间的相互作用,再结合先进的数学算法,建立一个直观的、操作简单的昆虫种群动态模型库,使模型结构与现实世界有最大的相似性。这样就可以实现昆虫种群动态的可视化、立体化、实时化和精确化的监测及预测。     

Invasion models of vegetation dynamics

Since communities change as a result of their successful invasion by new species it seems logical to attempt to predict future vegetation change by focussing on the invasion process. Several such invasion models are reviewed, and one particular model, based on dynamic game theory is developed further. This model can be used as an alternative to linear (e.g. Markov chain) models for the prediction of vegetation dynamics, and also to compare invasive abilities of species and resistance to invasion of communities. The main advantage of the model lies in the fact that it operates at a sufficiently high level of integration to allow for model calibration (in spite of the large number of underlying processes), and yet has an obvious population biological interpretation (in terms of the success of invading populations). The model can be calibrated using either time course data or experimental data, and it may be helpful in understanding what determines the fate of an invading population. It is used here to analyze two published vegetation dynamics data sets.This work was financially supported by operating grant A8115 from the Natural Sciences and Engineering Research Council of Canada.     

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.     

森林动态模型概论

讨论了模拟森林林分动态变化的模型,并把模型分为森林生长模型和演替模型。森林生长模型包括：全林分模型、林分级模型和单木模型;演替模型包括马尔可夫类模型和林窗模型。文中给出了演替模型的基本原理和适用性,在比较早期和最新发展的林窗模型后,叙述了林窗模型的新进展。生长和演替模型的结构和数据要求不同决定了它们的在时间和空间上的适应性,最后指出模型将向综合总体方向发展。     

Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants

Predicting the emergence of new pathogenic strains is a key goal of evolutionary epidemiology. However, the majority of existing studies have focussed on emergence at the population level, and not within a host. In particular, the coexistence of pre-existing and mutated strains triggers a heightened immune response due to the larger total pathogen population; this feedback can smother mutated strains before they reach an ample size and establish. Here, we extend previous work for measuring emergence probabilities in non-equilibrium populations, to within-host models of acute infections. We create a mathematical model to investigate the emergence probability of a fitter strain if it mutates from a self-limiting strain that is guaranteed to go extinct in the long-term. We show that ongoing immune cell proliferation during the initial stages of infection causes a drastic reduction in the probability of emergence of mutated strains; we further outline how this effect can be accurately measured. Further analysis of the model shows that, in the short-term, mutant strains that enlarge their replication rate due to evolving an increased growth rate are more favoured than strains that suffer a lower immune-mediated death rate (‘immune tolerance’), as the latter does not completely evade ongoing immune proliferation due to inter-parasitic competition. We end by discussing the model in relation to within-host evolution of human pathogens (including HIV, hepatitis C virus, and cancer), and how ongoing immune growth can affect their evolutionary dynamics.     

A Dynamic Energy Budget (DEB) Model for the Keystone Predator Pisaster ochraceus

We present a Dynamic Energy Budget (DEB) model for the quintessential keystone predator, the rocky-intertidal sea star Pisaster ochraceus. Based on first principles, DEB theory is used to illuminate underlying physiological processes (maintenance, growth, development, and reproduction), thus providing a framework to predict individual-level responses to environmental change. We parameterized the model for P. ochraceus using both data from the literature and experiments conducted specifically for the DEB framework. We devoted special attention to the model’s capacity to (1) describe growth trajectories at different life-stages, including pelagic larval and post-metamorphic phases, (2) simulate shrinkage when prey availability is insufficient to meet maintenance requirements, and (3) deal with the combined effects of changing body temperature and food supply. We further validated the model using an independent growth data set. Using standard statistics to compare model outputs with real data (e.g. Mean Absolute Percent Error, MAPE) we demonstrated that the model is capable of tracking P. ochraceus’ growth in length at different life-stages (larvae: MAPE = 12.27%; post-metamorphic, MAPE = 9.22%), as well as quantifying reproductive output index. However, the model’s skill dropped when trying to predict changes in body mass (MAPE = 24.59%), potentially because of the challenge of precisely anticipating spawning events. Interestingly, the model revealed that P. ochraceus reserves contribute little to total biomass, suggesting that animals draw energy from structure when food is limited. The latter appears to drive indeterminate growth dynamics in P. ochraceus. Individual-based mechanistic models, which can illuminate underlying physiological responses, offer a viable framework for forecasting population dynamics in the keystone predator Pisaster ochraceus. The DEB model herein represents a critical step in that direction, especially in a period of increased anthropogenic pressure on natural systems and an observed recent decline in populations of this keystone species.     

The Impact of Macroscopic Epistasis on Long-Term Evolutionary Dynamics

Genetic interactions can strongly influence the fitness effects of individual mutations, yet the impact of these epistatic interactions on evolutionary dynamics remains poorly understood. Here we investigate the evolutionary role of epistasis over 50,000 generations in a well-studied laboratory evolution experiment in Escherichia coli. The extensive duration of this experiment provides a unique window into the effects of epistasis during long-term adaptation to a constant environment. Guided by analytical results in the weak-mutation limit, we develop a computational framework to assess the compatibility of a given epistatic model with the observed patterns of fitness gain and mutation accumulation through time. We find that a decelerating fitness trajectory alone provides little power to distinguish between competing models, including those that lack any direct epistatic interactions between mutations. However, when combined with the mutation trajectory, these observables place strong constraints on the set of possible models of epistasis, ruling out many existing explanations of the data. Instead, we find that the data are consistent with a “two-epoch” model of adaptation, in which an initial burst of diminishing-returns epistasis is followed by a steady accumulation of mutations under a constant distribution of fitness effects. Our results highlight the need for additional DNA sequencing of these populations, as well as for more sophisticated models of epistasis that are compatible with all of the experimental data.     

Mathematical Modeling of Influenza A Virus Dynamics within Swine Farms and the Effects of Vaccination

Influenza A virus infections are widespread in swine herds across the world. Influenza negatively affects swine health and production, and represents a significant threat to public health due to the risk of zoonotic infections. Swine herds can act as reservoirs for potentially pandemic influenza strains. In this study, we develop mathematical models based on experimental data, representing typical breeding and wean-to-finish swine farms. These models are used to explore and describe the dynamics of influenza infection at the farm level, which are at present not well understood. In addition, we use the models to assess the effectiveness of vaccination strategies currently employed by swine producers, testing both homologous and heterologous vaccines. An important finding is that following an influenza outbreak in a breeding herd, our model predicts a persistently high level of infectious piglets. Sensitivity analysis indicates that this finding is robust to changes in both transmission rates and farm size. Vaccination does not eliminate influenza throughout the breeding farm population. In the wean-to-finish herd, influenza infection may persist in the population only if recovered individuals become susceptible to infection again. A homologous vaccine administered to the entire wean-to-finish population after the loss of maternal antibodies eliminates influenza, but a vaccine that only induces partial protection (heterologous vaccine) has little effect on influenza infection levels. Our results have important implications for the control of influenza in swine herds, which is crucial in order to reduce both losses for swine producers and the risk to public health.     

Bistability in a size-structured population model of cannibalistic fish--a continuation study

By numerical continuation of equilibria, we study a size-structured model for the dynamics of a cannibalistic fish population and its alternative resource. Because we model the cannibalistic interaction as dependent on the ratio of cannibal length and victim length, a cannibal experiences a size distribution of potential victims which depends on its own body size. We show how equilibria of the resulting infinite-dimensional dynamical system can be traced with an existing method for numerical continuation for physiologically structured population models. With this approach we found that cannibalism can induce bistability associated with a fold (or, saddle-node) bifurcation. The two stable states can be qualified as 'stunted' and 'piscivorous', respectively. We identify a new ecological mechanism for bistability, in which the energy gain from cannibalism plays a crucial role: Whereas in the stunted population state cannibals consume their victims, on average, while they are very small and yield little energy, in the piscivorous state cannibals consume their victims not before they have become much bigger, which results in a much higher mean yield of cannibalism. We refer to this mechanism as the 'Hansel and Gretel' effect. It is not related to any individual 'choice' or 'strategy', but depends purely on a difference in population size distribution. We argue that studying dynamics of size-structured population models with this new approach of equilibrium continuation extends the insight that can be gleaned from numerical simulations of the model dynamics.