Methods of modelling viral disease dynamics across the within- and between-host scales: the impact of virus dose on host population immunity

We study the epidemiology of a viral disease with dose-dependent replication and transmission by nesting a differential-equation model of the within-host viral dynamics inside a between-host epidemiological model. We use two complementary approaches for nesting the models: an agent-based (AB) simulation and a mean-field approximation called the growth-matrix (GM) model. We find that although infection rates and predicted case loads are somewhat different between the AB and GM models, several epidemiological parameters, e.g. mean immunity in the population and mean dose received, behave similarly across the methods. Further, through a comparison of our dose-dependent replication model against two control models that uncouple dose-dependent replication from transmission, we find that host immunity in a population after an epidemic is qualitatively different than when transmission depends on time-varying viral abundances within hosts. These results show that within-host dynamics and viral dose should not be neglected in epidemiological models, and that the simpler GM approach to model nesting provides a reasonable tradeoff between model complexity and accuracy of results.     

A unifying framework for chaos and stochastic stability in discrete population models

 In this paper we propose a general framework for discrete time one-dimensional Markov population models which is based on two fundamental premises in population dynamics. We show that this framework incorporates both earlier population models, like the Ricker and Hassell models, and experimental observations concerning the structure of density dependence. The two fundamental premises of population dynamics are sufficient to guarantee that the model will exhibit chaotic behaviour for high values of the natural growth and the density-dependent feedback, and this observation is independent of the particular structure of the model. We also study these models when the environment of the population varies stochastically and address the question under what conditions we can find an invariant probability distribution for the population under consideration. The sufficient conditions for this stochastic stability that we derive are of some interest, since studying certain statistical characteristics of these stochastic population processes may only be possible if the process converges to such an invariant distribution. Received 15 May 1995; received in revised form 17 April 1996     

THE TEMPO AND MODE OF EVOLUTION OF TRANSPOSABLE ELEMENTS AS REVEALED BY MOLECULAR PHYLOGENIES RECONSTRUCTED FROM MOSQUITO GENOMES

Although many mathematical models exist predicting the dynamics of transposable elements (TEs), there is a lack of available empirical data to validate these models and inherent assumptions. Genomes can provide a snapshot of several TE families in a single organism, and these could have their demographics inferred by coalescent analysis, allowing for the testing of theories on TE amplification dynamics. Using the available genomes of the mosquitoes Aedes aegypti and Anopheles gambiae, we indicate that such an approach is feasible. Our analysis follows four steps: (1) mining the two mosquito genomes currently available in search of TE families; (2) fitting, to selected families found in (1), a phylogeny tree under the general time‐reversible (GTR) nucleotide substitution model with an uncorrelated lognormal (UCLN) relaxed clock and a nonparametric demographic model; (3) fitting a nonparametric coalescent model to the tree generated in (2); and (4) fitting parametric models motivated by ecological theories to the curve generated in (3).     

Matching the forecast horizon with the relevant spatial and temporal processes and data sources

Most phenomenological, statistical models used to generate ecological forecasts take either a time-series approach, based on long-term data from one location, or a space-for-time approach, based on data describing spatial patterns across environmental gradients. However, the magnitude and even the sign of environment–response relationships detected using these two approaches often differs, leading to contrasting predictions about responses to future environmental change. Here we consider how the forecast horizon determines whether more accurate predictions come from the time-series approach, the space-for-time approach or a combination of the two. As proof of concept, we use simulated case studies to show that forecasts for short and long forecast horizons need to focus on different ecological processes, which are reflected in different kinds of data. First, we simulated population or community dynamics under stationary temperature using two simple, mechanistic models. Second, we fit statistical models to the simulated data using a time-series approach, a space-for-time approach or a weighted average. We then forecast the response to a temperature increase using the statistical models, and compared these forecasts to temperature effects simulated by the mechanistic models. We found that the time-series approach made accurate short-term predictions because it captured initial conditions and effects of fast processes such as birth and death. The space-for-time approach made more accurate long-term predictions because it better captured the influence of slower processes such as evolutionary and ecological selection. The weighted average made accurate predictions at all time scales, including intermediate time-scales where the other two approaches performed poorly. A weighted average of time-series and space-for-time approaches shows promise, but making this weighted model operational will require new research to predict the rate at which slow processes begin to influence dynamics.     

On the mechanistic underpinning of discrete-time population models with complex dynamics

We present a mechanistic underpinning for various discrete-time population models that can produce limit cycles and chaotic dynamics. Specific examples include the discrete-time logistic model and the Hassell model, which for a long time eluded convincing mechanistic interpretations, and also the Ricker- and Beverton-Holt models. We first formulate a continuous-time resource consumption model for the dynamics within a year, and from that we derive a discrete-time model for the between-year dynamics. Without influx of resources from the outside into the system, the resulting between-year dynamics is always overcompensating and hence may produce complex dynamics as well as extinction in finite time. We recover a connection between various standard types of continuous-time models for the resource dynamics within a year on the one hand and various standard types of discrete-time models for the population dynamics between years on the other. The model readily generalizes to several resource and consumer species as well as to more than two trophic levels for the within-year dynamics.     

A comparison of model linear chain molecules with constrained and flexible bond lengths under planar Couette and extensional flows

We compare directly under flow two commonly used coarse grained models of linear polymers, namely the flexible finitely extensible nonlinear elastic (FENE) chain, and the freely jointed tangent sphere chain, otherwise known as the freely jointed chain. The comparison is based on viscometric, structural and dynamical properties. We use non-equilibrium molecular dynamics to simulate steady-state systems under planar Couette flow and planar extensional flow. Computed properties include shear and elongational viscosities, normal stresses, radius of gyration and end-to-end distances, order parameters, alignment angles and spin angular velocities. In all computed properties we observe very little difference between the two molecular models. Therefore, the choice of either model is suitable, though there is a computational advantage in the use of the FENE model.     

The reliability of approximate reduction techniques in population models with two time scales

As a result of the complexity inherent in some natural systems, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often find multiregional models for structured populations in which individuals are classified regarding their age and their spatial location. Dealing with such structured populations leads to high dimensional models. Moreover, in many instances the dynamics of the system is controlled by processes whose time scales are very different from each other. For example, in multiregional models migration is often a fast process in comparison to the growth of the population.Approximate reduction techniques take advantage of the presence of different time scales in a system to introduce approximations that allow one to transform the original system into a simpler low dimensional system. In this way, the dynamics of the original system can be approximated in terms of that of the reduced system. This work deals with the study of that approximation. In particular, we work with a non-autonomous discrete time model previously presented in the literature and obtain different bounds for the error we incur when we describe the dynamics of the original system in terms of the reduced one.The results are illustrated by some numerical simulations corresponding to the reduction of a Leslie type model for a population structured in two age classes and living in a two patch system.     

Nutrient enrichment alters dynamics in experimental plant populations

Controlling weed populations requires an understanding of their underlying population dynamics which can be achieved through a combination of model development and long-term studies. In this paper, we develop models based on long-term data from experimental populations of the weedy annual plant Cardamine pensylvanica. Four replicate populations of C. pensylvanica were grown in growth chambers under three different nutrient levels but with all other environmental conditions held constant. We analyze the resulting time series using generalized additive models and perform stability analyses using Lyapunov exponents. Further, we test whether the proposed mechanism, delayed density dependence caused by maternal effects, is operating in our system by experimentally manipulating maternal density and assessing the resulting offspring quality. Our results show that that increasing the frequency of nutrients causes plant population dynamics to shift from stable to damped 2-point oscillations to longer cycles. This shift in population dynamics is due to a shift at high nutrients from populations being regulated by first order density feedbacks to being regulated by both first order and second order density feedbacks. A consequence of these first order and second order feedbacks was an increase in cycle lengths as demonstrated by the presence of complex eigenvalues. A short-term experiment confirmed that when grown under high nutrients, the density of maternal plants strongly affected offspring size, providing a mechanism whereby these second order density feedbacks could operate. Our results demonstrate that increasing nutrient frequency results in a qualitative shift in dynamics from stable to longer cycles.     

Hutchinson revisited: Patterns of density regulation and the coexistence of strong competitors

Ecologists have long been searching for mechanisms of species coexistence, particularly since G.E. Hutchinson raised the ‘paradox of the plankton’. A promising approach to solve this paradox and to explain the coexistence of many species with strong niche overlap is to consider over-compensatory density regulation with its ability to generate endogenous population fluctuations.Previous work has analysed the role of over-compensation in coexistence based on analytical approaches. Using a spatially explicit time-discrete simulation model, we systematically explore the dynamics and conditions for coexistence of two species. We go beyond the analytically accessible range of models by studying the whole range of density regulation from under- to very strong over-compensation and consider the impact of spatial structure and temporal disturbances. In particular, we investigate how coexistence can emerge in different types of population growth models.We show that two strong competitors are able to coexist if at least one species exhibits over-compensation. Analysing the time series of population dynamics reveals how the differential responses to density fluctuations of the two competitors lead to coexistence: The over-compensator generates density fluctuations but is the inferior competitor at strong amplitudes of those fluctuations; the competitor, therefore, becomes frequent and dampens the over-compensator's amplitudes, but it becomes inferior under dampened fluctuations.These species interactions cause a dynamic alternation of community states with long-term persistence of both species. We show that a variety of population growth models is able to reproduce this coexistence although the particular parameter ranges differ among the models. Spatial structure influences the probability of coexistence but coexistence is maintained for a broad range of dispersal parameters.The flexibility and robustness of coexistence through over-compensation emphasize the importance of nonlinear density dependence for species interactions, and they also highlight the potential of applying more flexible models than the classical Lotka-Volterra equations in community ecology.     

Simple discrete-time metapopulation models of patch occupancy

Simple models in theoretical ecology have a long-standing history of being used to understand how specific processes influence population dynamics as well as providing a foundation for future endeavors. The Levins model is the seminal example of this for continuous-time metapopulation dynamics. However, many natural populations have a distinct separation between processes and data is not collected continuously leading to the need for using a discrete-time model. Our goal is to develop a simple discrete-time metapopulation model of patch occupancy using difference equations. In our formulation, we consider the two fundamental processes of colonization and extinction that will be treated as sequential events and will only consider patch occupancy. To achieve this, we use a composition of two functions where one will reflect the extinction process and the other for the colonization process. Under some mild assumptions, we are able determine the dynamic behavior of the metapopulation. In addition, we provide numerous examples for the functions used to emulate the colonization and extinction processes. Our results illustrate that the dynamics of the model are tied to properties such as convexity and monotonicity of the colonization and extinction functions. In particular, if the model is non-monotone, then complex dynamics can arise such as cyclic and even chaotic behavior. Overall, our approach shows how certain properties of the colonization and extinction functions can influence metapopulation dynamics.     

Variables Aggregation in Time Varying Discrete Systems

In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists in describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced system with a few number of variables. We present a time discrete time varying model in which we distinguish two time scales. By using perturbation methods we transform the system to make the global variables appear and build up the aggregated system. The asymptotic relationships between the general and aggregated systems are explored in the cases of a cyclically varying environment and a changing environment in process of stabilization. We show that under quite general conditions the knowledge of the behavior of the aggregated system characterizes that of the general system. The general method is also applied to aggregate a multiregional time dependent Leslie model showing that the aggregated model has demographic rates depending on the equilibrium proportions of individuals in the different patches.     

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。     

How good are comparative models in the understanding of protein dynamics?

The 3D structure of a protein is essential to understand protein dynamics. If experimentally determined structure is unavailable, comparative models could be used to infer dynamics. However, the effectiveness of comparative models, compared to experimental structures, in inferring dynamics is not clear. To address this, we compared dynamics features of ~800 comparative models with their crystal structures using normal mode analysis. Average similarity in magnitude, direction, and correlation of residue motions is >0.8 (where value 1 is identical) indicating that the dynamics of models and crystal structures are highly similar. Accuracy of 3D structure and dynamics is significantly higher for models built on multiple and/or high sequence identity templates (>40%). Three-dimensional (3D) structure and residue fluctuations of models are closer to that of crystal structures than to templates (TM score 0.9 vs 0.7 and square inner product 0.92 vs 0.88). Furthermore, long-range molecular dynamics simulations on comparative models of RNase 1 and Angiogenin showed significant differences in the conformational sampling of conserved active-site residues that characterize differences in their activity levels. Similar analyses on two EGFR kinase variant models highlight the effect of mutations on the functional state-specific αC helix motions and these results corroborate with the previous experimental observations. Thus, our study adds confidence to the use of comparative models in understanding protein dynamics.     

A diverse view of protein dynamics from NMR studies of HIV-1 protease flaps

Internal motion in proteins fulfills a multitude of roles in biological processes. NMR spectroscopy has been applied to elucidate protein dynamics at the atomic level, albeit at a low resolution, and is often complemented by molecular dynamics simulation. However, it is critical to justify the consistency between simulation results and conclusions often drawn from multiple experiments in which uncertainties arising from assumed motional models may not be explicitly evaluated. To understand the role of the flaps of HIV-1 protease dimer in substrate recognition and protease function, many molecular dynamics simulations have been performed. The simulations have resulted in various proposed models of the flap dynamics, some of which are more consistent than others with our working model previously derived from experiments. However, using the working model to discriminate among the simulation results is not straightforward because the working model was derived from a combination of NMR experiments and crystal structure data. In this study, we use the NMR chemical shifts and relaxation data of the protease "monomer" rather than structural data to narrow down the possible conformations of the flaps of the "dimer". For the first time, we show that the tips of the flaps in the unliganded protease dimer interact with each other in solution. Accordingly, we discuss the consistency of the simulations with the model derived from all experimental data.     

植物群落动态的模型分析

植物群落的动态是植物群落学的中心问题之一,包括更新、波动、演替、进化等主要内容。空间格局对种群和群落的动态起着至关重要的作用,种群空间格局和群落空间结构是群落中各种过程相互作用的产物。模型是描述群落动态、认识植物群落组建和维持机理的有效工具。本文阐述和比较了描述群落动态的四种具有代表性的经验模型,即镶嵌循环模型、随意游走模型、同资源种团比例模型、空间抢先占有模型及其机理。四种经验模型的空间性及缺陷分别是:(1)“镶嵌循环模型”考虑到了相邻斑块之间的植被空间结合在群落动态中的作用,而另外三种模型没有考虑到这一点;(2)在一定程度上,四种植物群落动态模型对各自针对的植物群落可能是适合的,但要作为描述群落动态发展的一般性模型还需要不断完善和发展;因为四种模型均没有考虑到自然干扰和人类干扰对植物群落动态的影响。作者对将来植物群落动态的研究及实践意义做出以下展望:(1)在不同空间尺度上,更加有效地评价控制群落动态变化的各种过程的相对重要性,并进一步将它们之间的复杂相互作用整合到群落动态模型中;(2)充分认识植物群落中存在的各种自然环境条件和生物群体的结构配置对植物群落动态发展的重要性;(3)重视植物群落动态发展中自然干扰过程和人类干扰过程的整合以     

Linear analysis solves two puzzles in population dynamics: the route to extinction and extinction in coloured environments

In this paper, we give simple explanations to two unsolved puzzles that have emerged in recent theoretical studies in population dynamics. First, the tendency of some model populations to go extinct from high population densities, and second, the positive effect of autocorrelated environments on extinction risks for some model populations. Both phenomena are given general explanations by simple, linear, sto-chastic models. We emphasize the predictive and explanatory power of such models.     

Separating direct and indirect effects of global change: a population dynamic modeling approach using readily available field data

Two sources of complexity make predicting plant community response to global change particularly challenging. First, realistic global change scenarios involve multiple drivers of environmental change that can interact with one another to produce non‐additive effects. Second, in addition to these direct effects, global change drivers can indirectly affect plants by modifying species interactions. In order to tackle both of these challenges, we propose a novel population modeling approach, requiring only measurements of abundance and climate over time. To demonstrate the applicability of this approach, we model population dynamics of eight abundant plant species in a multifactorial global change experiment in alpine tundra where we manipulated nitrogen, precipitation, and temperature over 7 years. We test whether indirect and interactive effects are important to population dynamics and whether explicitly incorporating species interactions can change predictions when models are forecast under future climate change scenarios. For three of the eight species, population dynamics were best explained by direct effect models, for one species neither direct nor indirect effects were important, and for the other four species indirect effects mattered. Overall, global change had negative effects on species population growth, although species responded to different global change drivers, and single‐factor effects were slightly more common than interactive direct effects. When the fitted population dynamic models were extrapolated under changing climatic conditions to the end of the century, forecasts of community dynamics and diversity loss were largely similar using direct effect models that do not explicitly incorporate species interactions or best‐fit models; however, inclusion of species interactions was important in refining the predictions for two of the species. The modeling approach proposed here is a powerful way of analyzing readily available datasets which should be added to our toolbox to tease apart complex drivers of global change.     

生物协同学,Lorenz模型和种群动力学

由协同学方程出发,可以描述种群的大迁徙,由此又能够得到Lorenz模型,它可以描述两种种群的变化关系．当取绝热近似时,还可以导致种群动力学的不同模型．因此,生物协同学能够深刻揭示不同物种之间,既竞争又协同的复杂的非线性关系．