Density-structured models for plant population dynamics

Density-structured models are structured population models in which the state variable is the proportion of populations or sites in a small number of discrete density states. Although such models have rarely been used, they have the advantage that they are straightforward to parameterize, make few assumptions about population dynamics, and permit rapid data collection using coarse density assessment. In this article, we highlight their use in relating population dynamics to environmental variation and their robustness to measurement error. We show that density-structured models are able to accurately represent population dynamics under a wide range of conditions. We look at the effects of including a persistent seedbank and describe numerical approximations for the mean and variance of population size. For simulated data, we determine the extent to which the underlying continuous process may be inferred from density-structured data. Finally, we discuss issues of parameter estimation and applications for which these types of models may be useful.     

The diversity of population responses to environmental change

The current extinction and climate change crises pressure us to predict population dynamics with ever‐greater accuracy. Although predictions rest on the well‐advanced theory of age‐structured populations, two key issues remain poorly explored. Specifically, how the age‐dependency in demographic rates and the year‐to‐year interactions between survival and fecundity affect stochastic population growth rates. We use inference, simulations and mathematical derivations to explore how environmental perturbations determine population growth rates for populations with different age‐specific demographic rates and when ages are reduced to stages. We find that stage‐ vs. age‐based models can produce markedly divergent stochastic population growth rates. The differences are most pronounced when there are survival‐fecundity‐trade‐offs, which reduce the variance in the population growth rate. Finally, the expected value and variance of the stochastic growth rates of populations with different age‐specific demographic rates can diverge to the extent that, while some populations may thrive, others will inevitably go extinct.     

On using integral projection models to generate demographically driven predictions of species' distributions: development and validation using sparse data

Knowledge of species' geographic distributions is critical for understanding and forecasting population dynamics, responses to environmental change, biodiversity patterns, and conservation planning. While many suggestive correlative occurrence models have been used to these ends, progress lies in understanding the underlying population biology that generates patterns of range dynamics. Here, we show how to use a limited quantity of demographic data to produce demographic distribution models (DDMs) using integral projection models for size‐structured populations. By modeling survival, growth, and fecundity using regression, integral projection models can interpolate across missing size data and environmental conditions to compensate for limited data. To accommodate the uncertainty associated with limited data and model assumptions, we use Bayesian models to propagate uncertainty through all stages of model development to predictions. DDMs have a number of strengths: 1) DDMs allow a mechanistic understanding of spatial occurrence patterns; 2) DDMs can predict spatial and temporal variation in local population dynamics; 3) DDMs can facilitate extrapolation under altered environmental conditions because one can evaluate the consequences for individual vital rates. To illustrate these features, we construct DDMs for an overstory perennial shrub in the Proteaceae family in the Cape Floristic Region of South Africa. We find that the species' population growth rate is limited most strongly by adult survival throughout the range and by individual growth in higher rainfall regions. While the models predict higher population growth rates in the core of the range under projected climates for 2050, they also suggest that the species faces a threat along arid range margins from the interaction of more frequent fire and drying climate. The results (and uncertainties) are helpful for prioritizing additional sampling of particular demographic parameters along these gradients to iteratively refine projections. In the appendices, we provide fully functional R code to perform all analyses.     

How do plant ecologists use matrix population models?

Matrix projection models are among the most widely used tools in plant ecology. However, the way in which plant ecologists use and interpret these models differs from the way in which they are presented in the broader academic literature. In contrast to calls from earlier reviews, most studies of plant populations are based on < 5 matrices and present simple metrics such as deterministic population growth rates. However, plant ecologists also cautioned against literal interpretation of model predictions. Although academic studies have emphasized testing quantitative model predictions, such forecasts are not the way in which plant ecologists find matrix models to be most useful. Improving forecasting ability would necessitate increased model complexity and longer studies. Therefore, in addition to longer term studies with better links to environmental drivers, priorities for research include critically evaluating relative/comparative uses of matrix models and asking how we can use many short-term studies to understand long-term population dynamics.     

The rise of model protozoa

It is timely to evaluate the role of protozoa as model organisms given their diversity, abundance and versatility as well as the economic and ethical pressures placed on animal-based experimentation. We first define the term model organism and then examine through examples why protozoa make good models. Our examples reflect major issues including evolution, ecology, population and community biology, disease, the role of organelles, ageing, space travel, toxicity and teaching. We conclude by recognising that although protozoa may in some cases not completely mimic tissue- or whole-animal-level processes, they are extremely flexible and their use should be embraced. Finally, we offer advice on obtaining emergent model protozoa.     

Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design.

We construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The models are based on random walks where there may be a bias in the direction an individual moves when it encounters an interface. This sort of dispersal process is called skew Brownian motion. Our models take the form of diffusion equations with matching conditions across the interface between regions for population densities and fluxes. We combine the dispersal models with linear population growth models which assume that the population growth rate differs between regions of different habitat types. We use those models to study issues of refuge design. We specifically consider how the effectiveness of buffer zones depends on their size, quality, and the population's response to the interface between the buffer zone and the refuge.     

Saddle-Point Approximations,Integrodifference Equations,and Invasions

Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times. In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density.     

Extinction risk of a meta-population: aggregation approach

Aggregation of variables of a complex mathematical model with realistic structure gives a simplified model which is more suitable than the original one when the amount of data for parameter estimation is limited. Here we explore use of a formula derived for a single unstructured population (canonical model) in predicting the extinction time for a population living in multiple habitats. In particular we focus multiple populations each following logistic growth with demographic and environmental stochasticities, and examine how the mean extinction time depends on the migration and environmental correlation. When migration rate and/or environmental correlation are very large or very small, we may express the mean extinction time exactly using the formula with properly modified parameters. When parameters are of intermediate magnitude, we generate a Monte Carlo time series of the population size for the realistic structured model, estimate the "effective parameters" by fitting the time series to the canonical model, and then calculate the mean extinction time using the formula for a single population. The mean extinction time predicted by the formula was close to those obtained from direct computer simulation of structured models. We conclude that the formula for an unstructured single-population model has good approximation capability and can be applicable in estimating the extinction risk of the structured meta-population model for a limited data set.     

The Galilean turn in population ecology

The standard mathematical models in population ecology assume that a population's growth rate is a function of its environment. In this paper we investigate an alternative proposal according to which the rate of change of the growth rate is a function of the environment and of environmental change. We focus on the philosophical issues involved in such a fundamental shift in theoretical assumptions, as well as on the explanations the two theories offer for some of the key data such as cyclic populations. We also discuss the relationship between this move in population ecology and a similar move from first-order to second-order differential equations championed by Galileo and Newton in celestial mechanics.     

Modeling physical growth using mixed effects models

This article demonstrates the use of mixed effects models for characterizing individual and sample average growth curves based on serial anthropometric data. These models are advancement over conventional general linear regression because they effectively handle the hierarchical nature of serial growth data. Using body weight data on 70 infants in the Born in Bradford study, we demonstrate how a mixed effects model provides a better fit than a conventional regression model. Further, we demonstrate how mixed effects models can be used to explore the influence of environmental factors on the sample average growth curve. Analyzing data from 183 infant boys (aged 3–15 months) from rural South India, we show how maternal education shapes infant growth patterns as early as within the first 6 months of life. The presented analyses highlight the utility of mixed effects models for analyzing serial growth data because they allow researchers to simultaneously predict individual curves, estimate sample average curves, and investigate the effects of environmental exposure variables. Am J Phys Anthropol, 2013. © 2012 Wiley Periodicals, Inc.     

Estimation of fitness from energetics and life‐history data: An example using mussels

Changing environments have the potential to alter the fitness of organisms through effects on components of fitness such as energy acquisition, metabolic cost, growth rate, survivorship, and reproductive output. Organisms, on the other hand, can alter aspects of their physiology and life histories through phenotypic plasticity as well as through genetic change in populations (selection). Researchers examining the effects of environmental variables frequently concentrate on individual components of fitness, although methods exist to combine these into a population level estimate of average fitness, as the per capita rate of population growth for a set of identical individuals with a particular set of traits. Recent advances in energetic modeling have provided excellent data on energy intake and costs leading to growth, reproduction, and other life‐history parameters; these in turn have consequences for survivorship at all life‐history stages, and thus for fitness. Components of fitness alone (performance measures) are useful in determining organism response to changing conditions, but are often not good predictors of fitness; they can differ in both form and magnitude, as demonstrated in our model. Here, we combine an energetics model for growth and allocation with a matrix model that calculates population growth rate for a group of individuals with a particular set of traits. We use intertidal mussels as an example, because data exist for some of the important energetic and life‐history parameters, and because there is a hypothesized energetic trade‐off between byssus production (affecting survivorship), and energy used for growth and reproduction. The model shows exactly how strong this trade‐off is in terms of overall fitness, and it illustrates conditions where fitness components are good predictors of actual fitness, and cases where they are not. In addition, the model is used to examine the effects of environmental change on this trade‐off and on both fitness and on individual fitness components.     

A community model of ciliateTetrahymena and bacteriaE. coli: Part I. Individual-based models ofTetrahymena andE. coli populations

The dynamics of a microbial community consisting of a eucaryotic ciliateTetrahymena pyriformis and procaryoticEscherichia coli in a batch culture is explored by employing an individual-based approach. In this portion of the article, Part I, population models are presented. Because both models are individual-based, models of individual organisms are developed prior to construction of the population models. The individual models use an energy budget method in which growth depends on energy gain from feeding and energy sinks such as maintenance and reproduction. These models are not limited by simplifying assumptions about constant yield, constant energy sinks and Monod growth kinetics as are traditional models of microbal organisms. Population models are generated from individual models by creating distinct individual types and assigning to each type the number of real individuals they represent. A population is a compilation of individual types that vary in a phase of cell cycle and physiological parameters such as filtering rate for ciliates and maximum anabolic rate for bacteria. An advantage of the developed models is that they realistically describe the growth of the individual cells feeding on resource which varies in density and composition. Part II, the core of the project, integrates models into a dynamic microbial community and provides model analysis based upon available data.     

Biological collections and ecological/environmental research: a review,some observations and a look to the future

Housed worldwide, mostly in museums and herbaria, is a vast collection of biological specimens developed over centuries. These biological collections, and associated taxonomic and systematic research, have received considerable long‐term public support. The work remaining in systematics has been expanding as the estimated total number of species of organisms on Earth has risen over recent decades, as have estimated numbers of undescribed species. Despite this increasing task, support for taxonomic and systematic research, and biological collections upon which such research is based, has declined over the last 30‐40 years, while other areas of biological research have grown considerably, especially those that focus on environmental issues. Reflecting increases in research that deals with ecological questions (e.g. what determines species distribution and abundance) or environmental issues (e.g. toxic pollution), the level of research attempting to use biological collections in museums or herbaria in an ecological/environmental context has risen dramatically during about the last 20 years. The perceived relevance of biological collections, and hence the support they receive, should be enhanced if this trend continues and they are used prominently regarding such environmental issues as anthropogenic loss of biodiversity and associated ecosystem function, global climate change, and decay of the epidemiological environment. It is unclear, however, how best to use biological collections in the context of such ecological/environmental issues or how best to manage collections to facilitate such use. We demonstrate considerable and increasingly realized potential for research based on biological collections to contribute to ecological/environmental understanding. However, because biological collections were not originally intended for use regarding such issues and have inherent biases and limitations, they are proving more useful in some contexts than in others. Biological collections have, for example, been particularly useful as sources of information regarding variation in attributes of individuals (e.g. morphology, chemical composition) in relation to environmental variables, and provided important information in relation to species' distributions, but less useful in the contexts of habitat associations and population sizes. Changes to policies, strategies and procedures associated with biological collections could mitigate these biases and limitations, and hence make such collections more useful in the context of ecological/environmental issues. Haphazard and opportunistic collecting could be replaced with strategies for adding to existing collections that prioritize projects that use biological collections and include, besides taxonomy and systematics, a focus on significant environmental/ecological issues. Other potential changes include increased recording of the nature and extent of collecting effort and information associated with each specimen such as nearby habitat and other individuals observed but not collected. Such changes have begun to occur within some institutions. Institutions that house biological collections should, we think, pursue a mission of ‘understanding the life of the planet to inform its stewardship’ ( Krishtalka & Humphrey, 2000 ), as such a mission would facilitate increased use of biological collections in an ecological/environmental context and hence lead to increased appreciation, encouragement and support from the public for these collections, their associated research, and the institutions that house them.     

Estimating space-use and habitat preference from wildlife telemetry data

Management and conservation of populations of animals requires information on where they are, why they are there, and where else they could be. These objectives are typically approached by collecting data on the animals' use of space, relating these positional data to prevailing environmental conditions and employing the resulting statistical models to predict usage at other geographical regions. Technical advances in wildlife telemetry have accomplished manifold increases in the amount and quality of available data, creating the need for a statistical framework that can use them to make population‐level inferences for habitat preference and space‐use. This has been slow‐in‐coming because wildlife telemetry data are spatio‐temporally autocorrelated, often unbalanced, presence‐only observations of behaviourally complex animals, responding to a multitude of cross‐correlated environmental variables. We review the evolution of regression models for the analysis of space‐use and habitat preference and outline the essential features of a framework that emerges naturally from these foundations. This allows us to derive a relationship between usage of points in geographical space and preference of habitats in environmental space. Within this framework, we discuss eight challenges, inherent in the spatial analysis of telemetry data and, for each, we propose solutions that can work in tandem. Specifically, we propose a logistic, mixed‐effects approach that uses generalized additive transformations of the environmental covariates and is fitted to a response data‐set comprising the telemetry and simulated observations, under a case‐control design. We apply this framework to a non‐trivial case‐study using satellite‐tagged grey seals Halichoerus grypus from the east coast of Scotland. We perform model selection by cross‐validation and confront our final model's predictions with telemetry data from the same, as well as different, geographical regions. We conclude that, despite the complex behaviour of the study species, flexible empirical models can capture the environmental relationships that shape population distributions.     

Learning biology through research papers: a stimulus for question-asking by high-school students

Question-asking is a basic skill, required for the development of scientific thinking. However, the way in which science lessons are conducted does not usually stimulate question-asking by students. To make students more familiar with the scientific inquiry process, we developed a curriculum in developmental biology based on research papers suitable for high-school students. Since a scientific paper poses a research question, demonstrates the events that led to the answer, and poses new questions, we attempted to examine the effect of studying through research papers on students' ability to pose questions. Students were asked before, during, and after instruction what they found interesting to know about embryonic development. In addition, we monitored students' questions, which were asked orally during the lessons. Questions were scored according to three categories: properties, comparisons, and causal relationships. We found that before learning through research papers, students tend to ask only questions of the properties category. In contrast, students tend to pose questions that reveal a higher level of thinking and uniqueness during or following instruction with research papers. This change was not observed during or following instruction with a textbook. We suggest that learning through research papers may be one way to provide a stimulus for question-asking by high-school students and results in higher thinking levels and uniqueness.     

Assessing biodiversity loss due to land use with Life Cycle Assessment: are we there yet?

Ecosystems are under increasing pressure from human activities, with land use and land‐use change at the forefront of the drivers that provoke global and regional biodiversity loss. The first step in addressing the challenge of how to reverse the negative outlook for the coming years starts with measuring environmental loss rates and assigning responsibilities. Pinpointing the global pressures on biodiversity is a task best addressed using holistic models such as Life Cycle Assessment (LCA). LCA is the leading method for calculating cradle‐to‐grave environmental impacts of products and services; it is actively promoted by many public policies, and integrated as part of environmental information systems within private companies. LCA already deals with the potential biodiversity impacts of land use, but there are significant obstacles to overcome before its models grasp the full reach of the phenomena involved. In this review, we discuss some pressing issues that need to be addressed. LCA mainly introduces biodiversity as an endpoint category modeled as a loss in species richness due to the conversion and use of land over time and space. The functional and population effects on biodiversity are mostly absent due to the emphasis on species accumulation with limited geographic and taxonomical reach. Current land‐use modeling activities that use biodiversity indicators tend to oversimplify the real dynamics and complexity of the interactions of species among each other and with their habitats. To identify the main areas for improvement, we systematically reviewed LCA studies on land use that had findings related to global change and conservation ecology. We provide suggestion as to how to address some of the issues raised. Our overall objective was to encourage companies to monitor and take concrete steps to address the impacts of land use on biodiversity on a broader geographical scale and along increasingly globalized supply chains.     

Demographic factors and genetic variation influence population persistence under environmental change

Population persistence has been studied in a conservation context to predict the fate of small or declining populations. Persistence models have explored effects on extinction of random demographic and environmental fluctuations, but in the face of directional environmental change they should also integrate factors affecting whether a population can adapt. Here, we examine the population‐size dependence of demographic and genetic factors and their likely contributions to extinction time under scenarios of environmental change. Parameter estimates were derived from experimental populations of the rainforest species, Drosophila birchii, held in the lab for 10 generations at census sizes of 20, 100 and 1000, and later exposed to five generations of heat‐knockdown selection. Under a model of directional change in the thermal environment, rapid extinction of populations of size 20 was caused by a combination of low growth rate (r) and high stochasticity in r. Populations of 100 had significantly higher reproductive output, lower stochasticity in r and more additive genetic variance (V_A) than populations of 20, but they were predicted to persist less well than the largest size class. Even populations of 1000 persisted only a few hundred generations under realistic estimates of environmental change because of low V_A for heat‐knockdown resistance. The experimental results document population‐size dependence of demographic and adaptability factors. The simulations illustrate a threshold influence of demographic factors on population persistence, while genetic variance has a more elastic impact on persistence under environmental change.     

Predicting the past distribution of species climatic niches

Predicting past distributions of species climatic niches, hindcasting, by using climate envelope models (CEMs) is emerging as an exciting research area. CEMs are used to examine veiled evolutionary questions about extinctions, locations of past refugia and migration pathways, or to propose hypotheses concerning the past population structure of species in phylogeographical studies. CEMs are sensitive to theoretical assumptions, to model classes and to projections in non-analogous climates, among other issues. Studies hindcasting the climatic niches of species often make reference to these limitations. However, to obtain strong scientific inferences, we must not only be aware of these potential limitations but we must also overcome them. Here, I review the literature on hindcasting CEMs. I discuss the theoretical assumptions behind niche modelling, i.e. the stability of climatic niches through time and the equilibrium of species with climate. I also summarize a set of 'recommended practices' to improve hindcasting. The studies reviewed: (1) rarely test the theoretical assumptions behind niche modelling such as the stability of species climatic niches through time and the equilibrium of species with climate; (2) they only use one model class (72% of the studies) and one palaeoclimatic reconstruction (62.5%) to calibrate their models; (3) they do not check for the occurrence of non-analogous climates (97%); and (4) they do not use independent data to validate the models (72%). Ignoring the theoretical assumptions behind niche modelling and using inadequate methods for hindcasting CEMs may well entail a cascade of errors and naïve ecological and evolutionary inferences. We should also push integrative research lines linking macroecology, physiology, population biology, palaeontology, evolutionary biology and CEMs for a better understanding of niche dynamics across space and time.     

The role of algae in agriculture: a mathematical study

Synthetic fertilizers and livestock manure are nowadays widely used in agriculture to improve crop yield but nitrogen and phosphorous runoff resulting from their use compromises water quality and contributes to eutrophication phenomena in waterbeds within the countryside and ultimately in the ocean. Alternatively, algae could play an important role in agriculture where they can be used as biofertilizers and soil stabilizers. To examine the possible reuse of the detritus generated by dead algae as fertilizer for crops, we develop three mathematical models building upon each other. A system is proposed in which algae recover waste nutrients (nitrogen and phosphorus) for reuse in agricultural production. The results of our study show that in so doing, the crop yield may be increased and simultaneously the density of algae in the lake may be reduced. This could be a way to mitigate and possibly solve the environmental and economic issues nowadays facing agriculture.     

植物同化物分配及其模型研究综述

目前生态系统模型模拟中所用的大多数同化物分配模型是经验性的。同化物分配对植物的生长、竞争及结构的形成有重要的影响,是植物生长的关键,也是植物生长模型中的薄弱环节。总结了影响同化物分配的因素：生理过程和环境因子。指出植物作为一个有机的整体,所有的生理过程都对其有影响,维管束作为各器官间的连接系统,以及同化物的运输管道,其性质对同化物分配有重要的影响。综述了环境因子特别是环境水分条件对同化物分配的影响。总结了以往研究中发现的、主要的同化物分配规律,指出同化物分配的模式极其复杂,分配过程完全是根据环境以及生长阶段变化而变化的、随机应变的过程。 对于同化物分配模型按照经验模型,目的性模型,源汇关系模型进行了总结归纳,分析指出：经验性模型应用最多但机理性差;功能平衡模型在模拟营养生长阶段同化物在条与根之间的分配很成功,但应用于其它器官之间很困难;最优化模型适于模拟平衡态下同化物的分配;源汇关系模型机理性最强,可模拟任何器官间的同化物分配,应用范围最广泛。 同化物研究取得了很大的进展,但研究中仍存在很多不足：对于各相关过程的研究存在不平衡性;整体水平上同化物分配的机理仍缺乏深入研究;同化物分配对于环境的响应方面的研究相对较弱;缺乏多环境因素的研究;缺乏长期的实验观测研究。作者认为环境与同化物分配相互关系的研究将成为日后研究中的热点问题。