Demographic analysis of continuous-time life-history models

Impact of the antidepressant Bupropion on the Dynamic Energy Budget of Daphnia magna

Psychiatric drugs are considered among the emerging contaminants of concern in ecological risk assessment, due to their potential to disrupt homeostasis in aquatic organisms. Bupropion is an antidepressant that acts by selective reuptake inhibition of norepinephrine and dopamine. Little is known about this compound's effects on aquatic organisms, despite being detected in significant concentrations in both water and biota close to waste-water treatment plants and densely populated areas. Dynamic Energy Budget (DEB) models are flexible mechanistic tools that can be applied to understand toxic effects and extrapolate individual responses to higher biological levels and under untested environmental conditions. In this work, we used the stdDEB-TKTD (an application of the DEB theory to ecotoxicology) approach to investigate the possible physiological mode of action of Bupropion on the model organism Daphnia magna. Next, Dynamic Energy Budget Individual-Based Models (DEB-IBM) were used to extrapolate the results to the population level and to predict the combined effects of Bupropion exposure and food availability on the daphnids. Our results revealed an increasing negative effect of this antidepressant on the reproduction and survival of the animals with increasing concentration (0.004, 0.058, 0.58 and 58 μM). At the population level, we found that even the lowest used doses of Bupropion could reduce the population density and its reproductive output. The impacts are predicted to be stronger under limited food conditions.

Evolution of reproductive strategies in incipient multicellularity

Multicellular organisms potentially show a large degree of diversity in reproductive strategies, producing offspring with varying sizes and compositions compared to their unicellular ancestors. In reality, only a few of these reproductive strategies are prevalent. To understand why this could be the case, we develop a stage-structured population model to probe the evolutionary growth advantages of reproductive strategies in incipient multicellular organisms. The performance of reproductive strategies is evaluated by the growth rates of the corresponding populations. We identify the optimal reproductive strategy, leading to the largest growth rate for a population. Considering the effects of organism size and cellular interaction, we found that distinct reproductive strategies could perform uniquely or equally well under different conditions. If a single reproductive strategy is optimal, it is binary splitting, dividing into two parts. Our results show that organism size and cellular interaction can play crucial roles in shaping reproductive strategies in nascent multicellularity. Our model sheds light on understanding the mechanism driving the evolution of reproductive strategies in incipient multicellularity. Beyond multicellularity, our results imply that a crucial factor in the evolution of unicellular species’ reproductive strategies is organism size.

A review of bioenergetic modelling for marine mammal populations

Bioenergetic models describe the processes through which animals acquire energy from resources in the environment and allocate it to different life history functions. They capture some of the fundamental mechanisms regulating individuals, populations and ecosystems and have thus been used in a wide variety of theoretical and applied contexts. Here, I review the development of bioenergetic models for marine mammals and their application to management and conservation. For these long-lived, wide-ranging species, bioenergetic approaches were initially used to assess the energy requirements and prey consumption of individuals and populations. Increasingly, models are developed to describe the dynamics of energy intake and allocation and predict how resulting body reserves, vital rates and population dynamics might change as external conditions vary. The building blocks required to develop such models include estimates of intake rate, maintenance costs, growth patterns, energy storage and the dynamics of gestation and lactation, as well as rules for prioritizing allocation. I describe how these components have been parameterized for marine mammals and highlight critical research gaps. Large variation exists among available analytical approaches, reflecting the large range of life histories, management needs and data availability across studies. Flexibility in modelling strategy has supported tailored applications to specific case studies but has resulted in limited generality. Despite the many empirical and theoretical uncertainties that remain, bioenergetic models can be used to predict individual and population responses to environmental change and other anthropogenic impacts, thus providing powerful tools to inform effective management and conservation.

Quantifying impacts of plastic debris on marine wildlife identifies ecological breakpoints

Quantifying sublethal effects of plastics ingestion on marine wildlife is difficult, but key to understanding the ontogeny and population dynamics of affected species. We developed a method that overcomes the difficulties by modelling individual ontogeny under reduced energy intake and expenditure caused by debris ingestion. The predicted ontogeny is combined with a population dynamics model to identify ecological breakpoints: cessation of reproduction or negative population growth. Exemplifying this approach on loggerhead turtles, we find that between 3% and 25% of plastics in digestive contents causes a 2.5-20% reduction in perceived food abundance and total available energy, resulting in a 10-15% lower condition index and 10% to 88% lower total seasonal reproductive output compared to unaffected turtles. The reported plastics ingestion is insufficient to impede sexual maturation, but population declines are possible. The method is readily applicable to other species impacted by debris ingestion.

Population-level effects of acoustic disturbance in Atlantic cod: a size-structured analysis based on energy budgets

Anthropogenic underwater noise may negatively affect marine animals. Yet, while fishes are highly sensitive to sounds, effects of acoustic disturbances on fishes have not been extensively studied at the population level. In this study, we use a size-structured model based on energy budgets to analyse potential population-level effects of anthropogenic noise on Atlantic cod (Gadus morhua). Using the model framework, we assess the impact of four possible effect pathways of disturbance on the cod population growth rate. Through increased stress, changes in foraging and movement behaviour, and effects on the auditory system, anthropogenic noise can lead to (i) increased energy expenditure, (ii) reduced food intake, (iii) increased mortality, and (iv) reduced reproductive output. Our results show that population growth rates are particularly sensitive to changes in energy expenditure and food intake because they indirectly affect the age of maturation, survival and fecundity. Sub-lethal effects of sound exposure may thus affect populations of cod and fishes with similar life histories more than lethal effects of sound exposure. Moreover, anthropogenic noise may negatively affect populations when causing persistent increases of energy expenditure or decreases of food intake. Effects of specific acoustic pollutants on energy acquisition and expenditure should therefore be further investigated.

Habitat deterioration promotes the evolution of direct development in metamorphosing species

Although metamorphosis is widespread in the animal kingdom, several species have evolved life‐cycle modifications to avoid complete metamorphosis. Some species, for example, many salamanders and newts, have deleted the adult stage via a process called paedomorphosis. Others, for example, some frog species and marine invertebrates, no longer have a distinct larval stage and reach maturation via direct development. Here we study which ecological conditions can lead to the loss of metamorphosis via the evolution of direct development. To do so, we use size‐structured consumer‐resource models in conjunction with the adaptive‐dynamics approach. In case the larval habitat deteriorates, individuals will produce larger offspring and in concert accelerate metamorphosis. Although this leads to the evolutionary transition from metamorphosis to direct development when the adult habitat is highly favorable, the population will go extinct in case the adult habitat does not provide sufficient food to escape metamorphosis. With a phylogenetic approach we furthermore show that among amphibians the transition of metamorphosis to direct development is indeed, in line with model predictions, conditional on and preceded by the evolution of larger egg sizes.

Is there a Trivers-Willard effect for parental investment? Modelling evolutionarily stable strategies using a matrix population model with nonlinear mating

The Trivers-Willard hypothesis (TWH) states that parents in good condition preferentially produce the sex with a higher variation in reproductive success, whereas parents in bad condition favour the opposite sex. Theorists distinguish two variants of the TWH: (a) a biased sex-ratio at birth and (b) biased parental investment after birth. It has been argued before that the conditions stated by Trivers and Willard (good condition is inherited and affects reproductive success more strongly for one of the sexes) are sufficient for the sex-ratio version but insufficient for the investment version of the TWH. However, it has not yet been investigated how these conditions affect parental investment in high and low quality parents, depending on the life-cycle of a species. The present study aims to fill this gap by introducing a multi-stage matrix population model with nonlinear mating to describe the effects of parental investment after birth on the reproductive values of male and female individuals. Using methods from adaptive dynamics and evolutionary invasion analysis, evolutionary trajectories and evolutionarily stable strategies are derived for different parameterizations of the model. Simulation results demonstrate that the conditions given by Trivers and Willard produce a general bias of parental investment towards the sex with higher variance in reproductive value. This bias is stronger for low-quality parents than for high-quality parents and matches the expected marginal offspring reproductive values for parental investment.

Interacting cells driving the evolution of multicellular life cycles

Evolution of complex multicellular life began from the emergence of a life cycle involving the formation of cell clusters. The opportunity for cells to interact within clusters provided them with an advantage over unicellular life forms. However, what kind of interactions may lead to the evolution of multicellular life cycles? Here, we combine evolutionary game theory with a model for the emergence of multicellular groups to investigate how cell interactions can influence reproduction modes during the early stages of the evolution of multicellularity. In our model, the presence of both cell types is maintained by stochastic phenotype switching during cell division. We identify evolutionary optimal life cycles as those which maximize the population growth rate. Among all interactions captured by two-player games, the vast majority promotes two classes of life cycles: (i) splitting into unicellular propagules or (ii) fragmentation into two offspring clusters of equal (or almost equal) size. Our findings indicate that the three most important characteristics, determining whether multicellular life cycles will evolve, are the average performance of homogeneous groups, heterogeneous groups, and solitary cells.

Multicellular organisms are ubiquitous. But how did the first multicellular organisms arise? It is typically argued that this occurred due to benefits coming from interactions between cells. One example of such interactions is the division of labour. For instance, colonial cyanobacteria delegate photosynthesis and nitrogen fixation to different cells within the colony. In this way, the colony gains a growth advantage over unicellular cyanobacteria. However, not all cell interactions favour multicellular life. Cheater cells residing in a colony without any contribution will outgrow other cells. Then, the growing burden of cheaters may eventually destroy the colony. Here, we ask what kinds of interactions promote the evolution of multicellularity? We investigated all interactions captured by pairwise games and for each of them, we look for the evolutionarily optimal life cycle: How big should the colony grow and how should it split into offspring cells or colonies? We found that multicellularity can evolve with interactions far beyond cooperation or division of labour scenarios. More surprisingly, most of the life cycles found fall into either of two categories: A parent colony splits into two multicellular parts, or it splits into multiple independent cells.

Age × stage-classified demographic analysis: a comprehensive approach

This paper presents a comprehensive theory for the demographic analysis of populations in which individuals are classified by both age and stage. The earliest demographic models were age classified. Ecologists adopted methods developed by human demographers and used life tables to quantify survivorship and fertility of cohorts and the growth rates and structures of populations. Later, motivated by studies of plants and insects, matrix population models structured by size or stage were developed. The theory of these models has been extended to cover all the aspects of age-classified demography and more. It is a natural development to consider populations classified by both age and stage. A steady trickle of results has appeared since the 1960s, analyzing one or another aspect of age × stage-classified populations, in both ecology and human demography. Here, we use the vec-permutation formulation of multistate matrix population models to incorporate age- and stage-specific vital rates into demographic analysis. We present cohort results for the life table functions (survivorship, mortality, and fertility), the dynamics of intra-cohort selection, the statistics of longevity, the joint distribution of age and stage at death, and the statistics of life disparity. Combining transitions and fertility yields a complete set of population dynamic results, including population growth rates and structures, net reproductive rate, the statistics of lifetime reproduction, and measures of generation time. We present a complete analysis of a hypothetical model species, inspired by poecilogonous marine invertebrates that produce two kinds of larval offspring. Given the joint effects of age and stage, many familiar demographic results become multidimensional, so calculations of marginal and mixture distributions are an important tool. From an age-classified point of view, stage structure is a form of unobserved heterogeneity. From a stage-classified point of view, age structure is unobserved heterogeneity. In an age × stage-classified model, variance in demographic outcomes can be partitioned into contributions from both sources. Because these models are formulated as matrices, they are amenable to a complete sensitivity analysis. As more detailed and longer longitudinal studies are developed, age × stage-classified demography will become more common and more important.

Understanding the population consequences of disturbance

Managing the nonlethal effects of disturbance on wildlife populations has been a long-term goal for decision makers, managers, and ecologists, and assessment of these effects is currently required by European Union and United States legislation. However, robust assessment of these effects is challenging. The management of human activities that have nonlethal effects on wildlife is a specific example of a fundamental ecological problem: how to understand the population-level consequences of changes in the behavior or physiology of individual animals that are caused by external stressors. In this study, we review recent applications of a conceptual framework for assessing and predicting these consequences for marine mammal populations. We explore the range of models that can be used to formalize the approach and we identify critical research gaps. We also provide a decision tree that can be used to select the most appropriate model structure given the available data. Synthesis and applications: The implementation of this framework has moved the focus of discussion of the management of nonlethal disturbances on marine mammal populations away from a rhetorical debate about defining negligible impact and toward a quantitative understanding of long-term population-level effects. Here we demonstrate the framework's general applicability to other marine and terrestrial systems and show how it can support integrated modeling of the proximate and ultimate mechanisms that regulate trait-mediated, indirect interactions in ecological communities, that is, the nonconsumptive effects of a predator or stressor on a species' behavior, physiology, or life history.

Advanced quantitative methods in correlating sarcopenic muscle degeneration with lower extremity function biometrics and comorbidities

Sarcopenic muscular degeneration has been consistently identified as an independent risk factor for mortality in aging populations. Recent investigations have realized the quantitative potential of computed tomography (CT) image analysis to describe skeletal muscle volume and composition; however, the optimum approach to assessing these data remains debated. Current literature reports average Hounsfield unit (HU) values and/or segmented soft tissue cross-sectional areas to investigate muscle quality. However, standardized methods for CT analyses and their utility as a comorbidity index remain undefined, and no existing studies compare these methods to the assessment of entire radiodensitometric distributions. The primary aim of this study was to present a comparison of nonlinear trimodal regression analysis (NTRA) parameters of entire radiodensitometric muscle distributions against extant CT metrics and their correlation with lower extremity function (LEF) biometrics (normal/fast gait speed, timed up-and-go, and isometric leg strength) and biochemical and nutritional parameters, such as total solubilized cholesterol (SCHOL) and body mass index (BMI). Data were obtained from 3,162 subjects, aged 66–96 years, from the population-based AGES-Reykjavik Study. 1-D k-means clustering was employed to discretize each biometric and comorbidity dataset into twelve subpopulations, in accordance with Sturges’ Formula for Class Selection. Dataset linear regressions were performed against eleven NTRA distribution parameters and standard CT analyses (fat/muscle cross-sectional area and average HU value). Parameters from NTRA and CT standards were analogously assembled by age and sex. Analysis of specific NTRA parameters with standard CT results showed linear correlation coefficients greater than 0.85, but multiple regression analysis of correlative NTRA parameters yielded a correlation coefficient of 0.99 (P<0.005). These results highlight the specificities of each muscle quality metric to LEF biometrics, SCHOL, and BMI, and particularly highlight the value of the connective tissue regime in this regard.

Seasonal reproduction leads to population collapse and an Allee effect in a stage-structured consumer-resource biomass model when mortality rate increases

Many populations collapse suddenly when reaching low densities even if they have abundant food conditions, a phenomenon known as an Allee effect. Such collapses can have disastrous consequences, for example, for loss of biodiversity. In this paper, we formulate a stage-structured consumer-resource biomass model in which adults only reproduce at the beginning of each growing season, and investigate the effect of an increasing stage-independent background mortality rate of the consumer. As the main difference with previously studied continuous-time models, seasonal reproduction can result in an Allee effect and consumer population collapses at high consumer mortality rate. However, unlike the mechanisms reported in the literature, in our model the Allee effect results from the time difference between the maturation of juveniles and the reproduction of adults. The timing of maturation plays a crucial role because it not only determines the body size of the individuals at maturation but also influences the duration of the period during which adults can invest in reproductive energy, which together determine the reproductive output at the end of the season. We suggest that there exists an optimal timing of maturation and that consumer persistence is promoted if individuals mature later in the season at a larger body size, rather than maturing early, despite high food availability supporting rapid growth.

A numerical framework for computing steady states of structured population models and their stability

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

A Parent-Offspring Trade-Off Limits the Evolution of an Ontogenetic Niche Shift

Many free-living animal species, including the majority of fish, insects, and amphibians, change their food and habitat during their life. Even though these ontogenetic changes in niche are common, it is not well understood which ecological conditions have favored the evolution of these shifts. Using an adaptive dynamics approach, we show that it is evolutionarily advantageous to switch to an alternative food source in the course of ontogeny when this results in a higher intake rate for the switching consumers. Individuals are, however, not able to specialize on this new food source when this negatively affects the performance early in life on the original food source. Selection on these early life stages is so strong that in species with a complete diet shift, evolution results in large juveniles and adults that are maladapted to the alternative food source while their offspring are specialized on the original food source when young. These outcomes suggest strong selection to decouple the different life stages, such that they can maximize their performance on different food sources independently from each other. Metamorphosis could be a way to decouple the different life stages and therefore evolve in species that feed on multiple food sources during their life.

ECOLOGICAL MODELING OF TALITROIDES TOPITOTUM (CRUSTACEA: AMPHIPODA)

A compartmental model was developed to describe the temporal course of an exotic amphipod's population, Talitroides topitotum, in an Atlantic Forest habitat in Brazil. Extensive biological information — including breeding pattern, development and mortality rates, and temperature dependence of the parameters — were considered in the model. A genetic algorithm was used to estimate model parameters by comparing simulation results with field data. This allowed us to discuss the reproductive strategies adopted by this species and to analyze the potential influence of global climate changes on its populations dynamics.

Impacts of environmental variability on desiccation rate, plastic responses and population dynamics of Glossina pallidipes

Physiological responses to transient conditions may result in costly responses with little fitness benefits, and therefore, a trade-off must exist between the speed of response and the duration of exposure to new conditions. Here, using the puparia of an important insect disease vector, Glossina pallidipes, we examine this potential trade-off using a novel combination of an experimental approach and a population dynamics model. Specifically, we explore and dissect the interactions between plastic physiological responses, treatment-duration and -intensity using an experimental approach. We then integrate these experimental results from organismal water-balance data and their plastic responses into a population dynamics model to examine the potential relative fitness effects of simulated transient weather conditions on population growth rates. The results show evidence for the predicted trade-off for plasticity of water loss rate (WLR) and the duration of new environmental conditions. When altered environmental conditions lasted for longer durations, physiological responses could match the new environmental conditions, and this resulted in a lower WLR and lower rates of population decline. At shorter time-scales however, a mismatch between acclimation duration and physiological responses was reflected by reduced overall population growth rates. This may indicate a potential fitness cost due to insufficient time for physiological adjustments to take place. The outcomes of this work therefore suggest plastic water balance responses have both costs and benefits, and these depend on the time-scale and magnitude of variation in environmental conditions. These results are significant for understanding the evolution of plastic physiological responses and changes in population abundance in the context of environmental variability.

Predicting population dynamics from the properties of individuals: a cross-level test of dynamic energy budget theory

Individual-based models (IBMs) are increasingly used to link the dynamics of individuals to higher levels of biological organization. Still, many IBMs are data hungry, species specific, and time-consuming to develop and analyze. Many of these issues would be resolved by using general theories of individual dynamics as the basis for IBMs. While such theories have frequently been examined at the individual level, few cross-level tests exist that also try to predict population dynamics. Here we performed a cross-level test of dynamic energy budget (DEB) theory by parameterizing an individual-based model using individual-level data of the water flea, Daphnia magna, and comparing the emerging population dynamics to independent data from population experiments. We found that DEB theory successfully predicted population growth rates and peak densities but failed to capture the decline phase. Further assumptions on food-dependent mortality of juveniles were needed to capture the population dynamics after the initial population peak. The resulting model then predicted, without further calibration, characteristic switches between small- and large-amplitude cycles, which have been observed for Daphnia. We conclude that cross-level tests help detect gaps in current individual-level theories and ultimately will lead to theory development and the establishment of a generic basis for individual-based models and ecology.

Lifetime return on investment increases with leaf lifespan among 10 Australian woodland species

• Co-occurring species often differ in their leaf lifespan (LL) and it remains unclear how such variation is maintained in a competitive context. Here we test the hypothesis that leaves of long-LL species yield a greater return in carbon (C) fixed per unit C or nutrient invested by the plant than those of short-LL species. • For 10 sympatric woodland species, we assessed three-dimensional shoot architecture, canopy openness, leaf photosynthetic light response, leaf dark respiration and leaf construction costs across leaf age sequences. We then used the YPLANT model to estimate light interception and C revenue along the measured leaf age sequences. This was done under a series of simulations that incorporated the potential covariates of LL in an additive fashion. • Lifetime return in C fixed per unit C, N or P invested increased with LL in all simulations. • In contrast to other recent studies, our results show that extended LL confers a fundamental economic advantage by increasing a plant's return on investment in leaves. This suggests that time-discounting effects, that is, the compounding of income that arises from quick reinvestment of C revenue, are key in allowing short-LL species to succeed in the face of this economic handicap.

The meaning of birth and death (in macroevolutionary birth-death models)

Birth-death models are central to much macroevolutionary theory. The fundamental parameters of these models concern durations. Different species concepts realize different species durations because they represent different ideas of what birth (speciation) and death (extinction) mean. Here, we use Cenozoic macroperforate planktonic foraminifera as a case study to ask: what are the dynamical consequences of changing the definition of birth and death? We show strong evidence for biotic constraints on diversification using evolutionary species, but less with morphospecies. Discussing reasons for this discrepancy, we emphasize that clarity of species concept leads to clarity of meaning when interpreting macroevolutionary birth-death models.

Effects of growth curve plasticity on size-structured population dynamics

The physiological-structured population models assume that a fixed fraction of energy intake is utilized for individual growth and maintenance while the remaining for adult fertility. The assumption results in two concerns: energy loss for juveniles and a reproduction dilemma for adults. The dilemma results from the possibility that adults have to breed even if metabolic costs fail to be covered. We consider a size-structured population model, where standard metabolism is given top priority for utilizing energy intake and the surplus energy, if there is any, is distributed to individual growth and reproduction. Moreover, the portion of surplus energy for reproduction is size-dependent and increases monotonically with size. Using the newly developed parameter continuation, we demonstrate their disparate effects on population dynamics. Results show that the size-dependent mechanism of energy allocation primarily exerts destabilizing effects on the system but considerably promotes species coexistence, in comparison with the size-independent mechanism. We conclude that the size-dependent mechanism is, to a large extent, a dispensable component of model ingredients when ontogeny is explicitly taken into consideration.

Interplay between changing climate and species' ecology drives macroevolutionary dynamics

Ecological change provokes speciation and extinction, but our knowledge of the interplay among the biotic and abiotic drivers of macroevolution remains limited. Using the unparalleled fossil record of Cenozoic macroperforate planktonic foraminifera, we demonstrate that macroevolutionary dynamics depend on the interaction between species' ecology and the changing climate. This interplay drives diversification but differs between speciation probability and extinction risk: Speciation was more strongly shaped by diversity dependence than by climate change, whereas the reverse was true for extinction. Crucially, no single ecology was optimal in all environments, and species with distinct ecologies had significantly different probabilities of speciation and extinction. The ensuing macroevolutionary dynamics depend fundamentally on the ecological structure of species' assemblages.

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.

Effects of C60 nanoparticle exposure on earthworms (Lumbricus rubellus) and implications for population dynamics

Effects of C60 nanoparticles (nominal concentrations 0, 15.4 and 154 mg/kg soil) on mortality, growth and reproduction of Lumbricus rubellus earthworms were assessed. C60 exposure had a significant effect on cocoon production, juvenile growth rate and mortality. These endpoints were used to model effects on the population level. This demonstrated reduced population growth rate with increasing C60 concentrations. Furthermore, a shift in stage structure was shown for C60 exposed populations, i.e. a larger proportion of juveniles. This result implies that the lower juvenile growth rate due to exposure to C60 resulted in a larger proportion of juveniles, despite increased mortality among juveniles. Overall, this study indicates that C60 exposure may seriously affect earthworm populations. Furthermore, it was demonstrated that juveniles were more sensitive to C60 exposure than adults.

How to lift a model for individual behaviour to the population level?

The quick answer to the title question is: by bookkeeping; introduce as p(opulation)-state a measure telling how the individuals are distributed over their common i(ndividual)-state space, and track how the various i-processes change this measure. Unfortunately, this answer leads to a mathematical theory that is technically complicated as well as immature. Alternatively, one may describe a population in terms of the history of the population birth rate together with the history of any environmental variables affecting i-state changes, reproduction and survival. Thus, a population model leads to delay equations. This delay formulation corresponds to a restriction of the p-dynamics to a forward invariant attracting set, so that no information is lost that is relevant for long-term dynamics. For such equations there exists a well-developed theory. In particular, numerical bifurcation tools work essentially the same as for ordinary differential equations. However, the available tools still need considerable adaptation before they can be practically applied to the dynamic energy budget (DEB) model. For the time being we recommend simplifying the i-dynamics before embarking on a systematic mathematical exploration of the associated p-behaviour. The long-term aim is to extend the tools, with the DEB model as a relevant goal post.

Potential application of population models in the European ecological risk assessment of chemicals. II. Review of models and their potential to address environmental protection aims

Whereas current chemical risk assessment (RA) schemes within the European Union (EU) focus mainly on toxicity and bioaccumulation of chemicals in individual organisms, most protection goals aim at preserving populations of nontarget organisms rather than individuals. Ecological models are tools rarely recommended in official technical documents on RA of chemicals, but are widely used by researchers to assess risks to populations, communities and ecosystems. Their great advantage is the relatively straightforward integration of the sensitivity of species to chemicals, the mode of action and fate in the environment of toxicants, life-history traits of the species of concern, and landscape features. To promote the usage of ecological models in regulatory risk assessment, this study tries to establish whether existing, published ecological modeling studies have addressed or have the potential to address the protection aims and requirements of the chemical directives of the EU. We reviewed 148 publications, and evaluated and analyzed them in a database according to defined criteria. Published models were also classified in terms of 5 areas where their application would be most useful for chemical RA. All potential application areas are well represented in the published literature. Most models were developed to estimate population-level responses on the basis of individual effects, followed by recovery process assessment, both in individuals and at the level of metapopulations. We provide case studies for each of the proposed areas of ecological model application. The lack of clarity about protection goals in legislative documents made it impossible to establish a direct link between modeling studies and protection goals. Because most of the models reviewed here were not developed for regulatory risk assessment, there is great potential and a variety of ecological models in the published literature.

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023-1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman-Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254-274, 1984; de Roos et al. in J Math Biol 28:609-643, 1990) and a model introduced by Gurney-Nisbet (Theor Popul Biol 28:150-180, 1985) and Jones et al. (J Math Anal Appl 135:354-368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.

Dynamic energy budgets in syntrophic symbiotic relationships between heterotrophic hosts and photoautotrophic symbionts

In this paper we develop and investigate a dynamic energy budget (DEB) model describing the syntrophic symbiotic relationship between a heterotrophic host and an internal photoautotrophic symbiont. The model specifies the flows of matter and energy among host, symbiont and environment with minimal complexity and uses the concept of synthesizing units to describe smoothly the assimilation of multiple limiting factors, in particular inorganic carbon and nitrogen, and irradiance. The model has two passive regulation mechanisms: the symbiont shares only photosynthate that it cannot use itself, and the host delivers only excess nutrients to the symbiont. With parameter values plausible for scleractinian corals, we show that these two regulation mechanisms suffice to obtain a stable symbiotic relationship under constant ambient conditions, provided those conditions support sustenance of host and symbiont. Furthermore, the symbiont density in the host varies relatively little as a function of ambient food density, inorganic nitrogen and irradiance. This symbiont density tends to increase with light deprivation or nitrogen enrichment, either directly or via food. We also investigate the relative benefit each partner derives from the relationship and conclude that this relationship may shift from mutualism to parasitism as environmental conditions change.