Impact of demography on extinction/fixation events

In this article we consider diffusion processes modeling the dynamics of multiple allelic proportions (with fixed and varying population size). We are interested in the way alleles extinctions and fixations occur. We first prove that for the Wright–Fisher diffusion process with selection, alleles get extinct successively (and not simultaneously), until the fixation of one last allele. Then we introduce a very general model with selection, competition and Mendelian reproduction, derived from the rescaling of a discrete individual-based dynamics. This multi-dimensional diffusion process describes the dynamics of the population size as well as the proportion of each type in the population. We prove first that alleles extinctions occur successively and second that depending on population size dynamics near extinction, fixation can occur either before extinction almost surely, or not. The proofs of these different results rely on stochastic time changes, integrability of one-dimensional diffusion processes paths and multi-dimensional Girsanov’s tranform.

     

A paradox in discrete single species population dynamics with harvesting/thinning

We analyze a general time-discrete mathematical model of single species population dynamics with the intraspecific density effect and the harvesting/thinning effect. We harvest a portion of the population at a moment in each year. We investigate the condition under which the harvesting/thinning causes an eventual increase of its population at the equilibrium, and show that such a paradoxical increase could occur for the discrete single species population dynamics with a large family of density effect functions. Some typical models are analyzed in detail according to the possibility of the paradox emergence. Our result implies that the contest competition would never cause the paradox, while the scramble competition would be likely to cause it.     

Effect of substrate competition in kinetic models of metabolic networks

Substrate competition can be found in many types of biological processes, ranging from gene expression to signal transduction and metabolic pathways. Although several experimental and in silico studies have shown the impact of substrate competition on these processes, it is still often neglected, especially in modelling approaches. Using toy models that exemplify different metabolic pathway scenarios, we show that substrate competition can influence the dynamics and the steady state concentrations of a metabolic pathway. We have additionally derived rate laws for substrate competition in reversible reactions and summarise existing rate laws for substrate competition in irreversible reactions.     

Pioneer exclusion in a one-hump discrete pioneer-climax competitive system

One-hump maps as single species pioneer and climax ecological models confirm the expected pioneer exclusion in an uninterrupted two dimensional discrete pioneer-climax competition model. Other dynamic situations occur in such models. In particular a mutual exclusion principle is presented as a mathematical theorem.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Symmetric competition as a general model for single-species adaptive dynamics

Adaptive dynamics is a widely used framework for modeling long-term evolution of continuous phenotypes. It is based on invasion fitness functions, which determine selection gradients and the canonical equation of adaptive dynamics. Even though the derivation of the adaptive dynamics from a given invasion fitness function is general and model-independent, the derivation of the invasion fitness function itself requires specification of an underlying ecological model. Therefore, evolutionary insights gained from adaptive dynamics models are generally model-dependent. Logistic models for symmetric, frequency-dependent competition are widely used in this context. Such models have the property that the selection gradients derived from them are gradients of scalar functions, which reflects a certain gradient property of the corresponding invasion fitness function. We show that any adaptive dynamics model that is based on an invasion fitness functions with this gradient property can be transformed into a generalized symmetric competition model. This provides a precise delineation of the generality of results derived from competition models. Roughly speaking, to understand the adaptive dynamics of the class of models satisfying a certain gradient condition, one only needs a complete understanding of the adaptive dynamics of symmetric, frequency-dependent competition. We show how this result can be applied to number of basic issues in evolutionary theory.     

Coexistence of competing juvenile–adult structured populations

The Leslie-Gower model is a discrete time analog of the competition Lotka–Volterra model and is known to possess the same dynamic scenarios of that famous model. The Leslie–Gower model played a historically significant role in the history of competition theory in its application to classic laboratory experiments of two competing species of flour beetles (carried out by Park in the 1940s–1960s). While these experiments generally supported what became the Competitive Exclusion Principle, Park observed an anomalous coexistence case. Recent literature has discussed Park’s ‘coexistence case’ by means of non-Lotka–Volterra, non-equilibrium dynamics that occur in a high dimensional model with life cycle stages. We study this dynamic possibility in the lowest possible dimension, that is to say, by means of a model involving only two species each with two life cycle stages. We do this by extending the Leslie–Gower model so as to describe the competitive interaction of two species with juvenile and adult classes. We give a complete account of the global dynamics of the resulting model and show that it allows for non-equilibrium competitive coexistence as competition coefficients are increased. We also show that this phenomenon occurs in a general class of models for competing populations structured by juvenile and adult life cycle stages.     

Coexistence of competing juvenile-adult structured populations

The Leslie-Gower model is a discrete time analog of the competition Lotka-Volterra model and is known to possess the same dynamic scenarios of that famous model. The Leslie-Gower model played a historically significant role in the history of competition theory in its application to classic laboratory experiments of two competing species of flour beetles (carried out by Park in the 1940s-1960s). While these experiments generally supported what became the Competitive Exclusion Principle, Park observed an anomalous coexistence case. Recent literature has discussed Park's 'coexistence case' by means of non-Lotka-Volterra, non-equilibrium dynamics that occur in a high dimensional model with life cycle stages. We study this dynamic possibility in the lowest possible dimension, that is to say, by means of a model involving only two species each with two life cycle stages. We do this by extending the Leslie-Gower model so as to describe the competitive interaction of two species with juvenile and adult classes. We give a complete account of the global dynamics of the resulting model and show that it allows for non-equilibrium competitive coexistence as competition coefficients are increased. We also show that this phenomenon occurs in a general class of models for competing populations structured by juvenile and adult life cycle stages.     

Physiologically structured models – from versatile technique to ecological theory

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

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.     

Effects of delay,truncations and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models

Understanding effects of hypotheses about reproductive influences, reproductive schedules and the model mechanisms that lead to a loss of stability in a structured model population might provide information about the dynamics of natural population. To demonstrate characteristics of a discrete time, nonlinear, age structured population model, the transition from stability to instability is investigated. Questions about the stability, oscillations and delay processes within the model framework are posed. The relevant processes include delay of reproduction and truncation of lifetime, reproductive classes, and density dependent effects. We find that the effects of delaying reproduction is not stabilizing, but that the reproductive delay is a mechanism that acts to simplify the system dynamics. Density dependence in the reproduction schedule tends to lead to oscillations of large period and towards more unstable dynamics. The methods allow us to establish a conjecture of Levin and Goodyear about the form of the stability in discrete Leslie matrix models.This research was supported in part by the US Environmental Protection Agency under cooperation agreement CR-816081     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Dynamics of the integral measure of competition in plant communities of different uniformity

The effect of non-monotony of the correlation coefficient between crown projection area and the Voronoi polygon area in 90-year spruce sites in the Moscow Region arranged in ascending order of the variation coefficient of stem diameter is analyzed using the original 2D model of the plant community dynamics. Theoretical integral (over the entire community) indices of competition demonstrating non-monotone competition in relatively uniform communities—the Clements’ index (Clements et al., 1929) and the mean suppression of community plants—were analyzed. Direct computation demonstrated that the correlation coefficient can also be considered as an index of competition, and its dynamics is indeed non-monotone in uniform communities. This finding allows us to roughly evaluate the potential duration of biomass growth for free-growing spruce in the Moscow Region as 1000 years. Due to the limitations of the 2D resource model (nearly “symmetric” competition), certain model characteristics of communities (the variation coefficient and correlation characteristics of neighborhood) differed several times from natural ones although showing the correct trend of changes.     

A dynamic framework for the study of optimal birth intervals reveals the importance of sibling competition and mortality risks

Human reproductive patterns have been well studied, but the mechanisms by which physiology, ecology and existing kin interact to affect the life history need quantification. Here, we create a model to investigate how age‐specific interbirth intervals adapt to environmental and intrinsic mortality, and how birth patterns can be shaped by competition and help between siblings. The model provides a flexible framework for studying the processes underlying human reproductive scheduling. We developed a state‐based optimality model to determine age‐dependent and family‐dependent sets of reproductive strategies, including the state of the mother and her offspring. We parameterized the model with realistic mortality curves derived from five human populations. Overall, optimal birth intervals increase until the age of 30 after which they remain relatively constant until the end of the reproductive lifespan. Offspring helping each other does not have much effect on birth intervals. Increasing infant and senescent mortality in different populations decreases interbirth intervals. We show that sibling competition and infant mortality interact to lengthen interbirth intervals. In lower‐mortality populations, intense sibling competition pushes births further apart. Varying the adult risk of mortality alone has no effect on birth intervals between populations; competition between offspring drives the differences in birth intervals only when infant mortality is low. These results are relevant to understanding the demographic transition, because our model predicts that sibling competition becomes an important determinant of optimal interbirth intervals only when mortality is low, as in post‐transition societies. We do not predict that these effects alone can select for menopause.     

Competition between two species for two complementary or substitutable resources

The classical models of interspecific competition are phenomenological, that is, they purport to describe the trajectories followed by the abundances of both competitors, without specifying either mechanisms or the dynamics of what the competition is for, i.e. resources. Yet the conditions for the different outcomes of competition inferred from these models are most often interpreted in terms of resources. Here it is contended that the dynamics of resource supply and exploitation must be explicitly taken into account if these conditions are to be obtained in any accurate form. First, the distinction between perfectly substitutable, imperfectly substitutable and perfectly complementary resources is examined. This leads to an array of models intended to mirror interspecific exploitative competition for two resources for each category considered. Then, the conditions for the coexistence of the two competitors in each case are derived and presented with direct reference to the modes of resource use.     

A symbiont's dispersal strategy: condition-dependent dispersal underlies predictable variation in direct transmission among hosts

Direct horizontal transmission of pathogenic and mutualistic symbionts has profound consequences for host and symbiont fitness alike. While the importance of contact rates for transmission is widely recognized, the processes that underlie variation in transmission during contact are rarely considered. Here, we took a symbiont''s perspective of transmission as a form of dispersal and adopted the concept of condition-dependent dispersal strategies from the study of free-living organisms to understand and predict variation in transmission in the cleaning symbiosis between crayfish and ectosymbiotic branchiobdellidan worms. Field study showed that symbiont reproductive success was correlated with host size and competition among worms for microhabitats. Laboratory experiments demonstrated high variability in transmission among host contacts. Moreover, symbionts were more likely to disperse when host size and competition for microhabitat created a fitness environment below a discrete minimum threshold. A predictive model based on a condition-dependent symbiont dispersal strategy correctly predicted transmission in 95% of experimental host encounters and the exact magnitude of transmission in 67%, both significantly better than predictions that assumed a fixed transmission rate. Our work provides a dispersal-based understanding of symbiont transmission and suggests adaptive symbiont dispersal strategies can explain variation in transmission dynamics and complex patterns of host infection.     

Ecological,Evolutionary and Social Constraints on Reproductive Effort: Are Hoary Marmots Really Biennial Breeders?

Biennial breeding is a rare life-history trait observed in animal species living in harsh, unproductive environments. This reproductive pattern is thought to occur in 10 of 14 species in the genus Marmota, making marmots useful model organisms for studying its ecological and evolutionary implications. Biennial breeding in marmots has been described as an obligate pattern which evolved as a mechanism to mitigate the energetic costs of reproduction (Evolved Constraint hypothesis). However, recent anecdotal evidence suggests that it is a facultative pattern controlled by annual variation in climate and food availability (Environmental Constraint hypothesis). Finally, in social animals like marmots, biennial breeding could result from reproductive competition between females within social groups (Social Constraint hypothesis). We evaluated these three hypotheses using mark-recapture data from an 8-year study of hoary marmot (Marmota caligata) population dynamics in the Yukon. Annual variation in breeding probability was modeled using multi-state mark-recapture models, while other reproductive life-history traits were modeled with generalized linear mixed models. Hoary marmots were neither obligate nor facultative biennial breeders, and breeding probability was insensitive to evolved, environmental, or social factors. However, newly mature females were significantly less likely to breed than older individuals. Annual breeding did not result in increased mortality. Female survival and, to a lesser extent, average fecundity were correlated with winter climate, as indexed by the Pacific Decadal Oscillation. Hoary marmots are less conservative breeders than previously believed, and the evidence for biennial breeding throughout Marmota, and in other arctic/alpine/antarctic animals, should be re-examined. Prediction of future population dynamics requires an accurate understanding of life history strategies, and of how life history traits allow animals to cope with changes in weather and other demographic influences.     

Optimal life-history strategies in seasonal consumer-resource dynamics

The interplay between individual adaptive life histories and populations dynamics is an important issue in ecology. In this context, we considered a seasonal consumer-resource model with nonoverlapping generations. We focused on the consumers decision-making process through which they maximize their reproductive output via a differential investment into foraging for resources or reproducing. Our model takes a semi-discrete form, and is composed of a continuous time within-season part, similar to a dynamic model of energy allocation, and of a discrete time part, depicting the between seasons reproduction and mortality processes. We showed that the optimal foraging-reproduction strategies of the consumers may be "determinate" or "indeterminate" depending on the season length. More surprisingly, it depended on the consumers population density as well, with large densities promoting indeterminacy. A bifurcation analysis showed that the long-term dynamics produced by this model were quite rich, ranging from both populations' extinction, coexistence at some season-to-season equilibrium or on (quasi)-periodic motions, to initial condition-dependent dynamics. Interestingly, we observed that any long-term sustainable situation corresponds to indeterminate consumers' strategies. Finally, a comparison with a model involving typical nonoptimal consumers highlighted the stabilizing effects of the optimal life histories of the consumers.     

A trait-based approach for modelling microbial litter decomposition

Trait-based models are an emerging tool in ecology with the potential to link community dynamics, environmental responses and ecosystem processes. These models represent complex communities by defining taxa with trait combinations derived from prior distributions that may be constrained by trade-offs. Herein I develop a model that links microbial community composition with physiological and enzymatic traits to predict litter decomposition rates. This approach allows for trade-offs among traits that represent alternative microbial strategies for resource acquisition. The model predicts that optimal strategies depend on the level of enzyme production in the whole community, which determines resource availability and decomposition rates. There is also evidence for facilitation and competition among microbial taxa that co-occur on decomposing litter. These interactions vary with community investment in extracellular enzyme production and the magnitude of trade-offs affecting enzyme biochemical traits. The model accounted for 69% of the variation in decomposition rates of 15 Hawaiian litter types and up to 26% of the variation in enzyme activities. By explicitly representing diversity, trait-based models can predict ecosystem processes based on functional trait distributions in a community. The model developed herein illustrates that traits influencing microbial enzyme production are some of the key controls on litter decomposition rates.     

Evolutionarily stable mating decisions for sequentially searching females and the stability of reproductive isolation by assortative mating

We consider mating strategies for females who search for males sequentially during a season of limited length. We show that the best strategy rejects a given male type if encountered before a time‐threshold but accepts him after. For frequency‐independent benefits, we obtain the optimal time‐thresholds explicitly for both discrete and continuous distributions of males, and allow for mistakes being made in assessing the correct male type. When the benefits are indirect (genes for the offspring) and the population is under frequency‐dependent ecological selection, the benefits depend on the mating strategy of other females as well. This case is particularly relevant to speciation models that seek to explore the stability of reproductive isolation by assortative mating under frequency‐dependent ecological selection. We show that the indirect benefits are to be quantified by the reproductive values of couples, and describe how the evolutionarily stable time‐thresholds can be found. We conclude with an example based on the Levene model, in which we analyze the evolutionarily stable assortative mating strategies and the strength of reproductive isolation provided by them.