Human-environment feedback and the consistency of proenvironmental behavior.

Addressing global environmental crises such as anthropogenic climate change requires the consistent adoption of proenvironmental behavior by a large part of a population. Here, we develop a mathematical model of a simple behavior-environment feedback loop to ask how the individual assessment of the environmental state combines with social interactions to influence the consistent adoption of proenvironmental behavior, and how this feeds back to the perceived environmental state. In this stochastic individual-based model, individuals can switch between two behaviors, 'active' (or actively proenvironmental) and 'baseline', differing in their perceived cost (higher for the active behavior) and environmental impact (lower for the active behavior). We show that the deterministic dynamics and the stochastic fluctuations of the system can be approximated by ordinary differential equations and a Ornstein-Uhlenbeck type process. By definition, the proenvironmental behavior is adopted consistently when, at population stationary state, its frequency is high and random fluctuations in frequency are small. We find that the combination of social and environmental feedbacks can promote the spread of costly proenvironmental behavior when neither, operating in isolation, would. To be adopted consistently, strong social pressure for proenvironmental action is necessary but not sufficient-social interactions must occur on a faster timescale compared to individual assessment, and the difference in environmental impact must be small. This simple model suggests a scenario to achieve large reductions in environmental impact, which involves incrementally more active and potentially more costly behavior being consistently adopted under increasing social pressure for proenvironmentalism.

Drosophilids with darker cuticle have higher body temperature under light.

Cuticle pigmentation was shown to be associated with body temperature for several relatively large species of insects, but it was questioned for small insects. Here we used a thermal camera to assess the association between drosophilid cuticle pigmentation and body temperature increase when individuals are exposed to light. We compared mutants of large effects within species (Drosophila melanogaster ebony and yellow mutants). Then we analyzed the impact of naturally occurring pigmentation variation within species complexes (Drosophila americana/Drosophila novamexicana and Drosophila yakuba/Drosophila santomea). Finally we analyzed lines of D. melanogaster with moderate differences in pigmentation. We found significant differences in temperatures for each of the four pairs we analyzed. The temperature differences appeared to be proportional to the differently pigmented area: between Drosophila melanogaster ebony and yellow mutants or between Drosophila americana and Drosophila novamexicana, for which the whole body is differently pigmented, the temperature difference was around 0.6 °C ± 0.2 °C. By contrast, between D. yakuba and D. santomea or between Drosophila melanogaster Dark and Pale lines, for which only the posterior abdomen is differentially pigmented, we detected a temperature difference of about 0.14 °C ± 0.10 °C. This strongly suggests that cuticle pigmentation has ecological implications in drosophilids regarding adaptation to environmental temperature.

Multistage hematopoietic stem cell regulation in the mouse: A combined biological and mathematical approach.

We have reconciled steady-state and stress hematopoiesis in a single mathematical model based on murine in vivo experiments and with a focus on hematopoietic stem and progenitor cells. A phenylhydrazine stress was first applied to mice. A reduced cell number in each progenitor compartment was evidenced during the next 7 days through a drastic level of differentiation without proliferation, followed by a huge proliferative response in all compartments including long-term hematopoietic stem cells, before a return to normal levels. Data analysis led to the addition to the 6-compartment model, of time-dependent regulation that depended indirectly on the compartment sizes. The resulting model was finely calibrated using a stochastic optimization algorithm and could reproduce biological data in silico when applied to different stress conditions (bleeding, chemotherapy, HSC depletion). In conclusion, our multi-step and time-dependent model of immature hematopoiesis provides new avenues to a better understanding of both normal and pathological hematopoiesis.

Large fluctuations in multi-scale modeling for rest hematopoiesis.

Hematopoiesis is a biological phenomenon (process) of production of mature blood cells by cellular differentiation. It is based on amplification steps due to an interplay between renewal and differentiation in the successive cell types from stem cells to mature blood cells. We will study this mechanism with a stochastic point of view to explain unexpected fluctuations on the mature blood cell number, as surprisingly observed by biologists and medical doctors in a rest hematopoiesis. We consider three cell types: stem cells, progenitors and mature blood cells. Each cell type is characterized by its own dynamics parameters, the division rate and the renewal and differentiation probabilities at each division event. We model the global population dynamics by a three-dimensional stochastic decomposable branching process. We show that the amplification mechanism is given by the inverse of the small difference between the differentiation and renewal probabilities. Introducing a parameter K which scales simultaneously the size of the first component, the differentiation and renewal probabilities and the mature blood cell death rate, we describe the asymptotic behavior of the process for large K. We show that each cell type has its own size and time scales. Focusing on the third component, we prove that the mature blood cell population size, conveniently renormalized (in time and size), is expanded in an unusual way inducing large fluctuations. The proofs are based on a fine study of the different scales involved in the model and on the use of different convergence and average techniques in the proofs.

Inference with selection, varying population size, and evolving population structure: application of ABC to a forward-backward coalescent process with interactions.

Genetic data are often used to infer demographic history and changes or detect genes under selection. Inferential methods are commonly based on models making various strong assumptions: demography and population structures are supposed a priori known, the evolution of the genetic composition of a population does not affect demography nor population structure, and there is no selection nor interaction between and within genetic strains. In this paper, we present a stochastic birth-death model with competitive interactions and asexual reproduction. We develop an inferential procedure for ecological, demographic, and genetic parameters. We first show how genetic diversity and genealogies are related to birth and death rates, and to how individuals compete within and between strains. This leads us to propose an original model of phylogenies, with trait structure and interactions, that allows multiple merging. Second, we develop an Approximate Bayesian Computation framework to use our model for analyzing genetic data. We apply our procedure to simulated data from a toy model, and to real data by analyzing the genetic diversity of microsatellites on Y-chromosomes sampled from Central Asia human populations in order to test whether different social organizations show significantly different fertilities.

Bacterial metabolic heterogeneity: from stochastic to deterministic models.

We revisit the modeling of the diauxic growth of a pure microorganism on two distinct sugars which was first described by Monod. Most available models are deterministic and make the assumption that all cells of the microbial ecosystem behave homogeneously with respect to both sugars, all consuming the first one and then switching to the second when the first is exhausted. We propose here a stochastic model which describes what is called "metabolic heterogeneity". It allows to consider small populations as in microfluidics as well as large populations where billions of individuals coexist in the medium in a batch or chemostat. We highlight the link between the stochastic model and the deterministic behavior in real large cultures using a large population approximation. Then the influence of model parameter values on model dynamics is studied, notably with respect to the lag-phase observed in real systems depending on the sugars on which the microorganism grows. It is shown that both metabolic parameters as well as initial conditions play a crucial role on system dynamics.

Availability of the Molecular Switch XylR Controls Phenotypic Heterogeneity and Lag Duration during Escherichia coli Adaptation from Glucose to Xylose.

The glucose-xylose metabolic transition is of growing interest as a model to explore cellular adaption since these molecules are the main substrates resulting from the deconstruction of lignocellulosic biomass. Here, we investigated the role of the XylR transcription factor in the length of the lag phases when the bacterium Escherichia coli needs to adapt from glucose- to xylose-based growth. First, a variety of lag times were observed when different strains of E. coli were switched from glucose to xylose. These lag times were shown to be controlled by XylR availability in the cells with no further effect on the growth rate on xylose. XylR titration provoked long lag times demonstrated to result from phenotypic heterogeneity during the switch from glucose to xylose, with a subpopulation unable to resume exponential growth, whereas the other subpopulation grew exponentially on xylose. A stochastic model was then constructed based on the assumption that XylR availability influences the probability of individual cells to switch to xylose growth. The model was used to understand how XylR behaves as a molecular switch determining the bistability set-up. This work shows that the length of lag phases in E. coli is controllable and reinforces the role of stochastic mechanism in cellular adaptation, paving the way for new strategies for the better use of sustainable carbon sources in bioeconomy.IMPORTANCE For decades, it was thought that the lags observed when microorganisms switch from one substrate to another are inherent to the time required to adapt the molecular machinery to the new substrate. Here, the lag duration was found to be the time necessary for a subpopulation of adapted cells to emerge and become the main population. By identifying the molecular mechanism controlling the subpopulation emergence, we were able to extend or reduce the duration of the lags. This work is of special importance since it demonstrates the unexpected complexity of monoclonal populations during growth on mixed substrates and provides novel mechanistic insights with regard to bacterial cellular adaptation.

A birth-death model of ageing: from individual-based dynamics to evolutive differential inclusions.

Ageing's sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual's fitness. Thanks to the recently described (Smurf) 2-phase model of ageing (Tricoire and Rera in PLoS ONE 10(11):e0141920, 2015) we propose a fresh angle for modeling the evolution of ageing. Indeed, by coupling a dramatic loss of fertility with a high-risk of impending death-amongst other multiple so-called hallmarks of ageing-the Smurf phenotype allowed us to consider ageing as a couple of sharp transitions. The birth-death model (later called bd-model) we describe here is a simple life-history trait model where each asexual and haploid individual is described by its fertility period [Formula: see text] and survival period [Formula: see text]. We show that, thanks to the Lansing effect, the effect through which the "progeny of old parents do not live as long as those of young parents", [Formula: see text] and [Formula: see text] converge during evolution to configurations [Formula: see text] in finite time. To do so, we built an individual-based stochastic model which describes the age and trait distribution dynamics of such a finite population. Then we rigorously derive the adaptive dynamics models, which describe the trait dynamics at the evolutionary time-scale. We extend the Trait Substitution Sequence with age structure to take into account the Lansing effect. Finally, we study the limiting behaviour of this jump process when mutations are small. We show that the limiting behaviour is described by a differential inclusion whose solutions [Formula: see text] reach the diagonal [Formula: see text] in finite time and then remain on it. This differential inclusion is a natural way to extend the canonical equation of adaptive dynamics in order to take into account the lack of regularity of the invasion fitness function on the diagonal [Formula: see text].

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.

Feedback between environment and traits under selection in a seasonal environment: consequences for experimental evolution.

Batch cultures are frequently used in experimental evolution to study the dynamics of adaptation. Although they are generally considered to simply drive a growth rate increase, other fitness components can also be selected for. Indeed, recurrent batches form a seasonal environment where different phases repeat periodically and different traits can be under selection in the different seasons. Moreover, the system being closed, organisms may have a strong impact on the environment. Thus, the study of adaptation should take into account the environment and eco-evolutionary feedbacks. Using data from an experimental evolution on yeast Saccharomyces cerevisiae, we developed a mathematical model to understand which traits are under selection, and what is the impact of the environment for selection in a batch culture. We showed that two kinds of traits are under selection in seasonal environments: life-history traits, related to growth and mortality, but also transition traits, related to the ability to react to environmental changes. The impact of environmental conditions can be summarized by the length of the different seasons which weight selection on each trait: the longer a season is, the higher the selection on associated traits. Since phenotypes drive season length, eco-evolutionary feedbacks emerge. Our results show how evolution in successive batches can affect season lengths and strength of selection on different traits.

The effect of competition and horizontal trait inheritance on invasion, fixation, and polymorphism.

Horizontal transfer (HT) of heritable information or 'traits' (carried by genetic elements, plasmids, endosymbionts, or culture) is widespread among living organisms. Yet current ecological and evolutionary theory addressing HT is scant. We present a modeling framework for the dynamics of two populations that compete for resources and horizontally exchange (transfer) an otherwise vertically inherited trait. Competition influences individual demographics, thereby affecting population size, which feeds back on the dynamics of transfer. This feedback is captured in a stochastic individual-based model, from which we derive a general model for the contact rate, with frequency-dependent (FD) and density-dependent (DD) rates as special cases. Taking a large-population limit on the stochastic individual-level model yields a deterministic Lotka-Volterra competition system with additional terms accounting for HT. The stability analysis of this system shows that HT can revert the direction of selection: HT can drive invasion of a deleterious trait, or prevent invasion of an advantageous trait. Due to HT, invasion does not necessarily imply fixation. Two trait values may coexist in a stable polymorphism even if their invasion fitnesses have opposite signs, or both are negative. Addressing the question of how the stochasticity of individual processes influences population fluctuations, we identify conditions on competition and mode of transfer (FD versus DD) under which the stochasticity of transfer events overwhelms demographic stochasticity. Assuming that one trait is initially rare, we derive invasion and fixation probabilities and time. In the case of costly plasmids, which are transfered unilaterally, invasion is always possible if the transfer rate is large enough; under DD and for intermediate values of the transfer rate, maintenance of the plasmid in a polymorphic population is possible. In conclusion, HT interacts with ecology (competition) in non-trivial ways. Our model provides a basis to model the influence of HT on evolutionary adaptation.

Stochastic eco-evolutionary model of a prey-predator community.

We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the polymorphic evolutionary sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these different limits with an example of prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.

Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks.

How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effects of the trait are also expected to influence the generation and maintenance of neutral variation. We consider a large population limit with rare mutation, under the assumption that the neutral marker mutates faster than the trait under selection. We prove the convergence of the stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming-Viot Process (SFVP). When restricted to the trait space this process is the Trait Substitution Sequence first introduced by Metz et al. (1996). During the invasion of a favorable mutation, a genetical bottleneck occurs and the marker associated with this favorable mutant is hitchhiked. By rigorously analysing the hitchhiking effect and how the neutral diversity is restored afterwards, we obtain the condition for a time-scale separation; under this condition, we show that the marker distribution is approximated by a Fleming-Viot distribution between two trait substitutions. We discuss the implications of the SFVP for our understanding of the dynamics of neutral variation under eco-evolutionary feedbacks and illustrate the main phenomena with simulations. Our results highlight the joint importance of mutations, ecological parameters, and trait values in the restoration of neutral diversity after a selective sweep.

Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium.

We introduce a stochastic individual model for the spatial behavior of an animal population of dispersive and competitive species, considering various kinds of biological effects, such as heterogeneity of environmental conditions, mutual attractive or repulsive interactions between individuals or competition between them for resources. As a consequence of the study of the large population limit, global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model of competing species is proved. The main new feature of the corresponding integro-differential equation is the nonlocal nonlinearity appearing in the diffusion terms, which may depend on the spatial densities of all population types. Moreover, the diffusion matrix is generally not strictly positive definite and the cross-diffusion effect allows for influences growing linearly with the subpopulations' sizes. We prove uniqueness of the finite measure-valued solution and give conditions under which the solution takes values in a functional space. We then make the competition kernels converge to a Dirac measure and obtain the existence of a solution to a locally competitive version of the previous equation. The techniques are essentially based on the underlying stochastic flow related to the dispersive part of the dynamics, and the use of suitable dual distances in the space of finite measures.

Quantifying the mutational meltdown in diploid populations.

Mutational meltdown, in which demographic and genetic processes mutually reinforce one another to accelerate the extinction of small populations, has been poorly quantified despite its potential importance in conservation biology. Here we present a model-based framework to study and quantify the mutational meltdown in a finite diploid population that is evolving continuously in time and subject to resource competition. We model slightly deleterious mutations affecting the population demographic parameters and study how the rate of mutation fixation increases as the genetic load increases, a process that we investigate at two timescales: an ecological scale and a mutational scale. Unlike most previous studies, we treat population size as a random process in continuous time. We show that as deleterious mutations accumulate, the decrease in mean population size accelerates with time relative to a null model with a constant mean fixation time. We quantify this mutational meltdown via the change in the mean fixation time after each new mutation fixation, and we show that the meltdown appears less severe than predicted by earlier theoretical work. We also emphasize that mean population size alone can be a misleading index of the risk of population extinction, which could be better evaluated with additional information on demographic parameters.

A rigorous model study of the adaptive dynamics of Mendelian diploids.

Adaptive dynamics (AD) so far has been put on a rigorous footing only for clonal inheritance. We extend this to sexually reproducing diploids, although admittedly still under the restriction of an unstructured population with Lotka-Volterra-like dynamics and single locus genetics (as in Kimura's in Proc Natl Acad Sci USA 54: 731-736, 1965 infinite allele model). We prove under the usual smoothness assumptions, starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves on the mutational time scale (the 'long' time scale of the literature on the genetical foundations of ESS theory) as a jump process moving between homozygous states (the trait substitution sequence of the adaptive dynamics literature). Essential technical ingredients are a rigorous estimate for the probability of invasion in a dynamic diploid population, a rigorous, geometric singular perturbation theory based, invasion implies substitution theorem, and the use of the Skorohod M 1 topology to arrive at a functional convergence result. In the small mutational steps limit this process in turn gives rise to a differential equation in allele or in phenotype space of a type referred to in the adaptive dynamics literature as 'canonical equation'.

Lévy flights in evolutionary ecology.

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge's model of evolutionary punctuated equilibria. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved.

Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction.

We are interested in the long time behavior of a two-type density-dependent biological population conditioned on non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka-Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a d-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.

Trait Substitution Sequence process and Canonical Equation for age-structured populations.

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin-McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.