Influence of dispersal processes on the global dynamics of Emperor penguin, a species threatened by climate change

© The Author(s), 2017. This is the author's version of the work. It is posted here under a nonexclusive, irrevocable, paid-up, worldwide license granted to WHOI. It is made available for personal use, not for redistribution. The definitive version was published in Biological Conservation 212 (2017): 63-73, doi:10.1016/j.biocon.2017.05.017.Species endangered by rapid climate change may persist by tracking their optimal habitat; this depends on their dispersal characteristics. The Emperor Penguin (EP) is an Antarctic seabird threatened by future sea ice change, currently under consideration for listing under the US Endangered Species Act. Indeed, a climate-dependent-demographic model without dispersion projects that many EP colonies will decline by more than 50% from their current size by 2100, resulting in a dramatic global population decline. Here we assess whether or not dispersion could act as an ecological rescue, i.e. reverse the anticipated global population decline projected by a model without dispersion. To do so, we integrate de22 tailed dispersal processes in a metapopulation model|specifically, dispersal stages, dispersal distance, habitat structure, informed dispersal behaviors, and density-dependent dispersion rates. For EP, relative to a scenario without dispersion, dispersal can either offset or accelerate climate driven population declines; dispersal may increase the global population by up to 31% or decrease it by 65%, depending on the rate of emigration and distance individuals disperse. By developing simpler theoretical models, we demonstrate that the global population dynamic depends on the global landscape quality. In addition, the interaction among dispersal processes - dispersion rates, dispersal distance, and dispersal decisions - that influence landscape occupancy, impacts the global population dynamics. Our analyses bound the impact of between-colony emigration on global population size, and provides intuition as to the direction of population change depending on the EP dispersal characteristics. Our general model is flexible such that multiple dispersal scenarios could be implemented for a wide range of species to improve our understanding and predictions of species persistence under future global change.S. Jenouvrier acknowledges support from WHOI Unrestricted funds and Mission Blue / Biotherm; J. Garnier and L. Desvillettes acknowledge respectively the NONLOCAL project (ANR-14-CE25-0013) and the Kibord project (ANR-13-BS01-0004) from the French National Research Agency

A multi-scale epidemic model of Salmonella infection with heterogeneous shedding∗

Salmonella strains colonize the digestive tract of farm livestock, such as chickens or pigs, without affecting them, and potentially infect food products, representing a threat for human health ranging from food poisoning to typhoid fever. It has been shown that the ability to excrete the pathogen in the environment and contaminate other animals is variable. This heterogeneity in pathogen carriage and shedding results from interactions between the host’s immune response, the pathogen and the commensal intestinal microbiota. In this paper we propose a novel generic multiscale modeling framework of heterogeneous pathogen transmission in an animal population. At the intra-host level, the model describes the interaction between the commensal microbiota, the pathogen and the inflammatory response. Random fluctuations in the ecological dynamics of the individual microbiota and transmission at between-host scale are added to obtain a drift-diffusion PDE model of the pathogen distribution at the population level. The model is further extended to represent transmission between several populations. The asymptotic behavior as well as the impact of control strategies including cleaning and antimicrobial administration are investigated through numerical simulation

Asymptotic Analysis of Integro-differential Equations : populations dynamics and evolutionary models

Cette thèse est consacrée à l’étude de phénomènes de propagation et de concentration dans des modèles d’équations intégro-différentielles venant de la écologie. On étudie certaines équations de réaction-diffusion non locales apparaissant en dynamique de populations, ainsi que des modèles représentant l’évolution Darwinienne avec un mode de reproduction sexué.Dans une première partie, nous étudions la propagation spatiale pour une équation de réaction-diffusion ou la dispersion opère via un noyau de convolution à queue lourde. Nous mesurons de manière précise l’accélération du front de propagation de la solution. Nous proposons également une échelle adaptée pour mesurer les «petites» mutations. Dans les deux cas nous utilisons le formalisme des équations de Hamilton-Jacobi.Dans un second temps nous étudions un modèle de génétique quantitative, avec un mode de reproduction sexuée. Un petit paramètre mesure la déviation entre le trait des descendants est la moyenne des traits des parents. Dans le régime où ce paramètre est petit nous étudions l’existence de solutions stationnaires, puis le problème de Cauchy lié à ce modèle. Les solutions se concentrent autour des optima de sélection, sous la forme de perturbations de distributions Gaussiennes avec petite variance fixée par le paramètre. Notre analyse généralise le cas linéaire de la reproduction asexuée en utilisant des outils d’analyse perturbative. Enfin dans une dernière partie nous fournissons des simulations numériques et des méthodes mathématiques pour étudier la dynamique interne des équilibres dans le régime de petite variance, pour les deux modes de reproduction : asexué et sexué.This manuscript tackles propagation and concentration phenomena in different integro-differential equations with a background in ecology. We study non local reaction-diffusion equations from population dynamics, and models for Darwinian evolution with a sexual or asexual mode of reproduction, with a preference for the former.In a first part, we study spatial propagation for a reaction diffusion equation where dispersion acts through a fat tailed kernel. We measure accurately the acceleration of the propagation front of the population. We propose as well a scaling well adapted to “small mutations” when we consider the model in the context of adaptative dynamics. This scaling is very natural following the previous spatial investigation. In both cases we look at the long time behavior and we use the Hamilton-Jacobi framework. Then we turn our attention towards a quantitative genetics model, with a sexual mode of reproduction, imposed by the “infinitesimal operator”. In this non-linear setting, a small parameter tunes the deviation between the phenotypic trait of the offspring and the mean of the traits of the parents. In the regime where this parameter is small, we prove existence of stationary solutions, and their local uniqueness. We also provide an example of non-uniqueness in the case where the selection function admits several extrema. We prove that the solution concentrates around the points of minimum of the selection function. The analysis is carried by the small perturbations of special profiles : Gaussian distributions with small variance fixed by the parameter.We then study the stability of the Cauchy problem associated to the previous model. This time we prove that at all times, for a well prepared initial data, the solutions is arbitrary close to a Gaussian distribution with small variance. The proof follows the framework of the previous : we use perturbative analysis tools, but this time an even more precise description of the correctors is needed and we linearize the equation to obtain it. In a final part we show numerical simulations and different mathematical approaches to study inside dynamics of phenotypic lineages in the regime of small variance, with a moving environement

THE CAUCHY PROBLEM FOR THE INFINITESIMAL MODEL IN THE REGIME OF SMALL VARIANCE

We study the asymptotic behavior of solutions of the Cauchy problem associated to a quantitative genetics model with a sexual mode of reproduction. It combines trait-dependent mortality and a nonlinear integral reproduction operator "the infinitesimal model" with a parameter describing the standard deviation between the offspring and the mean parental traits. We show that under mild assumptions upon the mortality rate m, when the deviations are small, the solutions stay close to a Gaussian profile with small variance, uniformly in time. Moreover we characterize accurately the dynamics of the mean trait in the population. Our study extends previous results on the existence and uniqueness of stationary solutions for the model. It relies on perturbative analysis techniques together with a sharp description of the correction measuring the departure from the Gaussian profile

Analyse asymptotique d'équations intégro-différentielles : modèles d'évolution et de dynamique des populations

This manuscript tackles propagation and concentration phenomena in different integro-differential equations with a background in ecology. We study non local reaction-diffusion equations from population dynamics, and models for Darwinian evolution with a sexual or asexual mode of reproduction, with a preference for the former.In a first part, we study spatial propagation for a reaction diffusion equation where dispersion acts through a fat tailed kernel. We measure accurately the acceleration of the propagation front of the population. We propose as well a scaling well adapted to “small mutations” when we consider the model in the context of adaptative dynamics. This scaling is very natural following the previous spatial investigation. In both cases we look at the long time behavior and we use the Hamilton-Jacobi framework. Then we turn our attention towards a quantitative genetics model, with a sexual mode of reproduction, imposed by the “infinitesimal operator”. In this non-linear setting, a small parameter tunes the deviation between the phenotypic trait of the offspring and the mean of the traits of the parents. In the regime where this parameter is small, we prove existence of stationary solutions, and their local uniqueness. We also provide an example of non-uniqueness in the case where the selection function admits several extrema. We prove that the solution concentrates around the points of minimum of the selection function. The analysis is carried by the small perturbations of special profiles : Gaussian distributions with small variance fixed by the parameter.We then study the stability of the Cauchy problem associated to the previous model. This time we prove that at all times, for a well prepared initial data, the solutions is arbitrary close to a Gaussian distribution with small variance. The proof follows the framework of the previous : we use perturbative analysis tools, but this time an even more precise description of the correctors is needed and we linearize the equation to obtain it. In a final part we show numerical simulations and different mathematical approaches to study inside dynamics of phenotypic lineages in the regime of small variance, with a moving environement.Cette thèse est consacrée à l’étude de phénomènes de propagation et de concentration dans des modèles d’équations intégro-différentielles venant de la écologie. On étudie certaines équations de réaction-diffusion non locales apparaissant en dynamique de populations, ainsi que des modèles représentant l’évolution Darwinienne avec un mode de reproduction sexué.Dans une première partie, nous étudions la propagation spatiale pour une équation de réaction-diffusion ou la dispersion opère via un noyau de convolution à queue lourde. Nous mesurons de manière précise l’accélération du front de propagation de la solution. Nous proposons également une échelle adaptée pour mesurer les «petites» mutations. Dans les deux cas nous utilisons le formalisme des équations de Hamilton-Jacobi.Dans un second temps nous étudions un modèle de génétique quantitative, avec un mode de reproduction sexuée. Un petit paramètre mesure la déviation entre le trait des descendants est la moyenne des traits des parents. Dans le régime où ce paramètre est petit nous étudions l’existence de solutions stationnaires, puis le problème de Cauchy lié à ce modèle. Les solutions se concentrent autour des optima de sélection, sous la forme de perturbations de distributions Gaussiennes avec petite variance fixée par le paramètre. Notre analyse généralise le cas linéaire de la reproduction asexuée en utilisant des outils d’analyse perturbative. Enfin dans une dernière partie nous fournissons des simulations numériques et des méthodes mathématiques pour étudier la dynamique interne des équilibres dans le régime de petite variance, pour les deux modes de reproduction : asexué et sexué

Ancestral lineages in mutation-selection equilibria with moving optimum

We investigate the evolutionary dynamics of a population structured in phenotype, subjected to trait dependent selection with a linearly moving optimum and an asexual mode of reproduction. Our model consists of a non-local and non-linear parabolic PDE. Our main goal is to measure the history of traits when the population stays around an equilibrium. We define an ancestral process based on the idea of neutral fractions. It allows us to derive quantitative information upon the evolution of diversity in the population along time. First, we study the long-time asymptotics of the ancestral process. We show that the very few fittest individuals drive adaptation. We then tackle the adaptive dynamics regime, where the effect of mutations is asymptotically small. In this limit, we provide an interpretation for the minimizer of some related optimization problem, an Hamilton Jacobi equation, as the typical ancestral lineage. We check the theoretical results against individual based simulations

ASYMPTOTIC ANALYSIS OF A QUANTITATIVE GENETICS MODEL WITHNONLINEAR INTEGRAL OPERATOR

International audienceWe study the asymptotic behavior of stationary solutions to a quantitative genetics model with trait-dependent mortality and sexual reproduction. The infinitesimal model accounts for the mixing of parental phenotypes at birth.Our asymptotic analysis encompasses the case when deviations between the offspring and the mean parental trait are typically small. Under suitable regularity and growth conditions on the mortality rate, we prove existence and local uniqueness of a stationary profile that get concentrated around a local optimum of mortality, with a Gaussian shape having small variance. Our approach is based on perturbative analysis techniques that require to describe accurately the correction to the Gaussian leading order profile. Our result extends previous results obtained with an asexual mode of reproduction, but using an alternative methodology

Influence of dispersal processes on the global dynamics of Emperor penguin, a species threatened by climate change

International audienceSpecies endangered by rapid climate change may persist by tracking their optimal habitat; this depends on theirdispersal characteristics. The Emperor penguin (EP) is an Antarctic seabird threatened by future sea ice change,currently under consideration for listing under the US Endangered Species Act. Indeed, a climate-dependentdemographicmodel without dispersion projects that many EP colonies will decline by more than 50% from theircurrent size by 2100, resulting in a dramatic global population decline. Here we assess whether or not dispersioncould act as an ecological rescue, i.e. reverse the anticipated global population decline projected by a modelwithout dispersion. To do so, we integrate detailed dispersal processes in a metapopulation model—specifically,dispersal stages, dispersal distance, habitat structure, informed dispersal behaviors, and density-dependentdispersion rates. For EP, relative to a scenario without dispersion, dispersal can either offset or accelerate climatedriven population declines; dispersal may increase the global population by up to 31% or decrease it by 65%,depending on the rate of emigration and distance individuals disperse. By developing simpler theoretical models,we demonstrate that the global population dynamic depends on the global landscape quality. In addition, theinteraction among dispersal processes - dispersion rates, dispersal distance, and dispersal decisions - that influencelandscape occupancy, impacts the global population dynamics. Our analyses bound the impact of between-colony emigration on global population size, and provide intuition as to the direction of populationchange depending on the EP dispersal characteristics. Our general model is flexible such that multiple dispersalscenarios could be implemented for a wide range of species to improve our understanding and predictions ofspecies persistence under future global change

Adaptation in a heterogeneous environment II: To be three or not to be

We propose a model to describe the adaptation of a phenotypically structured population in a H-patch environment connected by migration, with each patch associated with a different phenotypic optimum, and we perform a rigorous mathematical analysis of this model. We show that the large-time behaviour of the solution (persistence or extinction) depends on the sign of a principal eigenvalue, λ_H , and we study the dependency of λ_H with respect to H. This analysis sheds new light on the effect of increasing the number of patches on the persistence of a population, which has implications in agroecology and for understanding zoonoses; in such cases we consider a pathogenic population and the patches correspond to different host species. The occurrence of a springboard effect, where the addition of a patch contributes to persistence, or on the contrary the emergence of a detrimental effect by increasing the number of patches on the persistence, depends in a rather complex way on the respective positions in the phenotypic space of the optimal phenotypes associated with each patch. From a mathematical point of view, an important part of the difficulty in dealing with H ≥ 3, compared to H = 1 or H = 2, comes from the lack of symmetry. Our results, which are based on a fixed point theorem, comparison principles, integral estimates, variational arguments, rearrangement techniques, and numerical simulations, provide a better understanding of these dependencies. In particular, we propose a precise characterisation of the situations where the addition of a third patch increases or decreases the chances of persistence, compared to a situation with only two patches

Adaptation in General Temporally Changing Environments

International audienceWe analyze a nonlocal partial differential equation (PDE) model describing the dynamics of adaptation of a phenotypically structured population, under the effects of mutation and selection, in a changing environment. Previous studies have analyzed the large-time behavior of such models, with particular forms of environmental changes---either linearly changing or periodically fluctuating. We use here a completely different mathematical approach, which allows us to consider very general forms of environmental variations and to give an analytic description of the full trajectories of adaptation, including the transient phase, before a stationary behavior is reached. The main idea behind our approach is to study a bivariate distribution of two “fitness components" that contains enough information to describe the distribution of fitness at any time. This distribution solves a degenerate parabolic equation that is dealt with by defining a multidimensional cumulant generating function associated with the distribution and solving the associated transport equation. We apply our results to several examples and check their accuracy using stochastic individual-based simulations as a benchmark. These examples illustrate the importance of being able to describe the transient dynamics of adaptation to understand the development of drug resistance in pathogens