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We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L $\infty$ norm of these solutions.

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

The emergence and the spread of multi-adapted pathogens is often viewed as a slow process resulting from the incremental accumulation of single adaptations. In bacteria, for instance, multidrug resistance to antibiotics may result from the sequential acquisition of single drug resistance to different antibiotics. In phytopathogens, the ability to infect different resistant varieties of crops may also result from the accumulation of distinct virulence genes. Here we use a general epidemiological model to analyse the evolution of pathogen adaptations throughout an epidemic spreading in a heterogeneous host population where selection varies periodically in space. This spatially heterogeneous selection may result from the use of different drugs, different vaccines or different crop varieties in agriculture. We study both the transient evolution of pathogen adaptation at the front of the epidemic and the long-term evolution far behind the epidemic front. We identify five different types of epidemic profiles that may arise from different combinations of spatial heterogeneity and the cost of multi-adaptation. In particular, we show that multi-adaptation can drive epidemic spread, while the evolution of single-adaptation may only occur in a second phase, when the pathogen specializes on local selective pressures. Indeed, a generalist pathogen with multiple adaptations can outpace the spread of a coalition of specialist pathogens when selection varies frequently in space. This result is amplified in finite host populations because demographic stochasticty can lead to the extinction of maladapted pathogens specialised to a local selective pressure. Our work has important implications for the management of multiple drugs and vaccines against pathogens but also for the optimal deployment of resistant varieties in agriculture.

We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between "no acceleration" and "acceleration". In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets., arXiv admin note: text overlap with arXiv:1505.04626

We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.

We consider nonlinear mutation selection models, known as replicator-mutator equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: we propose a new milder sufficient condition for the existence of a principal eigenfunction, and we provide what is to our knowledge the first quantification of the spectral gap. We also recover existing results in the non-symmetric case, through a new approach.

We analyze a replicator-mutator model arising in the context of directed evolution [23], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [13] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena., 19 pages, 7 figures

We explore the interaction between two genetic incompatibilities (underdominant loci in diploid organisms) in a population occupying a one-dimensional space. We derive a system of partial differential equations describing the dynamics of allele frequencies and linkage disequilibrium between the two loci, and use a quasi-linkage equilibrium approximation in order to reduce the number of variables. We investigate the solutions of this system and demonstrate the existence of a solution in which the two clines in allele frequency remain stacked together. In the case of asymmetric incompatibilities (i.e. when one homozygote is favored over the other at each locus), these stacked clines propagate in the form of a traveling wave. We obtain an approximation for the speed of this wave which, in particular, is decreased by recombination between the two loci but is always larger than the speed of "one cline alone".

International audience; We consider a nonlocal parabolic equation describing the dynamics of a population structured by a spatial position and a phenotypic trait, submitted to dispersion , mutations and growth. The growth term may be of the Fisher-KPP type but may also be subject to an Allee effect which can be weak (non-KPP monostable nonlinearity, possibly degenerate) or strong (bistable nonlinearity). The type of growth depends on the value of a variable θ : the Allee threshold, which is considered here as an evolutionary trait. After proving the well-posedness of the Cauchy problem, we study the long time behavior of the solutions. Due to the richness of the model and the interplay between the various phenomena and the nonlocality of the growth term, the outcomes (extinction vs. persistence) are various and in sharp contrast with earlier results of the existing literature on local reaction-diffusion equations.

We consider a bistable ($0\textless{}\theta\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $\theta$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large.

International audience; We consider the homogeneous integro-differential equation$\partial _t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist when $f$ belongs to one of the three main class of non-linearities (bistable, ignition or monostable), and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On the other hand, when the dispersion kernel $J$ has heavy tails and the non-linearity $f$ is non degenerate, i.e $f'(0)>0$, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a general monostable non-linearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of $f$, i.e. $f'(0)=0$. First, for algebraic decaying kernels, we prove the exact separation between existence and non existence of travelling waves. This in turn provides the exact separation between non acceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.

International audience; We first study the so-called Heat equation with two families of elliptic operators whichare fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equationwith operators including the "large" eigenvalues has strong similarities with a Heatequation in lower dimension whereas, surprisingly, for operators including "small"eigenvalues it shares some properties with some transport equations. In particular, forthese operators, the Heat equation (which is nonlinear) not only does not have theproperty that "disturbances propagate with infinite speed" but may lead to quenchingin finite time. Last, based on our analysis of the Heat equations (for which we providea large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.

International audience; We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic equations), 35B40 (Asymptotic behavior of solutions).

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data., Comment: 14 pages

International audience; We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

We show that the solution to a nonlocal reaction–diffusion equation, present in evolutionary genetics, can be related explicitly to the solution of the heat equation with the same initial data. As a consequence, we show different possible scenario for the solution: it can be either well-defined for all time, or become extinct in finite time, or even be defined for no positive time. In the former case, we give the leading-order asymptotic behavior of the solution for large time, which is universal.

International audience; Several mathematical models are proposed to understand spatial patchy vegetation patterns arising in drylands. In this paper, we consider the system with nonlocal dispersal of plants (through a redistribution kernel for seeds) proposed by Pueyo et al. (Oikos 117:1522-1532, 2008) as a model for vegetation in water-limited ecosystems. It consists in two reaction diffusion equations for surface water and soil water, combined with an integro-differential equation for plants. For this system, under suitable assumptions, we prove well-posedness using the Schauder fixed point theorem. In addition, we consider the stationary problem from the viewpoint of vegetated pattern formation, and show a transition of vegetation patterns when parameter values (rainfall, seed dispersal range, seed germination rate) in the system vary. The influence of the shape of the redistribution kernel is also discussed.

We consider a class of non-local reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schr\"odinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur., Comment: 24 pages, 7 figures

International audience; We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between " no acceleration and acceleration ". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.

We consider the nonlocal diffusion equation ∂ t u = J ⁎ u − u + u 1 + p in the whole of R N . We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel J near the origin, which is linked to the tails of J. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation ∂ t u = Δ u + u 1 + p . On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of J. As an application of the result in population dynamics models, we discuss the hair trigger effect for ∂ t u = J ⁎ u − u + u 1 + p ( 1 − u ) .

This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is non-positive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is non-negative (inverted harmonic potential), solutions always become extinct in finite time., Comment: 12 pages

We consider a periodic reaction diffusion system which, because of competition between u and v , does not enjoy the comparison principle. It also takes into account mutations, allowing u to switch to v and vice versa. Such a system serves as a model in evolutionary epidemiology where two types of pathogens compete in a heterogeneous environment while mutations can occur, thus allowing coexistence. We first discuss the existence of nontrivial positive steady states, using some bifurcation technics. Then, to sustain the possibility of invasion when nontrivial steady states exist, we construct pulsating fronts. As far as we know, this is the first such construction in a KPP situation where comparison arguments are not available.

International audience; Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.

We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $||x|| \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. Moreover, we obtain a new estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.

We investigate the singular limit, as \({\varepsilon \to 0}\), of the Allen-Cahn equation \({u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}\), with f a balanced bistable nonlinearity. We consider rather general initial data u0 that is independent of \({{\varepsilon}}\). It is known that this equation converges to the generalized motion by mean curvature — in the sense of viscosity solutions—defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions \({u^{\varepsilon}}\) are sandwiched between two sharp “interfaces” moving by mean curvature, provided that these “interfaces” sandwich at t = 0 an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.

International audience; We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial _t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with {\it slow exponential decay}. We prove that the sharp interface limit moves by a constant speed, which dramatically depends on the tails of the initial data. By performing a fine analysis of both the generation and motion of interface, we provide a new estimate of the thickness of the transition layers.

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} %[4]of the well-known spreading properties \cite{Wein02}, %[32], \cite{Ber-Ham-02}, %[9],and the solution of a Hamilton-Jacobi equation.

International audience; We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of thepropagation phenomena on the direction. We prove that the(minimal) speed of the underlying pulsating fronts dependscontinuously on the direction of propagation, and so does itsassociated profile provided it is unique up to time shifts. Wealso prove that the spreading properties \cite{Wein02} areactually uniform with respect to thedirection.

International audience; We consider the mass conserving Allen-Cahn equation proposed in \cite{Bra-Bre}: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with \cite{Che-Hil-Log}). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solutions. Then, equipped with these approximate solutions, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that classical solutions of the latter exist. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.

International audience; We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is \lq\lq above and away\rq\rq\, from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.

International audience; We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c < c^*$

International audience; We consider a model of cellular detonations in gases. They consist in conservation laws with a non-local pseudo-differential operator whose symbol is asymptotically $|\xi|^\lam$, where $0

International audience; In this note, we give a positive answer to a question addressed in \cite{Nad-Per-Tan}. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are \lq\lq rapid\rq\rq) of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with {\it fat tails}.

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.

International audience; In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property.

We consider an Allen–Cahn type equation of the form u t = Δ u + e −2 f e ( x , t , u ) , where e is a small parameter and f e ( x , t , u ) = f ( u ) − e g e ( x , t , u ) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u 0 that is independent of e , we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order e 2 | ln e | , and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e . This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where g e ≡ 0 . Next we consider systems of reaction–diffusion equations of the form { u t = Δ u + e −2 f e ( u , v ) , v t = D Δ v + h ( u , v ) , which include the FitzHugh–Nagumo system as a special case. Given a rather general initial data ( u 0 , v 0 ) , we show that the component u develops a steep transition layer and that all the above-mentioned results remain true for the u -component of these systems.

International audience; We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.

Previous theory on adaptation to a changing environment has identified genetic variance as a key factor. Contemporary climate change renews the interest for such studies, which now attempt to include the complexity of life and climate change. Rapid evolution of flowering time has been documented in several species as a response to climate change. Assortative mating, i.e. mating restricted to individuals with similar phenotypes, is frequent and obligate for flowering time, the intensity of natural selection can differ between sexes, and the intensity of stabilizing selection on flowering can vary with the duration of seasons, and which fluctuates across years. These features are however largely ignored by extent theory on adaptation to changing environments. In a first chapter, we have studied the effects of assortative mating for flowering date on evolutionary responses to climate change, with the aim to evaluate whether assortative mating, compared to random mating, can explain fast evolutionary responses of flowering phenology to climate change. To this end, an individual-based quantitative genetics model simulates climate change and the evolution of mean individual flowering date in an isolated population. In most scenarios, and despite its negative effect on genetic polymorphism, assortative mating maintains higher genetic variance at equilibrium than random mating, and therefore allows populations to better track climate change and to have a better fitness. An analytic model, based on the infinitesimal model of trait heritability, confirms those results. The second chapter integrates more elements of realism, common in plant and animal populations: sex-specific natural selection and sexual dimorphism. The analytical model is an extension of the previous one, and generalizes results by including sexual dimorphism, and two frequently observed types of assortative mating in animals and plants: assortative preference of females for male phenotypes, and temporal assortative mating for flowering date. The model shows that (i) assortative mating generates sexual selection, which increases the effect of natural selection on females and decreases the effect of natural selection on males, (ii) when sexual dimorphism is large, assortative mating further generates directional sexual selection on male phenotypes, which can lead to the evolution of trait values overshooting the interval between the male and female optima; (iii) in some conditions, which occur both under random and assortative mating, female maladaptation can be smaller in a changing environment than in a constant environment; (iv) assortative mating can help populations to better track climate change than random mating, when selection on females is stronger than that on males and/or sexual dimorphism is not too large, and/or climate change is fast enough. The robustness of results has been tested with an individual-based model. The third chapter studies the effects of the fluctuations of the strength of selection on the long-term responses of populations. To this end, we have used both analytical approximations and a numerical exploration of the infinitesimal model. Fluctuations in the strength of selection are modeled by fluctuations of the width of the fitness function assumed to be Gaussian. Such fluctuations increase the mean strength of selection and therefore decrease genetic variance and adaptive lag. Fluctuations of the strength of selection however have a demographic cost, and decrease the long-run growth rate of populations in most cases. Taken together, these results suggest that: (i) assortative mating improves adaptation to climate change only under specific circumstances, (ii) rapid evolutionary responses to climate change do not necessarily mitigate its negative consequences on demography.; Les précédentes études théoriques de l’adaptation de populations à un changement environnemental ont identifié la variance génétique comme un élément clef. Le changement climatique contemporain a ravivé l’intérêt de ces études qui tentent maintenant d’intégrer la complexité du vivant et du changement climatique. Des évolutions rapides des dates de floraison ont été documentées chez plusieurs espèces en lien avec le changement climatique. L’appariement entre des individus qui se ressemblent – homogamie – est fréquent, et obligatoire pour la date de floraison, la sélection naturelle peut être différente sur les fonctions mâles et femelles et l’intensité de la sélection entre années diffère selon la durée des saisons. Ces caractéristiques sont généralement ignorées par la théorie sur l’adaptation à un environnement changeant. Dans un premier chapitre, nous avons évalué si l’homogamie peut être responsable des réponses évolutives rapides des phénologies de floraison au changement climatique. Un modèle individu-centré de génétique quantitative simule un changement climatique et l’évolution des dates moyennes de floraison individuelle dans une population isolée. Dans la plupart des scénarios, malgré ses effets négatifs sur le polymorphisme génétique, l’homogamie maintient plus de variance génétique dans un environnement changeant que la panmixie, et permet ainsi aux populations de mieux suivre le changement climatique et d’avoir une valeur sélective plus grande. Un modèle analytique, basé sur le modèle infinitésimal d’héritabilité des traits, confirme ces résultats. Le deuxième chapitre intègre deux éléments supplémentaires, fréquents dans les populations de plantes et d’animaux : la sélection naturelle sexe-spécifique et le dimorphisme sexuel. Le modèle analytique construit est une extension du précédent, et généralise les résultats en incluant le dimorphisme sexuel, et deux modes courants d’homogamie: la préférence homogame des femelles pour certains phénotypes mâles, et, l’homogamie temporelle pour la date de floraison.. Le modèle montre que (i) l’homogamie produit de la sélection sexuelle qui intensifie l’effet de la sélection naturelle sur les femelles et diminue celui de la sélection naturelle sur les mâles; (ii) en présence d’un fort dimorphisme sexuel, cette sélection sexuelle engendre une sélection directionnelle sur les mâles, et peut conduire à une évolution de valeurs de traits en dehors de l’intervalle défini par les optimums mâles et femelles; (iii) dans certaines conditions, la mal-adaptation des femelles peut être plus petite dans un environnement changeant que constant, en homogamie comme en panmixie ; (iv) l’homogamie facilite l’adaptation des femelles à un climat changeant, seulement si la sélection sur les femelles est plus forte que celle sur les mâles, et/ou le dimorphisme sexuel n’est pas trop fort et/ou le changement climatique est rapide. La robustesse de ces résultats a été testée à l’aide d’un modèle individu-centré. Le troisième chapitre étudie les effets des fluctuations de l’intensité de la sélection sur les réponses génétiques à long terme des populations. Nous avons utilisé des approximations analytiques et une exploration numérique du modèle infinitésimal. Les fluctuations de l’intensité de la sélection sont modélisées par des fluctuations de la largeur de la fonction de sélection supposée Gaussienne. Ces fluctuations augmentent l’intensité moyenne de la sélection, et diminuent la variance génétique et le retard adaptatif des populations. Les fluctuations ont un coût démographique et diminuent le taux de croissance à long-terme des populations dans la plupart des scénarios. L’ensemble de ces résultats suggère que (i) l’homogamie ne facilite les réponses évolutives au changement climatique que dans certaines conditions seulement, (ii) des réponses évolutives rapides ne sont pas nécessairement un gage d’atténuation des conséquences démographiques du changement climatique.

In this thesis we consider several nonlocal partial differential equations and integro-differential equations, arising from evolutionary biology models. We aim at performing a rigorous mathematical analysis of extinction, survival and invasion phenomena of these models that lead to relevant biological insights. Firstly, we look at a population facing a linear environmental gradient, that is the optimal trait is linearly dependent of the spatial position (say the temperature along a north-south axis). We show that, under certain conditions on the initial data, the solution spreads in space by accelerating. We also give precise estimates of the asymptotic position of the level sets of the solution. Secondly, we consider a model with a nonlinear environmental gradient. Using perturbative techniques, we construct steady states and, when the gradient is periodic, pulsating fronts. Our analysis reveals how the distribution of the population at equilibrium and its invasion dynamics are affected by the nonlinear gradient. Finally, we introduce new models with nonlocal, heterogeneous, anisotropic diffusion. We investigate their connections with local diffusion models found in the literature, as well as their steady states. Our approach involves the notion of “deciding factors”, and sheds light on the “preferential position of individuals”.; Dans cette thèse nous considérons des équations aux dérivées partielles non-locales, et des équations intégro-différentielles, qui servent de modèles pour la biologie évolutive. L'objectif est de faire une analyse mathématique rigoureuse des phénomènes d'extinction, de survie et d'invasion dans ces modèles puis d'en retirer une interprétation biologique pertinente. Dans un premier temps, nous envisageons une population affrontant un gradient environnemental linéaire, i.e. le trait optimal dépend linéairement de la position en espace (par exemple la température selon l'axe Nord-Sud). On montre que, sous certaines conditions sur la donnée initiale, la solution se propage en espace en accélérant. Nous donnons également une estimation fine de la position asymptotique des ensembles de niveau de la solution. Dans un deuxième temps, nous considérons un modèle avec un gradient environnemental non-linéaire. Par des techniques de perturbation, nous construisons des états stationnaires et, lorsque le gradient est périodique, des fronts pulsatoires. Notre analyse révèle ainsi comment la répartition à l'équilibre de la population et sa dynamique d'invasion sont affectées par le gradient non-linéaire. Enfin, nous introduisons de nouveaux modèles de diffusion non-locale, hétérogène et anisotrope. Nous en étudions les liens avec des modèles de diffusion locale présents dans la littérature ainsi que les états stationnaires. Notre approche fait intervenir la notion de “points décideurs”, et apporte un éclairage nouveau sur la “position préférentielle des individus”.

In this thesis we study the existence of a travelling wave solutions for an integro-differential system of equations from evolutionary epidemiology. We use ideas from dynamical system ideas theory coupled with estimates of the asymptotic behaviour of profiles. We prove that the wave solutions have a rather simple structure. This analysis allows us to reduce the infinite dimensional travelling wave profile system of equations to a four dimensional ODE system. The latter is used to prove the existence of travelling wave solutions for any wave speed larger than a minimal wave speed c?, provided that the epidemic threshold R0, which is expressed as a function of the principal eigenvalue of a certain integral operator, is strictly greater than 1. This same threshold condition is also used to prove that any travelling wave connects two determined stationary states. In the second part, we study the propagation properties of the solutions for the same spatially distributed system of equations, when the initial density of infected plants is a compactly supported function with the space variable x. When R0 > 1, we prove that spreading occurs with a definite spreading speed that coincides with the minimal speed c? of the travelling wave solutions discussed in the first part. Moreover, the solution of the Cauchy problem asymptotically converges to some specific function for which the moving frame variable x and the phenotype one y are separated.; Dans cette thèse nous étudions l’existence d’une onde progressive pour un système d’équations intégro-différentiels provenant de l’épidémiologie évolutive. Nous utilisons des idées issues de la théorie des systèmes dynamiques couplées à des estimations sur le comportement asymptotique des profils. Nous prouvons que les ondes progressives ont une structure assez simple découplant les variables de propagation spatio-temporelle des variables de trait phénotypique. Cette analyse nous permet de réduire le système d’équations des profils d’ondes progressives à dimension infinie à un système d’EDO à quatre dimensions. Nous prouvons l’existence d’ondes progressives pour toute vitesse d’onde supérieure à une vitesse minimale c?, pourvu que le seuil épidémique R0, qui s’exprime en fonction de la valeur propre principale d’un certain opérateur intégral, soit strictement supérieur à 1. Cette même condition de seuil est également utilisée pour démontrer que toute onde progressive relie deux états stationnaires déterminés. Dans une deuxième partie, nous étudions les propriétés de propagation des solutions pour le même système d’équations spatialement distribué, avec une densité initiale de plantes infectées à support compact spatialement en x. Lorsque R0 > 1, nous prouvons que la propagation se produit avec une vitesse de propagation qui coïncide avec la vitesse minimale c? des ondes progressives étudiées dans la première partie. De plus, la solution du problème de Cauchy converge asymptotiquement vers une fonction spécifique pour laquelle la variable x du repère mobile et celle du phénotype y sont séparées.