Lionel Roques

Various models describe asexual evolution by mutation, selection and drift. Some focus directly on fitness, typically modelling drift but ignoring or simplifying both epistasis and the distribution of mutation effects (travelling wave models). Others follow the dynamics of quantitative traits determining fitness (Fisher’s geometrical model), imposing a complex but fixed form of mutation effects and epistasis, and often ignoring drift. In all cases, predictions are typically obtained in high or low mutation rate limits and for long-term stationary regimes, thus loosing information on transient behaviors and the effect of initial conditions. Here, we connect fitness-based and trait-based models into a single framework, and seek explicit solutions even away from stationarity. The expected fitness distribution is followed over time via its cumulant generating function, using a deterministic approximation that neglects drift. In several cases, explicit trajectories for the full fitness distribution are obtained, for arbitrary mutation rates and standing variance. For non-epistatic mutation, especially with beneficial mutations, this approximation fails over the long term but captures the early dynamics, thus complementing stationary stochastic predictions. The approximation also handles several diminishing return epistasis models (e.g. with an optimal genotype): it can then apply at and away from equilibrium. General results arise at equilibrium, where fitness distributions display a ‘phase transition’ with mutation rate. Beyond this phase transition, in Fisher’s geometrical model, the full trajectory of fitness and trait distributions takes simple form, robust to details of the mutant phenotype distribution. Analytical arguments are explored for why and when the deterministic approximation applies. Significance statement: How fast do asexuals evolve in new environments? Asexual fitness dynamics are well documented empirically. Various corresponding theories exist, to which they may be compared, but most typically describe stationary regimes, thus losing information on the shorter timescale of experiments, and on the impact of the initial conditions set by the experimenter. Here, a general deterministic approximation is proposed that encompasses many previous models as subcases, and shows surprising accuracy when compared to stochastic simulations. It can yield predictions over both short and long timescales, hopefully fostering the quantitative test of alternative models, using data from experimental evolution in asexuals.

We propose and develop a general approach based on reaction-diffusion equations for modelling a species dynamics in a realistic two-dimensional (2D) landscape crossed by linear one-dimensional (1D) corridors, such as roads, hedgerows or rivers. Our approach is based on a hybrid “2D/1D model”, i.e, a system of 2D and 1D reaction-diffusion equations with homogeneous coefficients, in which each equation describes the population dynamics in a given 2D or 1D element of the landscape. Using the example of the range expansion of the tiger mosquito Aedes albopictus in France and its main highways as 1D corridors, we show that the model can be fitted to realistic observation data. We develop a mechanistic-statistical approach, based on the coupling between a model of population dynamics and a probabilistic model of the observation process. This allows us to bridge the gap between the data (3 levels of infestation, at the scale of a French department) and the output of the model (population densities at each point of the landscape), and to estimate the model parameter values using a maximum-likelihood approach. Using classical model comparison criteria, we obtain a better fit and a better predictive power with the 2D/1D model than with a standard homogeneous reaction-diffusion model. This shows the potential importance of taking into account the effect of the corridors (highways in the present case) on species dynamics. With regard to the particular case of A. albopictus, the conclusion that highways played an important role in species range expansion in mainland France is consistent with recent findings from the literature.

Having a precise knowledge of the dispersal ability of a population in a heterogeneous environment is of critical importance in agroecology and conservation biology as it can provide management tools to limit the effects of pests or to increase the survival of endangered species. In this paper, we propose a mechanistic-statistical method to estimate space-dependent diffusion parameters of spatially-explicit models based on stochastic differential equations, using genetic data. Dividing the total population into subpopulations corresponding to different habitat patches with known allele frequencies, the expected proportions of individuals from each subpopulation at each position is computed by solving a system of reaction–diffusion equations. Modelling the capture and genotyping of the individuals with a statistical approach, we derive a numerically tractable formula for the likelihood function associated with the diffusion parameters. In a simulated environment made of three types of regions, each associated with a different diffusion coefficient, we successfully estimate the diffusion parameters with a maximum-likelihood approach. Although higher genetic differentiation among subpopulations leads to more accurate estimations, once a certain level of differentiation has been reached, the finite size of the genotyped population becomes the limiting factor for accurate estimation.

The main goal of Multiland is to generate neutral landscapes made of several types of regions, with an exact control of the proportions occupied by each type of region. An important feature of the software is that it allows a control of the landscape fragmentation. It is intended to theoretical studies on the effect of landscape structure in applied sciences. It has been developed in the framework of the PEERLESS ANR project "Predictive Ecological Engineering for Landscape Ecosystem Services and Sustainability".

Having a precise knowledge of the dispersal ability of a population in a heterogeneous environment is of critical importance in agroecology and conservation biology as it can provide management tools to limit the effects of pests or to increase the survival of endangered species. In this paper, we propose a mechanistic-statistical method to estimate space-dependent diffusion parameters of spatially-explicit models based on stochastic differential equations, using genetic data. Dividing the total population into subpopulations corresponding to different habitat patches with known allele frequencies, the expected proportions of individuals from each subpopulation at each position is computed by solving a system of reaction-diffusion equations. Modelling the capture and genotyping of the individuals with a statistical approach, we derive a numerically tractable formula for the likelihood function associated with the diffusion parameters. In a simulated environment made of three types of regions, each associated with a different diffusion coefficient, we successfully estimate the diffusion parameters with a maximum-likelihood approach. Although higher genetic differentiation among subpopulations leads to more accurate estimations, once a certain level of differentiation has been reached, the finite size of the genotyped population becomes the limiting factor for accurate estimation.

We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE's domain of definition D , using only temperature information in a small subdomain E ⊂ D . This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample's remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.

In this note, we discuss parameter estimation for population models based on partial differential equations (PDEs). Parametric estimation is first considered in the perspective of inverse problems (i.e., when the observation of the solution of a PDE is exactly observed or noise‐free). Then, adopting the point of view of statistics, we turn to parametric estimation for PDEs using more realistic noisy measurements. The approach that we describe uses mechanistic‐statistical models which combine (1) a PDE‐based submodel describing the dynamic under study and (2) a stochastic submodel describing the observation process. This Note is expected to contribute to bridge the gap between modelers using PDEs and population ecologists collecting and analyzing spatio‐temporal data.