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, [Formula: see text], and we study the dependency of [Formula: see text] 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 [Formula: see text], compared to [Formula: see text] or [Formula: see text], 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.

Seed predation-induced Allee effects, seed dispersal and masting jointly drive the diversity of seed sources during population expansion

The environmental factors affecting plant reproduction and effective dispersal, in particular biotic interactions, have a strong influence on plant expansion dynamics, but their demographic and genetic consequences remain an understudied body of theory. Here, we use a mathematical model in a one-dimensional space and on a single reproductive period to describe the joint effects of predispersal seed insect predators foraging strategy and plant reproduction strategy (masting) on the spatio-temporal dynamics of seed sources diversity in the colonisation front of expanding plant populations. We show that certain foraging strategies can result in a higher seed predation rate at the colonisation front compared to the core of the population, leading to an Allee effect. This effect promotes the contribution of seed sources from the core to the colonisation front, with long-distance dispersal further increasing this contribution. As a consequence, our study reveals a novel impact of the predispersal seed predation-induced Allee effect, which mitigates the erosion of diversity in expanding populations. We use rearrangement inequalities to show that masting has a buffering role: it mitigates this seed predation-induced Allee effect. This study shows that predispersal seed predation, plant reproductive strategies and seed dispersal patterns can be intermingled drivers of the diversity of seed sources in expanding plant populations, and opens new perspectives concerning the analysis of more complex models such as integro-difference or reaction-diffusion equations.

An ecological-economic model of land-use decisions, agricultural production and biocontrol

The regulation of agricultural pests by their natural enemies is a key step in the agroecological transition. The level of biocontrol seems, however, to highly depend on the agronomic and ecological context. It is thus important to identify the conditions under which this ecosystem service is efficient as well as the magnitude of its effects. An actual reduction of pesticide use depends on a change in farmers' decisions, calling for the consideration of economic dimensions. We develop a dynamic agroecological-economic model representing land-use and agricultural intensity decisions as well as the dynamics of a crop pest and a natural enemy. Biocontrol is assessed considering both private benefits (increase in farmers' profit) and public benefits (reduction of pesticide use) with respect to a situation without a natural enemy. We provide a theoretical assessment of the magnitude of biocontrol over a wide range of agronomic contexts (spatially explicit maps of agricultural production potential, with heterogeneous distribution and control of spatial fragmentation) and ecological contexts, described through various parameter values of a reaction-diffusion model. The contexts in which biocontrol plays a significant role are identified, and the role of key parameters is discussed. Our open-access model offers a tool to investigate alternative specifications.

A research agenda for scaling up agroecology in European countries

A profound transformation of agricultural production methods has become unavoidable due to the increase in the world's population, and environmental and climatic challenges. Agroecology is now recognized as a challenging model for agricultural systems, promoting their diversification and adaptation to environmental and socio-economic contexts, with consequences for the entire agri-food system and the development of rural and urban areas. Through a prospective exercise performed at a large interdisciplinary institute, INRAE, a research agenda for agroecology was built that filled a gap through its ambition and interdisciplinarity. It concerned six topics. For genetics, there is a need to study genetic aspects of complex systems (e.g., mixtures of genotypes) and to develop breeding methods for them. For landscapes, challenges lie in effects of heterogeneity at multiple scales, in multifunctionality and in the design of agroecological landscapes. Agricultural equipment and digital technologies show high potential for monitoring dynamics of agroecosystems. For modeling, challenges include approaches to complexity, consideration of spatial and temporal dimensions and representation of the cascade from cropping practices to ecosystem services. The agroecological transition of farms calls for modeling and observational approaches as well as for creating new design methods. Integration of agroecology into food systems raises the issues of product specificity, consumer behavior and organization of markets, standards and public policies. In addition, transversal priorities were identified: (i) generating sets of biological data, through research and participatory mechanisms, that are appropriate for designing agroecological systems and (ii) collecting and using coherent sets of data to enable assessment of vulnerability, resilience and risk in order to evaluate the performance of agroecological systems and to contribute to scaling up. The main lessons learned from this collective exercise can be useful for the entire scientific community engaged in research into agroecology.

Emerging strains of watermelon mosaic virus in Southeastern France: model-based estimation of the dates and places of introduction

Where and when alien organisms are successfully introduced are central questions to elucidate biotic and abiotic conditions favorable to the introduction, establishment and spread of invasive species. We propose a modelling framework to analyze multiple introductions by several invasive genotypes or genetic variants, in competition with a resident population, when observations provide knowledge on the relative proportions of each variant at some dates and places. This framework is based on a mechanistic-statistical model coupling a reaction–diffusion model with a probabilistic observation model. We apply it to a spatio-temporal dataset reporting the relative proportions of five genetic variants of watermelon mosaic virus (WMV, genus Potyvirus, family Potyviridae) in infections of commercial cucurbit fields. Despite the parsimonious nature of the model, it succeeds in fitting the data well and provides an estimation of the dates and places of successful introduction of each emerging variant as well as a reconstruction of the dynamics of each variant since its introduction.

A parsimonious approach for spatial transmission and heterogeneity in the COVID-19 propagation

Raw data on the number of deaths at a country level generally indicate a spatially variable distribution of COVID-19 incidence. An important issue is whether this pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we fit them to French mortality data with a mixed probabilistic-ODE approach. Using statistical criteria, we select the model with non-local transmission corresponding to a diffusion on the graph of counties that depends on the geographic proximity, with time-dependent contact rate and spatially constant parameters. This suggests that in a geographically middle size centralized country such as France, once the epidemic is established, the effect of global processes such as restriction policies and sanitary measures overwhelms the effect of local factors. Additionally, this approach reveals the latent epidemiological dynamics including the local level of immunity, and allows us to evaluate the role of non-local interactions on the future spread of the disease.

Towards unified and real-time analyses of outbreaks at country-level during pandemics

The management of public health and the preparedness for health emergencies partly rely on the collection and analysis of surveillance data, which become crucial in the context of an emergency such as the pandemic caused by COVID-19. For COVID-19, typically, numerous national and global initiatives have been set up from this perspective. Here, we propose to develop a shared vision of the country-level outbreaks during a pandemic, by enhancing, at the international scale, the foundations of the analysis of surveillance data and by adopting a unified and real-time approach to monitor and forecast the outbreak across time and across the world. This proposal, rolled out as a web platform, should contribute to strengthen epidemiological understanding, sanitary democracy as well as global and local responses to pandemics.

COVID-19 mortality dynamics: The future modelled as a (mixture of) past(s)

Discrepancies in population structures, decision making, health systems and numerous other factors result in various COVID-19-mortality dynamics at country scale, and make the forecast of deaths in a country under focus challenging. However, mortality dynamics of countries that are ahead of time implicitly include these factors and can be used as real-life competing predicting models. We precisely propose such a data-driven approach implemented in a publicly available web app timely providing mortality curves comparisons and real-time short-term forecasts for about 100 countries. Here, the approach is applied to compare the mortality trajectories of second-line and front-line European countries facing the COVID-19 epidemic wave. Using data up to mid-April, we show that the second-line countries generally followed relatively mild mortality curves rather than fast and severe ones. Thus, the continuation, after mid-April, of the COVID-19 wave across Europe was likely to be mitigated and not as strong as it was in most of the front-line countries first impacted by the wave (this prediction is corroborated by posterior data).

Impact of Lockdown on the Epidemic Dynamics of COVID-19 in France

The COVID-19 epidemic was reported in the Hubei province in China in December 2019 and then spread around the world reaching the pandemic stage at the beginning of March 2020. Since then, several countries went into lockdown. Using a mechanistic-statistical formalism, we estimate the effect of the lockdown in France on the contact rate and the effective reproduction number R e of the COVID-19. We obtain a reduction by a factor 7 (R e = 0.47, 95%-CI: 0.45-0.50), compared to the estimates carried out in France at the early stage of the epidemic. We also estimate the fraction of the population that would be infected by the beginning of May, at the official date at which the lockdown should be relaxed. We find a fraction of 3.7% (95%-CI: 3.0-4.8%) of the total French population, without taking into account the number of recovered individuals before April 1st, which is not known. This proportion is seemingly too low to reach herd immunity. Thus, even if the lockdown strongly mitigated the first epidemic wave, keeping a low value of R e is crucial to avoid an uncontrolled second wave (initiated with much more infectious cases than the first wave) and to hence avoid the saturation of hospital facilities.

Using Early Data to Estimate the Actual Infection Fatality Ratio from COVID-19 in France

The number of screening tests carried out in France and the methodology used to target the patients tested do not allow for a direct computation of the actual number of cases and the infection fatality ratio (IFR). The main objective of this work is to estimate the actual number of people infected with COVID-19 and to deduce the IFR during the observation window in France. We develop a `mechanistic-statistical' approach coupling a SIR epidemiological model describing the unobserved epidemiological dynamics, a probabilistic model describing the data acquisition process and a statistical inference method. The actual number of infected cases in France is probably higher than the observations: we find here a factor ×8 (95%-CI: 5-12) which leads to an IFR in France of 0.5% (95%-CI: 0.3-0.8) based on hospital death counting data. Adjusting for the number of deaths in nursing homes, we obtain an IFR of 0.8% (95%-CI: 0.45-1.25). This IFR is consistent with previous findings in China (0.66%) and in the UK (0.9%) and lower than the value previously computed on the Diamond Princess cruse ship data (1.3%).

Parameter estimation for energy balance models with memory

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.

A statistical-reaction-diffusion approach for analyzing expansion processes

In this article, we propose a method for analyzing the spatial variations in the range expansion of the pine processionary moth (PPM), an invasive species in France. Based on binary measurements - the presence or absence of PPM nests - the proposed method allows us to infer the local effect of the environment on PPM population expansion. This effect is estimated at each position x using a parameter F(x) that corresponds to the local PPM fitness. The data type and the two stage PPM life cycle make estimating this parameter difficult. To overcome these difficulties we adopt a mechanistic-statistical approach that combines a statistical model for the observation process with a hierarchical,reaction-diffusion based mechanistic model for the expansion process. Bayesian inference of the parameter F(x) reveals that PPM fitness is spatially heterogeneous and highlights the existence of large regions associated with lower fitness. The factors underlying this lower fitness are yet to be determined.

Mathematical analysis of the optimal habitat configurations for species persistence

We study a reaction-diffusion model in a binary environment made of habitat and non-habitat regions. Environmental heterogeneity is expressed through the species intrinsic growth rate coefficient. It was known that, for a fixed habitat abundance, species survival depends on habitat arrangements. Our goal is to describe the spatial configurations of habitat that maximise the chances of survival. Through numerical computations, we find that they are of two main types - ball-shaped or stripe-shaped. We formally prove that these optimal shapes depend on the habitat abundance and on the amplitude of the growth rate coefficient. We deduce from these observations that the optimal shape of the habitat realises a compromise between reducing the detrimental habitat edge effects and taking advantage of the domain boundary effects. In the case of an infinite-periodic environment, we prove that the optimal habitat shapes can be deduced from those in the case of a bounded domain.

Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments

This paper presents a study of a nonlinear reaction-diffusion population model in fragmented environments. The model is set on R(N), with periodic heterogeneous coefficients obtained using stochastic processes. Using a criterion of species persistence based on the notion of principal eigenvalue of an elliptic operator, we provided a precise numerical analysis of the interactions between habitat fragmentation and species persistence. The obtained results clearly indicated that species persistence strongly tends to decrease with habitat fragmentation. Moreover, comparing two stochastic models of landscape pattern generation, we observed that in addition to local fragmentation, a more global effect of the position of the habitat patches also influenced species persistence.

Analysis of the periodically fragmented environment model : I – Species persistence

This paper is concerned with the study of the stationary solutions of the equation [Equation: see text] where the diffusion matrix A and the reaction term f are periodic in x. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.