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.

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.

Adaptation in a heterogeneous environment I: persistence versus extinction.

Understanding how a diversity of plants in agroecosystems affects the adaptation of pathogens is a key issue in agroecology. We analyze PDE systems describing the dynamics of adaptation of two phenotypically structured populations, under the effects of mutation, selection and migration in a two-patch environment, each patch being associated with a different phenotypic optimum. We consider two types of growth functions that depend on the n-dimensional phenotypic trait: either local and linear or nonlocal nonlinear. In both cases, we obtain existence and uniqueness results as well as a characterization of the large-time behaviour of the solution (persistence or extinction) based on the sign of a principal eigenvalue. We show that migration between the two environments decreases the chances of persistence, with in some cases a 'lethal migration threshold' above which persistence is not possible. Comparison with stochastic individual-based simulations shows that the PDE approach accurately captures this threshold. Our results illustrate the importance of cultivar mixtures for disease prevention and control.

Linking niche theory to ecological impacts of successful invaders: insights from resource fluctuation-specialist herbivore interactions.

Theories of species coexistence and invasion ecology are fundamentally connected and provide a common theoretical framework for studying the mechanisms underlying successful invasions and their ecological impacts. Temporal fluctuations in resource availability and differences in life-history traits between invasive and resident species are considered as likely drivers of the dynamics of invaded communities. Current critical issues in invasion ecology thus relate to the extent to which such mechanisms influence coexistence between invasive and resident species and to the ability of resident species to persist in an invasive-dominated ecosystem. We tested how a fluctuating resource, and species trait differences may explain and help predict long-term impacts of biological invasions in forest specialist insect communities. We used a simple invasion system comprising closely related invasive and resident seed-specialized wasps (Hymenoptera: Torymidae) competing for a well-known fluctuating resource and displaying divergent diapause, reproductive and phenological traits. Based on extensive long-term field observations (1977-2010), we developed a combination of mechanistic and statistical models aiming to (i) obtain a realistic description of the population dynamics of these interacting species over time, and (ii) clarify the respective contributions of fluctuation-dependent and fluctuation-independent mechanisms to long-term impact of invasion on the population dynamics of the resident wasp species. We showed that a fluctuation-dependent mechanism was unable to promote coexistence of the resident and invasive species. Earlier phenology of the invasive species was the main driver of invasion success, enabling the invader to exploit an empty niche. Phenology also had the greatest power to explain the long-term negative impact of the invasive on the resident species, through resource pre-emption. This study provides strong support for the critical role of species differences in interspecific competition outcomes within animal communities. Our mechanistic-statistical approach disentangles the critical drivers of novel species assemblages resulting from intentional and non-intentional introductions of non-native species.

Allee effect promotes diversity in traveling waves of colonization.

Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves.

Success rate of a biological invasion in terms of the spatial distribution of the founding population.

We analyze the role of the spatial distribution of the initial condition in reaction-diffusion models of biological invasion. Our study shows that, in the presence of an Allee effect, the precise shape of the initial (or founding) population is of critical importance for successful invasion. Results are provided for one-dimensional and two-dimensional models. In the one-dimensional case, we consider initial conditions supported by two disjoint intervals of length L/2 and separated by a distance α. Analytical as well as numerical results indicate that the critical size L*(α) of the population, where the invasion is successful if and only if L > L*(α), is a continuous function of α and tends to increase with α, at least when α is not too small. This result emphasizes the detrimental effect of fragmentation. In the two-dimensional case, we consider more general, stochastically generated initial conditions u0, and we provide a new and rigorous definition of the rate of fragmentation of u0. We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of u0 and the fragmentation rate of u0. Our results show that the outcome of an invasion is almost completely determined by these two parameters. Moreover, we observe that the minimum abundance required for successful invasion tends to increase in a non-linear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.

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.

Patchy patterns due to group dispersal.

The presence of multiple foci in population patterns may be due to various processes arising in the population dynamics. Group dispersal, which has been lightly investigated for airborne species, is one of these processes. We built a stochastic model generating the dispersal of groups of particles. This model may be viewed as an extension of classical dispersal models based on parametric kernels. It has a hierarchical structure: at the first stage group centers are drawn under a classical dispersal kernel; at the second stage the particles are diffused around their group centers. Analytic and simulation results show that group dispersal is a sufficient condition to generate patterns with multiple foci, i.e. patchy patterns, even if the population can remain particularly concentrated.

Spreading speeds in slowly oscillating environments.

In this paper, we derive exact asymptotic estimates of the spreading speeds of solutions of some reaction-diffusion models in periodic environments with very large periods. Contrarily to the other limiting case of rapidly oscillating environments, there was previously no explicit formula in the case of slowly oscillating environments. The knowledge of these two extremes permits to quantify the effect of environmental fragmentation on the spreading speeds. On the one hand, our analytical estimates and numerical simulations reveal speeds which are higher than expected for Shigesada-Kawasaki-Teramoto models with Fisher-KPP reaction terms in slowly oscillating environments. On the other hand, spreading speeds in very slowly oscillating environments are proved to be 0 in the case of models with strong Allee effects; such an unfavorable effect of aggregation is merely seen in reaction-diffusion models.

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.

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.

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).

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%).

Population persistence under high mutation rate: From evolutionary rescue to lethal mutagenesis.

Populations may genetically adapt to severe stress that would otherwise cause their extirpation. Recent theoretical work, combining stochastic demography with Fisher's geometric model of adaptation, has shown how evolutionary rescue becomes unlikely beyond some critical intensity of stress. Increasing mutation rates may however allow adaptation to more intense stress, raising concerns about the effectiveness of treatments against pathogens. This previous work assumes that populations are rescued by the rise of a single resistance mutation. However, even in asexual organisms, rescue can also stem from the accumulation of multiple mutations in a single genome. Here, we extend previous work to study the rescue process in an asexual population where the mutation rate is sufficiently high so that such events may be common. We predict both the ultimate extinction probability of the population and the distribution of extinction times. We compare the accuracy of different approximations covering a large range of mutation rates. Moderate increase in mutation rates favors evolutionary rescue. However, larger increase leads to extinction by the accumulation of a large mutation load, a process called lethal mutagenesis. We discuss how these results could help design "evolution-proof" antipathogen treatments that even highly mutable strains could not overcome.

Biological invasions: deriving the regions at risk from partial measurements.

We consider the problem of forecasting the regions at higher risk for newly introduced invasive species. Favourable and unfavourable regions may indeed not be known a priori, especially for exotic species whose hosts in native range and newly-colonised areas can be different. Assuming that the species is modelled by a logistic-like reaction-diffusion equation, we prove that the spatial arrangement of the favourable and unfavourable regions can theoretically be determined using only partial measurements of the population density: (1) a local 'spatio-temporal' measurement, during a short time period and, (2) a 'spatial' measurement in the whole region susceptible to colonisation. We then present a stochastic algorithm which is proved analytically, and then on several numerical examples, to be effective in deriving these regions.

Modelling the impact of an invasive insect via reaction-diffusion.

An exotic, specialist seed chalcid, Megastigmus schimitscheki, has been introduced along with its cedar host seeds from Turkey to southeastern France during the early 1990s. It is now expanding in plantations of Atlas Cedar (Cedrus atlantica). We propose a model to predict the expansion and impact of this insect. This model couples a time-discrete equation for the ovo-larval stage with a two-dimensional reaction-diffusion equation for the adult stage, through a formula linking the solution of the reaction-diffusion equation to a seed attack rate. Two main diffusion operators, of Fokker-Planck and Fickian types, are tested. We show that taking account of the dependence of the insect mobility with respect to spatial heterogeneity, and choosing the appropriate diffusion operator, are critical factors for obtaining good predictions.

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.

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.

Beneficial mutation-selection dynamics in finite asexual populations: a free boundary approach.

Using a free boundary approach based on an analogy with ice melting models, we propose a deterministic PDE framework to describe the dynamics of fitness distributions in the presence of beneficial mutations with non-epistatic effects on fitness. Contrarily to most approaches based on deterministic models, our framework does not rely on an infinite population size assumption, and successfully captures the transient as well as the long time dynamics of fitness distributions. In particular, consistently with stochastic individual-based approaches or stochastic PDE approaches, it leads to a constant asymptotic rate of adaptation at large times, that most deterministic approaches failed to describe. We derive analytic formulas for the asymptotic rate of adaptation and the full asymptotic distribution of fitness. These formulas depend explicitly on the population size, and are shown to be accurate for a wide range of population sizes and mutation rates, compared to individual-based simulations. Although we were not able to derive an analytic description for the transient dynamics, numerical computations lead to accurate predictions and are computationally efficient compared to stochastic simulations. These computations show that the fitness distribution converges towards a travelling wave with constant speed, and whose profile can be computed analytically.