Moment equations in spatial evolutionary ecology

The molding of intraspecific trait variation by selection under ecological inheritance

Organisms continuously modify their environment, often impacting the fitness of future conspecifics due to ecological inheritance. When this inheritance is biased toward kin, selection favors modifications that increase the fitness of downstream individuals. How such selection shapes trait variation within populations remains poorly understood. Using mathematical modelling, we investigate the coevolution of multiple traits in a group-structured population when these traits affect the group environment, which is then bequeathed to future generations. We examine when such coevolution favors polymorphism as well as the resulting associations among traits. We find in particular that two traits become associated when one trait affects the environment while the other influences the likelihood that future kin experience this environment. To illustrate this, we model the coevolution of (a) the attack rate on a local renewable resource, which deteriorates environmental conditions, with (b) dispersal between groups, which reduces the likelihood that kin suffers from such deterioration. We show this often leads to the emergence of two highly differentiated morphs: one that readily disperses and depletes local resources, and another that maintains these resources and tends to remain philopatric. More broadly, we suggest that ecological inheritance can contribute to phenotypic diversity and lead to complex polymorphism.

Regulated bacterial interaction networks: A mathematical framework to describe competitive growth under inclusion of metabolite cross-feeding

When bacterial species with the same resource preferences share the same growth environment, it is commonly believed that direct competition will arise. A large variety of competition and more general 'interaction' models have been formulated, but what is currently lacking are models that link monoculture growth kinetics and community growth under inclusion of emerging biological interactions, such as metabolite cross-feeding. In order to understand and mathematically describe the nature of potential cross-feeding interactions, we design experiments where two bacterial species Pseudomonas putida and Pseudomonas veronii grow in liquid medium either in mono- or as co-culture in a resource-limited environment. We measure population growth under single substrate competition or with double species-specific substrates (substrate 'indifference'), and starting from varying cell ratios of either species. Using experimental data as input, we first consider a mean-field model of resource-based competition, which captures well the empirically observed growth rates for monocultures, but fails to correctly predict growth rates in co-culture mixtures, in particular for skewed starting species ratios. Based on this, we extend the model by cross-feeding interactions where the consumption of substrate by one consumer produces metabolites that in turn are resources for the other consumer, thus leading to positive feedback in the species system. Two different cross-feeding options were considered, which either lead to constant metabolite cross-feeding, or to a regulated form, where metabolite utilization is activated with rates according to either a threshold or a Hill function, dependent on metabolite concentration. Both mathematical proof and experimental data indicate regulated cross-feeding to be the preferred model to constant metabolite utilization, with best co-culture growth predictions in case of high Hill coefficients, close to binary (on/off) activation states. This suggests that species use the appearing metabolite concentrations only when they are becoming high enough; possibly as a consequence of their lower energetic content than the primary substrate. Metabolite sharing was particularly relevant at unbalanced starting cell ratios, causing the minority partner to proliferate more than expected from the competitive substrate because of metabolite release from the majority partner. This effect thus likely quells immediate substrate competition and may be important in natural communities with typical very skewed relative taxa abundances and slower-growing taxa. In conclusion, the regulated bacterial interaction network correctly describes species substrate growth reactions in mixtures with few kinetic parameters that can be obtained from monoculture growth experiments.

Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact

Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Eco-evolutionary model on spatial graphs reveals how habitat structure affects phenotypic differentiation

Differentiation mechanisms are influenced by the properties of the landscape over which individuals interact, disperse and evolve. Here, we investigate how habitat connectivity and habitat heterogeneity affect phenotypic differentiation by formulating a stochastic eco-evolutionary model where individuals are structured over a spatial graph. We combine analytical insights into the eco-evolutionary dynamics with numerical simulations to understand how the graph topology and the spatial distribution of habitat types affect differentiation. We show that not only low connectivity but also heterogeneity in connectivity promotes neutral differentiation, due to increased competition in highly connected vertices. Habitat assortativity, a measure of habitat spatial auto-correlation in graphs, additionally drives differentiation under habitat-dependent selection. While assortative graphs systematically amplify adaptive differentiation, they can foster or depress neutral differentiation depending on the migration regime. By formalising the eco-evolutionary and spatial dynamics of biological populations on graphs, our study establishes fundamental links between landscape features and phenotypic differentiation.

Emerging dynamics from high-resolution spatial numerical epidemics

Simulating nationwide realistic individual movements with a detailed geographical structure can help optimise public health policies. However, existing tools have limited resolution or can only account for a limited number of agents. We introduce Epidemap, a new framework that can capture the daily movement of more than 60 million people in a country at a building-level resolution in a realistic and computationally efficient way. By applying it to the case of an infectious disease spreading in France, we uncover hitherto neglected effects, such as the emergence of two distinct peaks in the daily number of cases or the importance of local density in the timing of arrival of the epidemic. Finally, we show that the importance of super-spreading events strongly varies over time.

Accounting for Space—Quantification of Cell-To-Cell Transmission Kinetics Using Virus Dynamics Models

Mathematical models based on ordinary differential equations (ODE) that describe the population dynamics of viruses and infected cells have been an essential tool to characterize and quantify viral infection dynamics. Although an important aspect of viral infection is the dynamics of viral spread, which includes transmission by cell-free virions and direct cell-to-cell transmission, models used so far ignored cell-to-cell transmission completely, or accounted for this process by simple mass-action kinetics between infected and uninfected cells. In this study, we show that the simple mass-action approach falls short when describing viral spread in a spatially-defined environment. Using simulated data, we present a model extension that allows correct quantification of cell-to-cell transmission dynamics within a monolayer of cells. By considering the decreasing proportion of cells that can contribute to cell-to-cell spread with progressing infection, our extension accounts for the transmission dynamics on a single cell level while still remaining applicable to standard population-based experimental measurements. While the ability to infer the proportion of cells infected by either of the transmission modes depends on the viral diffusion rate, the improved estimates obtained using our novel approach emphasize the need to correctly account for spatial aspects when analyzing viral spread.

Class Structure, Demography, and Selection: Reproductive-Value Weighting in Nonequilibrium, Polymorphic Populations

In natural populations, individuals of a given genotype may belong to different classes. Such classes can represent different age groups, developmental stages, or habitats. Class structure has important evolutionary consequences because the fitness of individuals with the same genetic background may vary depending on their class. As a result, demographic transitions between classes can cause fluctuations in the trait mean that need to be removed when estimating selection on a trait. Intrinsic differences between classes are classically taken into account by weighting individuals by class-specific reproductive values, defined as the relative contribution of individuals in a given class to the future of the population. These reproductive values are generally constant weights calculated from a constant projection matrix. Here, I show for large populations and clonal reproduction that reproductive values can be defined as time-dependent weights satisfying dynamical demographic equations that depend only on the average between-class transition rates over all genotypes. Using these time-dependent demographic reproductive values yields a simple Price equation where the nonselective effects of between-class transitions are removed from the dynamics of the trait. This generalizes previous theory to a large class of ecological scenarios, taking into account density dependence, ecological feedbacks, arbitrary strength of selection, and arbitrary trait distributions. I discuss the role of reproductive values for prospective and retrospective analyses of the dynamics of phenotypic traits.

Theoretical Approaches in Evolutionary Ecology: Environmental Feedback as a Unifying Perspective

Evolutionary biology and ecology have a strong theoretical underpinning, and this has fostered a variety of modeling approaches. A major challenge of this theoretical work has been to unravel the tangled feedback loop between ecology and evolution. This has prompted the development of two main classes of models. While quantitative genetics models jointly consider the ecological and evolutionary dynamics of a focal population, a separation of timescales between ecology and evolution is assumed by evolutionary game theory, adaptive dynamics, and inclusive fitness theory. As a result, theoretical evolutionary ecology tends to be divided among different schools of thought, with different toolboxes and motivations. My aim in this synthesis is to highlight the connections between these different approaches and clarify the current state of theory in evolutionary ecology. Central to this approach is to make explicit the dependence on environmental dynamics of the population and evolutionary dynamics, thereby materializing the eco-evolutionary feedback loop. This perspective sheds light on the interplay between environmental feedback and the timescales of ecological and evolutionary processes. I conclude by discussing some potential extensions and challenges to our current theoretical understanding of eco-evolutionary dynamics.

Spatial evolutionary epidemiology of spreading epidemics

Most spatial models of host-parasite interactions either neglect the possibility of pathogen evolution or consider that this process is slow enough for epidemiological dynamics to reach an equilibrium on a fast timescale. Here, we propose a novel approach to jointly model the epidemiological and evolutionary dynamics of spatially structured host and pathogen populations. Starting from a multi-strain epidemiological model, we use a combination of spatial moment equations and quantitative genetics to analyse the dynamics of mean transmission and virulence in the population. A key insight of our approach is that, even in the absence of long-term evolutionary consequences, spatial structure can affect the short-term evolution of pathogens because of the build-up of spatial differentiation in mean virulence. We show that spatial differentiation is driven by a balance between epidemiological and genetic effects, and this quantity is related to the effect of kin competition discussed in previous studies of parasite evolution in spatially structured host populations. Our analysis can be used to understand and predict the transient evolutionary dynamics of pathogens and the emergence of spatial patterns of phenotypic variation.

Dynamic, spatial models of parasite transmission in wildlife: Their structure, applications and remaining challenges

Individual differences in contact rate can arise from host, group and landscape heterogeneity and can result in different patterns of spatial spread for diseases in wildlife populations with concomitant implications for disease control in wildlife of conservation concern, livestock and humans. While dynamic disease models can provide a better understanding of the drivers of spatial spread, the effects of landscape heterogeneity have only been modelled in a few well-studied wildlife systems such as rabies and bovine tuberculosis. Such spatial models tend to be either purely theoretical with intrinsic limiting assumptions or individual-based models that are often highly species- and system-specific, limiting the breadth of their utility. Our goal was to review studies that have utilized dynamic, spatial models to answer questions about pathogen transmission in wildlife and identify key gaps in the literature. We begin by providing an overview of the main types of dynamic, spatial models (e.g., metapopulation, network, lattice, cellular automata, individual-based and continuous-space) and their relation to each other. We investigate different types of ecological questions that these models have been used to explore: pathogen invasion dynamics and range expansion, spatial heterogeneity and pathogen persistence, the implications of management and intervention strategies and the role of evolution in host-pathogen dynamics. We reviewed 168 studies that consider pathogen transmission in free-ranging wildlife and classify them by the model type employed, the focal host-pathogen system, and their overall research themes and motivation. We observed a significant focus on mammalian hosts, a few well-studied or purely theoretical pathogen systems, and a lack of studies occurring at the wildlife-public health or wildlife-livestock interfaces. Finally, we discuss challenges and future directions in the context of unprecedented human-mediated environmental change. Spatial models may provide new insights into understanding, for example, how global warming and habitat disturbance contribute to disease maintenance and emergence. Moving forward, better integration of dynamic, spatial disease models with approaches from movement ecology, landscape genetics/genomics and ecoimmunology may provide new avenues for investigation and aid in the control of zoonotic and emerging infectious diseases.