We study the population dynamics of a predator-prey system with two types in each species. Within a species, predator or prey, dynamics are described by a neutral competitive Lotka-Volterra model, i.e., birth, death and competition parameters are equal for both types. Additionally, we assume that the intra- and inter-type competition parameters are equal. The predator-prey interaction is defined by a matching-types model where predators of type i exclusively interact with prey of type i. The individual-based model is described by a birth-death process with immigration, where immigration reflects mutations between the types of the same species. We completely describe the deterministic dynamics arising as a large population limit of this birth-death process. We find that depending on the parameters, potential equilibria are the coexistence of all four types, coexistence of a non-matching or matching pair of predators and prey, or the extinction of the predator or prey species resulting in a line of two-type equilibria. When mutations are sufficiently rare, then the predator-prey dynamics are described by successive jumps between the different deterministic equilibria on this mutational time scale. These jumps describe eco-evolutionary cycles of repeated prey or predator invasions and declines. When coexistence of all the types is possible, we show that these cycles accumulate on this time scale. Lastly, to prove that after the accumulation point the system converges to the coexistence equilibrium, we consider a slightly modified model with unequal intra- and inter-type competition parameters. This modified setting allows us to conclude that after the accumulation point all four populations remain macroscopic and converge to the coexistence equilibrium.

We revisit the large population limit of our epidemic model with infection age dependent infectivity and progressive immunity waning, under the assumption that the supremum in t of the random infectivity function has a finite expectation, while the previous proofs assumed that this supremum admits a deterministic upper bound.

This note corrects a mistake in the author's above-mentionned published work regarding the intersection local time of two independent random walks in a random environment. The random walks each behave as nearest neighbour random walks in random conductances in the same random environment. The proof of the main result in the original work relied on the derivation of the scaling limit of an additive functionnal of the environment at the locations where the two independent random walks meet, which was shown to be given by a constant times the intersection local time of two Brownian motions. This note shows how the value of the constant denoted by γ in this work should be updated to correct the statement. The derivation of this constant uses new results on the scaling limit of the occupation time of reversible random walks in a random environment.

Abstract Fitness landscapes offer a compact representation of adaptation, yet are rarely inferred from sparse multi-environment data. We present a Bayesian approach to infer an effective multi-host phenotypic fitness landscape from cross-inoculation assays by linking successful infection probabilities to Fisher’s geometrical model and to an explicit decomposition of establishment routes. The model estimates (i) the distance matrix among host-specific phenotypic optima, (ii) target-host permissiveness through the widths of fitness peaks, (iii) target-specific differences in the efficiency with which phenotypic suitability translates into successful infection, and (iv) the conditional probabilities that successful infections are attributed to direct establishment, rescue from standing variation in the source inoculum, or de novo rescue in the target host. We apply the approach to an experimental evolution dataset for endive necrotic mosaic virus evolved on five Asteraceae hosts and challenged in a full cross-inoculation design. The inferred landscape can be visualized as a phenotypic map of the host community, revealing pronounced heterogeneity in target-host permissiveness and a geometry broadly concordant with host phylogeny. By grounding assay-derived distances in an explicit mechanistic model, the approach provides a parsimonious representation of multi-host constraints that can be used to discuss establishment barriers and potential springboard hosts in heterogeneous communities. More broadly, it offers a general method for inferring effective fitness landscapes from sparse multi-environment data.

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection. In contrast to classical compartment models, after each infection, the infectivity is a random function of the time elapsed since one's infection. Similarly, recovered individuals become gradually susceptible after some time according to a random susceptibility function. We study the large population asymptotic behaviour of the model, by proving a functional law of large numbers (FLLN) and investigating the endemic equilibria properties of the limit. The limit depends on the law of the susceptibility random functions but only on the mean infectivity functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a special case of our FLLN limit.% for a particular set of infectivity and susceptibility random functions and initial conditions. For the endemic equilibria, if $ R_0 $ is lower than (or equal to) some threshold, the epidemic does not last forever and eventually disappears from the population, while if $ R_0 $ is larger than this threshold, the epidemic will not disappear and there exists an endemic equilibrium. The value of this threshold turns out to depend on the harmonic mean of the susceptibility a long time after an infection, a fact which was not previously known.

We derive a central limit theorem for a spatial Lambda-Fleming-Viot model with fluctuating population size. At each reproduction, a proportion of the population dies and is replaced by a not necessarily equal mass of new individuals. The mass depends on the local population size and a function thereof. Additionally, as new individuals have a single parental type, with growing population size, events become more frequent and of smaller impact, modelling the successful reproduction of a higher number of individuals. From the central limit theorem we derive a Wright-Mal & eacute;cot formula quantifying the asymptotic probability of identity by descent and thus isolation by distance. The formula reflects that ancestral lineages are attracted by centres of population mass and coalesce with a rate inversely proportional to the population size. Notably, we obtain this information despite the varying population size rendering the dual process intractable.

We consider a continuous-time Bienaymé-Galton-Watson process with logistic competition in a regime of weak competition, or equivalently of a large carrying capacity. Individuals reproduce at random times independently of each other but die at a rate which increases with the population size. When individuals reproduce, they produce a random number of offspring, drawn according to some probability distribution on the natural integers. We keep track of the number of descendants of the initial individuals by adding neutral markers to the individuals, which are inherited by one's offspring. We then consider several scaling limits of the measure-valued process describing the distribution of neutral markers in the population, as well as the population size, when the competition parameter tends to zero. Three regimes emerge, depending on the tail of the offspring distribution. When the offspring distribution admits a second moment (actually a 2+δ moment for some positive δ), the fluctuations of the population size around its carrying capacity are small and the neutral types asymptotically follow a Fleming-Viot process. When the offspring distribution has a power-law decay with exponent α∈ (1,2), the population size remains most of the time close to its carrying capacity with some (short-lived) fluctuations, and the neutral types evolve in the limit according to a generalised Λ-Fleming-Viot process. When the exponent α is equal to 1, the time scale of the fluctuations changes drastically, as well as the order of magnitude of the population size. In that case the limiting dynamics of the neutral markers is given by the dual of the Bolthausen-Sznitman coalescent.

Within-host pathogen dynamics can be difficult to track, involving pathogen expansions and restrictions by host barriers bottlenecks, ultimately determining the outcome of the infection. We use a neutral barcoding approach to monitor the root entry bottleneck during infections of tomato plants Solanum lycopersicum by the bacterial phytopathogen Ralstonia pseudosolanacearum . In our context, classical bottleneck analyses frameworks showed to be poorly accurate. By investigating data with novel bottleneck models, we conclude that the overlooked time dimension is crucial to understand root colonisation but also allows the resolution of previously inaccessible infection metrics. We define a new type of “time-resolved” bottlenecks describing infection scenarios where entry of pathogen occurs continuously with growth of pathogen in the host. Application of the “time-resolved” bottleneck framework to two published animal infection datasets confirms that it is not host-specific but may provide new conceptual understanding to many experimental studies that assumed bottlenecks to be instantaneous. ### Competing Interest Statement The authors have declared no competing interest. Ministère de l’Enseignement Supérieur et de la Recherche, https://ror.org/03sjk9a61 Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, https://ror.org/003vg9w96, STAMP Agence Nationale de la Recherche, ANR-23-CE40-0023-01 Muséum national d’Histoire naturelle

Within-host pathogen dynamics are shaped by successive host-imposed barriers that generate bottlenecks and ultimately determine infection outcome, yet these processes remain difficult to study in vivo . Here, we use neutral genetic barcoding to quantify the root-entry bottleneck during infection of tomato by the bacterial soil-borne phytopathogen Ralstonia pseudosolanacearum . We establish an updated theoretical framework showing that the temporal structure of pathogen entry, often overlooked, is a key determinant of root colonisation. We introduce the concept of time-resolved bottleneck, capturing infection scenarios in which entry occurs continuously while the pathogen population simultaneously expands within the host. We develop quantitative tools that exploit barcode dynamics to disentangle the respective roles of motility, growth, and spatial competition, enabling direct inference of in vivo pathogen fluxes and growth rates and linking host susceptibility to mechanistic features of colonisation. Extending beyond plants, reanalysis of two independent animal infection datasets reveals previously unrecognised within-host development patterns. Together, these results show that time-resolved bottleneck is a general principle of bacterial infection dynamics and propose a new standard for barcoding-based studies: a host- and pathogen-agnostic framework that is directly applicable to a majority of existing STAMP(-R)-style datasets, distributed as an open, annotated R/Python notebook to facilitate community reuse.

The dynamics of a general structured population is modelled using a general stochastic differential equation (SDE) with an infinite decomposability property. This property allows the population to be divided into an arbitrary number of allelic components, also known as stochastic neutral fractions. When demographic noise is small, a fast-slow principle provides a general formula for the effective population size in structured populations. To illustrate this approach, we revisit several examples from the literature, including expansion fronts.

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $\Lambda$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and V\'eber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.

In this work we propose a measure-valued stochastic process representing the dynamics of a virus population, structured by phenotypic traits and geographical space, and where viruses are transported between spatial locations by mechanical vectors. As a first example of the use of this model, we show how to use this model to infer results on the probability of extinction of the virus population. Later, by combining various scalings on population sizes, speed of diffusion of vectors, and other relevant model parameters, we show the emergence of two systems of integro-differential equations as Macroscopic descriptions of the system. Under the existence of densities at time zero, we also show the propagation of this property for later times, and derive the strong formulation of the limiting systems of IDEs. These strong formulations, in a sense, correspond to spatial Lotka-Volterra competition models with mutation and vector-borne dispersal.

We introduce a modified spatial Λ-Fleming-Viot process to model the ancestry of individuals in a population occupying a continuous spatial habitat divided into two areas by a sharp discontinuity of the dispersal rate and effective population density. We derive an analytical formula for the expected number of shared haplotype segments between two individuals depending on their sampling locations. This formula involves the transition density of a skew diffusion which appears as a scaling limit of the ancestral lineages of individuals in this model. We then show that this formula can be used to infer the dispersal parameters and the effective population density of both regions, using a composite likelihood approach, and we demonstrate the efficiency of this method on a range of simulated data sets.

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $\Lambda$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and V\'eber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.

This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $L$ distinct patches (with migrations between them) and $K$ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d. The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $D$, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.

Many populations can somehow adapt to rapid environmental changes. To understand this fast evolution, we investigate the genealogy of individuals inside those populations. More precisely, we use a deterministic model to describe the phenotypic density of a population under selection when the fitness optimum moves at constant speed. We study the inside dynamics of this population using the neutral fractions approach. We then define a Markov process characterizing the distribution of ancestral phenotypic lineages inside the equilibrium. This construction yields qualitative as well as quantitative properties on the phenotype of typical ancestors. In particular, we show that in asexual populations typical ancestors of present individuals carried traits much closer to the fitness optimum than most individuals alive at the same time. We also investigate more deeply the asymptotic regime of small mutation effects. In this regime, we obtain an explicit formula for the typical ancestral lineage using the description of the solutions of the Hamilton Jacobi equation as a minimizer of an optimization problem. In addition, we compare our deterministic results on lineages with the lineages of stochastic models.

In this paper, we consider a mathematical model for the evolution of neutral genetic diversity in a spatial continuum including mutations, genetic drift and either short range or long range dispersal. The model we consider is the spatial Lambda-Fleming-Viot process introduced by Barton, Etheridge and Veber, which describes the state of the population at any time by a measure on R-d x [0, 1], where R-d is the geographical space and [0, 1] is the space of genetic types. In both cases (short range and long range dispersal), we prove a functional central limit theorem for the process as the population density becomes large and under some space-time rescaling. We then deduce from these two central limit theorems a formula for the asymptotic probability of identity of two individuals picked at random from two given spatial locations. In the case of short range dispersal, we recover the classical Wright-Malecot formula, which is widely used in demographic inference for spatially structured populations. In the case of long range dispersal we obtain a new formula which could open the way for a better appraisal of long range dispersal in inference methods.

This paper presents a new view of household epidemic models, where the interaction between the households is of mean field type. We thus obtain in the limit of infinitely many households a nonlinear Markov process solution of a McKean - Vlasov type Poisson driven SDE, and a propagation of chaos result. We also define a basic reproduction number R0, and show that if R0>1, then the nonlinear Markov process has a unique non trivial ergodic invariant probability measure, whereas if R0<=1, it converges to 0 as t tends to infinity.

In this paper, we use a deterministic epidemic model with memory to estimate the state of the COVID-19 epidemic in France, from early March until mid-December 2020. Our model is in the SEIR class, which means that when a susceptible individual (S) becomes infected, he/she is first exposed (E), i.e. not yet contagious. Then he/she becomes infectious (I) for a certain length of time, during which he/she may infect susceptible individuals around him/her, and finally becomes removed (R), that is, either immune or dead. The specificity of our model is that it assumes a very general probability distribution for the pair of exposed and infectious periods. The law of large numbers limit of such a model is a model with memory (the future evolution of the model depends not only upon its present state, but also upon its past). We present theoretical results linking the (unobserved) parameters of the model to various quantities which are more easily measured during the early stages of an epidemic. We then apply these results to estimate the state of the COVID-19 epidemic in France, using available information on the infection fatality ratio and on the distribution of the exposed and infectious periods. Using the hospital data published daily by Santé Publique France, we gather some information on the delay between infection and hospital admission, intensive care unit (ICU) admission and hospital deaths, and on the proportion of people who have been infected up to the end of 2020.

In this survey paper, we review the recent advances in individual based non–Markovian epidemic models. They include epidemic models with a constant infectivity rate, varying infectivity rate or infection-age dependent infectivity, infection-age recovery rate (or equivalently, general law of infectious period), as well as varying susceptibility/immunity. We focus on the scaling limits with a large population, functional law of large numbers (FLLN) and functional central limit theorems (FCLT), while the large and moderate deviations for some Markovian epidemic models are also reviewed. In the FLLN, the limits are a set of Volterra integral equations, and in the FCLT, the limits are stochastic Volterra integral equations driven by Gaussian processes. We relate our deterministic limits to the results in the seminal papers by Kermack and McKendrick published in 1927, 1932 and 1933, where the varying infectivity and susceptibility/immunity were already considered. We also discuss some extensions, including models with heterogeneous population, spatial models and control problems, as well as open problems.