arXiv subjects

Sylvain Billiard

Publications and source records attributed to Sylvain Billiard.

14 recordsLinked to original sources

arXiv2025

Dynamics of Two Species with Density-Dependent Interactions in a Mutualistic Context

Mutualistic interactions, where individuals from different species can benefit from each other, are widespread across ecosystems. This study develops a general deterministic model of mutualism involving two populations, assuming that mutualism may involve both costs and benefits for the interacting individuals, leading to density-dependent effects on the dynamics of the two species. This framework aims at generalizing pre-existing models, by allowing the ecological interactions to transition from mutualistic to parasitic when the respective densities of interacting species change. Through ordinary differential equations and phase portrait analysis, we derive general principles governing these systems, identifying sufficient conditions for the emergence of certain dynamic behaviors. In particular, we show that limit cycles can arise when interactions include parasitic phases but are absent in strictly mutualistic regimes. This framework provides a general approach for characterizing the population dynamics of interacting species and highlights the effect of the transitions from mutualism to parasitism due to density dependence.

q-bio.PE↗

arXiv2025

Spinal study of a population model for colonial species with interactions and environmental noise

We introduce and study a stochastic model for the dynamics of colonial species, which reproduce through fission or fragmentation. The fission rate depends on the relative sizes of colonies in the population, and the growth rate of colonies is influenced by intrinsic and environmental stochasticities. Our setting thus captures the effect of an external noise, correlating the trait dynamics of all colonies. In particular, we study the effect of the strength of this correlation on the distribution of resources between colonies. We then extend this model to a large class of structured branching processes with interactions in which the particle type evolves according to a diffusion. The branching and death rates are general functions of the whole population. In this framework, we derive a $ψ$-spine construction and a Many-to-One formula, extending previous works on interacting branching processes. Using this spinal construction, we also propose an alternative simulation method and illustrate its efficiency on the colonial population model. The extended framework we propose can model various ecological systems with interactions, and individual and environmental noises.

math.PR↗

arXiv2025

An individual-based stochastic model reveals strong constraints on allometric relationships with minimal metabolic and ecological assumptions

We design a stochastic individual-based model structured in energy, for single species consuming an external resource, where populations are characterized by a typical energy at birth in $\mathbb{R}^{*}_{+}$. The resource is maintained at a fixed amount, so we benefit from a branching property at the population level. Thus, we focus on individual trajectories, constructed as Piecewise Deterministic Markov Processes, with random jumps modelling births and deaths in the population; and a continuous and deterministic evolution of energy between jumps. We are mainly interested in the case where metabolic (i.e. energy loss for maintenance), growth, birth and death rates depend on the individual energy over time, and follow allometric scalings (i.e. power laws). Our goal is to determine in a bottom-up approach what are the possible allometric coefficients (i.e. exponents of these power laws) under elementary -- and ecologically relevant -- constraints, for our model to be valid for the whole spectrum of possible body sizes. We show in particular that assuming an allometric coefficient $α$ related to metabolism strongly constrains the range of possible values for the allometric coefficients $β$, $δ$, $γ$, respectively related to birth, death and growth rates. We further identify and discuss the precise and minimal ecological mechanisms that are involved in these strong constraints on allometric scalings.

math.PR↗

arXiv2022

Continuous limits of large plant-pollinator random networks and some applications

We study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on the densities of pollinators. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interaction graph and the quantities of resources exchanged between species. Our model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, we show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.

math.PR↗

arXiv2021

Identifying conversion efficiency as a key mechanism underlying food webs evolution : a step forward, or backward ?

Body size or mass is one of the main factors underlying food webs structure. A large number of evolutionary models have shown that indeed, the adaptive evolution of body size (or mass) can give rise to hierarchically organised trophic levels with complex between and within trophic interactions. However, these models generally make strong arbitrary assumptions on how traits evolve, casting doubts on their robustness. In particular, biomass conversion efficiency is always considered independent of the predator and prey size, which contradicts with the literature. In this paper, we propose a general model encompassing most previous models which allows to show that relaxing arbitrary assumptions gives rise to unrealistic food webs. We then show that considering biomass conversion efficiency dependent on species size is certainly key for food webs adaptive evolution because realistic food webs can evolve, making obsolete the need of arbitrary constraints on traits' evolution. We finally conclude that, on the one hand, ecologists should pay attention to how biomass flows into food webs in models. On the other hand, we question more generally the robustness of evolutionary models for the study of food webs.

q-bio.PE↗

arXiv2020

Inference with selection, varying population size and evolving population structure: Application of ABC to a forward-backward coalescent process with interactions

Genetic data are often used to infer demographic history and changes or detect genes under selection. Inferential methods are commonly based on models making various strong assumptions: demography and population structures are supposed \textit{a priori} known, the evolution of the genetic composition of a population does not affect demography nor population structure, and there is no selection nor interaction between and within genetic strains. In this paper, we present a stochastic birth-death model with competitive interactions and asexual reproduction. We develop an inferential procedure for ecological, demographic and genetic parameters. We first show how genetic diversity and genealogies are related to birth and death rates, and to how individuals compete within and between strains. {This leads us to propose an original model of phylogenies, with trait structure and interactions, that allows multiple merging}. Second, we develop an Approximate Bayesian Computation framework to use our model for analyzing genetic data. We apply our procedure to simulated data from a toy model, and to real data by analyzing the genetic diversity of microsatellites on Y-chromosomes sampled from Central Asia human populations in order to test whether different social organizations show significantly different fertility.

q-bio.PE↗

arXiv2019

Convergence of knowledge in a cultural evolution model with population structure, random social learning and credibility biases

Understanding how knowledge is created and propagates within groups is crucial to explain how human populations have evolved through time. Anthropologists have relied on different theoretical models to address this question. In this work, we introduce a mathematically oriented model that shares properties with individual based approaches, inhomogeneous Markov chains and learning algorithms, such as those introduced in [F. Cucker, S. Smale, Bull. Amer. Math. Soc., 39 (1), 2002] and [F. Cucker, S. Smale and D.~X Zhou, Found. Comput. Math., 2004]. After deriving the model, we study some of its mathematical properties, and establish theoretical and quantitative results in a simplified case. Finally, we run numerical simulations to illustrate some properties of the model.

math.DS↗

arXiv2019

Stochastic dynamics of three competing clones: Conditions and times for invasion, coexistence and fixation

In large clonal populations, several clones generally compete which results in complex evolutionary and ecological dynamics: experiments show successive selective sweeps of favorable mutations as well as long-term coexistence of multiple clonal strains. The mechanisms underlying either coexistence or fixation of several competing strains have rarely been studied altogether. Conditions for coexistence have mostly been studied by population and community ecology, while rates of invasion and fixation have mostly been studied by population genetics. In order to provide a global understanding of the complexity of the dynamics observed in large clonal populations, we develop a stochastic model where three clones compete. Competitive interactions can be intransitive and we suppose that strains enter the population via mutations or rare immigrations. We first describe all possible final states of the population, including stable coexistence of two or three strains, or the fixation of a single strain. Second, we give estimate of the invasion and fixation times of a favorable mutant (or immigrant) entering the population in a single copy. We show that invasion and fixation can be slower or faster when considering complex competitive interactions. Third, we explore the parameter space assuming prior distributions of reproduction, death and competitive rates and we estimate the likelihood of the possible dynamics. We show that when mutations can affect competitive interactions, even slightly, stable coexistence is likely. We discuss our results in the context of the evolutionary dynamics of large clonal populations.

q-bio.PE↗

arXiv2017

Stochastic dynamics for adaptation and evolution of microorganisms

We present a model for the dynamics of a population of bacteria with a continuum of traits, who compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait with possible mutations. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We consider a stochastic individual-based pure jump process taking values in the space of point measures, and whose jump events describe the individual reproduction, transfer and death mechanisms. In a large population scale, the stochastic process is proved to converge to the solution of a nonlinear integro-differential equation. When there are only two different traits and no mutation, this equation reduces to a non-standard two-dimensional dynamical system. We show how crucial the forms of the transfer rates are for the long-term behavior of its solutions. We describe the dynamics of invasion and fixation when one of the two traits is initially rare, and compute the invasion probabilities. Then, we study the process under the assumption of rare mutations. We prove that the stochastic process at the mutation time scale converges to a jump process which describes the successive invasions of successful mutants. We show that the horizontal transfer can have a major impact on the distribution of the successive mutational fixations, leading to dramatically different behaviors, from expected evolution scenarios to evolutionary suicide. Simulations are given to illustrate these phenomena.

math.PR↗

arXiv2016

Perennial life-histories and demographic advantages may play contradictory roles in the evolution of plant mating systems

When predicting the fate and consequences of recurring deleterious mutations in self-fertilising populations most models developed make the assumption that populations have discrete non-overlapping generations. This makes them biologically irrelevant when considering perennial species with over-lapping generations and where mating occurs independently of the age group. Previous models studying the effect of perennial life-histories on the genetic properties of populations in the presence of self-fertilisation have done so considering age-dependent selection and have found that, contrary to empirical observations, perennial populations should exhibit lower levels of inbreeding depression. Here we propose a simple deterministic model in continuous time with selection at different fitness traits and feedback between population fitness and size. We find that a perennial life-history can result in high levels of inbreeding depression in spite of inbreeding, due to higher frequencies of heterozygous individuals at the adult stage. We also propose that there may be demographic advantages for self-fertilisation that are independent of reproductive success.

q-bio.PE↗

arXiv2015

The effect of competition and horizontal trait inheritance on invasion, fixation and polymorphism

Horizontal transfer (HT) of heritable information or `traits' (carried by genetic elements, endosymbionts, or culture) is widespread among living organisms. Yet current ecological and evolutionary theory addressing HT is limited. We present a modeling framework for the dynamics of two populations that compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We capture this feedback with a stochastic individual-based model, from which we derive a deterministic approximation for large populations. The interaction between horizontal transfer and competition makes possible the stable (or bi-stable) polymorphic maintenance of deleterious traits (including costly plasmids). When transfer rates are of a general density-dependent form, transfer stochasticity contributes strongly to population fluctuations. For an initially rare trait, we describe the probabilistic dynamics of invasion and fixation. Acceleration of fixation by HT is faster when competition is weak in the resident population. Thus, HT can have a major impact on the distribution of mutational effects that are fixed, and our model provides a basis for a general theory of the influence of HT on eco-evolutionary dynamics and adaptation.

q-bio.PE↗

arXiv2015

The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference

Clonal interference, competition between multiple co-occurring beneficial mutations, has a major role in adaptation of asexual populations. We provide a simple individual based stochastic model of clonal interference taking into account a wide variety of competitive interactions which can be found in nature. In particular, we relax the classical assumption of transitivity between the different mutations. It allows us to predict genetic patterns, such as coexistence of several mutants or emergence of Rock-Paper-Scissors cycles, which were not explained by existing models. In addition, we call into questions some classical preconceived ideas about fixation time and fixation probability of competing mutations.

math.PR↗

arXiv2014

Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks

How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. Population dynamics are driven by births and deaths, mutations at birth, and competition between individuals. Trait values influence ecological processes (demographic events, competition), and competition generates selection on trait variation, thus closing the eco-evolutionary feedback loop. The demographic effects of the trait are also expected to influence the generation and maintenance of neutral variation. We consider a large population limit with rare mutation, under the assumption that the neutral marker mutates faster than the trait under selection. We prove the convergence of the stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming-Viot Process (SFVP). When restricted to the trait space this process is the Trait Substitution Sequence first introduced by Metz et al. (1996). During the invasion of a favorable mutation, a genetical bottleneck occurs and the marker associated with this favorable mutant is hitchhiked. By rigorously analysing the hitchhiking effect and how the neutral diversity is restored afterwards, we obtain the condition for a time-scale separation; under this condition, we show that the marker distribution is approximated by a Fleming-Viot distribution between two trait substitutions. We discuss the implications of the SFVP for our understanding of the dynamics of neutral variation under eco-evolutionary feedbacks and illustrate the main phenomena with simulations. Our results highlight the joint importance of mutations, ecological parameters, and trait values in the restoration of neutral diversity after a selective sweep.

math.PR↗

arXiv2011

A general stochastic model for sporophytic self-incompatibility

Disentangling the processes leading populations to extinction is a major topic in ecology and conservation biology. The difficulty to find a mate in many species is one of these processes. Here, we investigate the impact of self-incompatibility in flowering plants, where several inter-compatible classes of individuals exist but individuals of the same class cannot mate. We model pollen limitation through different relationships between mate availability and fertilization success. After deriving a general stochastic model, we focus on the simple case of distylous plant species where only two classes of individuals exist. We first study the dynamics of such a species in a large population limit and then, we look for an approximation of the extinction probability in small populations. This leads us to consider inhomogeneous random walks on the positive quadrant. We compare the dynamics of distylous species to self-fertile species with and without inbreeding depression, to obtain the conditions under which self-incompatible species could be less sensitive to extinction while they can suffer more pollen limitation.

q-bio.PE↗