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Michel Benaïm

Publications and source records attributed to Michel Benaïm.

At least 19 recordsLinked to original sources
arXiv2026

Sufficient condition for dispersal-induced growth on dynamic networks

We consider a population spreading across a finite number of sites. Individuals can move from one site to the other according to a network (oriented links between the sites) that vary periodically over time. On each site, the population experiences a growth rate which is also periodically time varying. Recently, this kind of models have been extensively studied, using various technical tools to derive precise necessary and sufficient conditions on the parameters of the system (ie the local growth rate on each site, the time period and the strength of migration between the sites) for the population to grow. In the present paper, we take a completely different approach: using elementary comparison results between linear systems, we give sufficient condition for the growth of the population This condition is easy to check and can be applied in a broad class of examples. In particular, in the case when all sites are sinks (ie, in the absence of migration, the population become extinct in each site), we prove that when our condition of growth if satisfied, the population grows when the time period is large and for values of the migration strength that are exponentially small with respect to the time period, which answers positively to a conjecture stated by Katriel.

math.DS↗
arXiv2026

Selection principles for quasi-stationary distributions and reinforcement processes

Let $P$ be a sub-Markov matrix on a finite set $S$, representing the transition probabilities of a Markov chain on \(S\) absorbed at a cemetery point $\partial\notin S$. We consider a reinforced process \((X_n,μ_n)\) defined as follows: $(X_n)$ behaves like a chain with kernel $P$ until it dies, and when it dies at time $n$, it is instantaneously ``resurrected'' at a point sampled according to its weighted past occupation measure $$ μ_n = \frac{1}{W_n} \left( w_0μ_0+\sum_{k=1}^n w_kδ_{X_k} \right), \qquad W_n=\sum_{k=0}^n w_k, $$ where the positive weights $w_k$ satisfy certain technical assumptions, a typical example being given by $w_k = k^q$, with $q\geq -1$. When $P$ is irreducible, the behaviour of $(μ_n)$ is well understood \cite{AFP}, \cite{bansaye2022non}: it converges almost surely toward the unique quasi-stationary distribution (QSD) of $P$. The purpose of this paper is to investigate the general situation where $P$ is not irreducible. Under generic assumptions on $P$, there are finitely many QSDs. We prove that the asymptotic selection depends on the summability of the inverse cumulative weights $1/W_n$. If $$ \sum_{n\geq 0}\frac1{W_n}=\infty, $$ then $(μ_n)$ almost surely converges toward the QSD associated with the largest Perron value. If instead $$ \sum_{n\geq0}\frac1{W_n}<\infty, $$ then each QSD is selected with positive probability. In particular, for polynomial weights $w_0=1$ and $w_k=k^q$, $k\geq1$, this gives almost sure selection of the QSD with largest Perron value for $-1\leq q\leq 0$, whereas each quasi-stationary distribution is selected with positive probability for $q>0$.

math.PR↗
arXiv2026

On the occupation measure of evolution models with vanishing mutations

We study the almost sure convergence of the occupation measure of evolution models where mutation rates decrease over time. We show that if the mutation parameter vanishes at a controlled rate, then the empirical occupation measure converges almost surely to a specific invariant distribution of a limiting Markov chain. Our results are obtained through the analysis of a larger class of time-inhomogeneous Markov chains with finite state space, where the control on the mutation parameter is explained by the energy barrier of the limit process. Additionally, we derive an explicit convergence rate, explained through the tree-optimality gap, that may be of independent interest.

math.PR↗
arXiv2025

Stochastic persistence and extinction for degenerate stochastic Rosenzweig-MacArthur model

We consider the classical two-dimensional Rosenzweig-MacArthur prey-predator model with a degenerate noise, whereby only the prey variable is subject to small environmental fluctuations. This model has already been introduced in arXiv:1806.08450 and partially investigated by exhibiting conditions ensuring persistence. In this paper, we extend the results to study the conditions for persistence, the uniqueness of an invariant probability measure supported on the interior of $\mathbb R^2_+$ with a smooth density, and convergence in Total variation at a polynomial rate. Our contribution lies in providing a convergence rate in the case of persistence, as well as detailing the situations involving the extinction of one or both species. We also specify all the proofs of the intermediary results supporting our conclusions that are lacking in arXiv:1806.08450.

math.PR↗
arXiv2024

On invariant distributions of Feller Markov chains with applications to dynamical systems with random switching

We introduce simple conditions ensuring that invariant distributions of a Feller Markov chain on a compact Riemannian manifold are absolutely continuous with a lower semi-continuous, continuous or smooth density with respect to the Riemannian measure. This is applied to Markov chains obtained by random composition of maps and to piecewise deterministic Markov processes obtained by random switching between flows.

math.PR↗
arXiv2024

Degenerate processes killed at the boundary of a domain

We investigate certain properties of degenerate Feller processes that are killed when exiting a relatively compact set. Our main result provides general conditions ensuring that such a process possesses a (possibly non unique) quasi stationary distribution. Conditions ensuring uniqueness and exponential convergence are discussed. The results are applied to nonelliptic and hypoelliptic stochastic differential equations.

math.PR↗
arXiv2024

The asymptotic behavior of fraudulent algorithms

Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \geq 2,$ satisfying $\min U=0$ and let $\mathcal{U} = \{x \in M \: : U(x) = 0\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(β)}:=(X^{(β)}(t))_{t\geq 0}$ defined on $N = M \setminus \mathcal{U},$ whose generator is$U Δ\cdot-β\langle \nabla U,\nabla \cdot\rangle$, where $β\in\RR$ is a real parameter.We show that for $β>\frac{m}{2}-1,$ $X^{(β)}(t)$ converges a.s.\ as $t \rightarrow \infty$, toward a point $p \in \mathcal{U}$ and that each $p \in \mathcal{U}$ has a positive probability to be selected. On the other hand, for $β< \frac{m}{2}-1,$ the law of $(X^{(β)}(t))$ converges in total variation (at an exponential rate) toward the probability measure $π_β$ having density proportional to $U(x)^{-1-β}$ with respect to the Riemannian measure.

math.PR↗
arXiv2023

A note on the top Lyapunov exponent of linear cooperative systems

In a recent paper [Asymptotic of the largest Floquet multiplier for cooperative matrices Annales de la Faculté des Sciences de Toulouse, Tome XXXI, no 4 (2022)] P. Carmona gives an asymptotic formulae for the top Lyapunov exponent of a linear T-periodic cooperative differential equation, in the limit T goes to infinity. This short note discusses and extends this result.

math.DS↗
arXiv2023

Untangling the role of temporal and spatial variations in persistance of populations

We consider a population distributed between two habitats, in each of which it experiences a growth rate that switches periodically between two values, $1- \varepsilon > 0$ or $ - (1 + \varepsilon) < 0$. We study the specific case where the growth rate is positive in one habitat and negative in the other one for the first half of the period, and conversely for the second half of the period, that we refer as the $(\pm 1)$ model. In the absence of migration, the population goes to $0$ exponentially fast in each environment. In this paper, we show that, when the period is sufficiently large, a small dispersal between the two patches is able to produce a very high positive exponential growth rate for the whole population, a phenomena called inflation. We prove in particular that the threshold of the dispersal rate at which the inflation appears is exponentially small with the period. We show that inflation is robust to random perturbation, by considering a model where the values of the growth rate in each patch are switched at random times: we prove, using theory of Piecewise Deterministic Markov Processes (PDMP) that inflation occurs for low switching rate and small dispersal. Finally, we provide some extensions to more complicated models, especially epidemiological and density dependent models.

math.DS↗
arXiv2023

Regularity of the stationary density for systems with fast random switching

We consider the piecewise-deterministic Markov process obtained by randomly switching between the flows generated by a finite set of smooth vector fields on a compact set. We obtain Hörmander-type conditions on the vector fields guaranteeing that the stationary density is: $C^k$ whenever the jump rates are sufficiently fast, for any $k<\infty$; unbounded whenever the jump rates are sufficiently slow and lower semi-continuous regardless of the jump rates. Our proofs are probabilistic, relying on a novel application of stopping times.

math.PR↗
arXiv2021

Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

We consider a Lotka-Volterra food chain model with possibly intra-specific competition in a stochastic environment represented by stochastic differential equations. In the non-degenerate setting, this model has already been studied by A. Hening and D. Nguyen. They provided conditions for stochastic persistence and extinction. In this paper, we extend their results to the degenerate situation in which the top or the bottom species is subject to random perturbations. Under the persistence condition, there exists a unique invariant probability measure supported by the interior of $\mathbb{R}_+^n$ having a smooth density. Moreover, we study a more general model, in which we give new conditions which make it possible to characterise the convergence of the semi-group towards the unique invariant probability measure either at an exponential rate or at a polynomial one. This will be used in the stochastic Lotka-Volterra food chain to see that if intra-specific competition occurs for all species, the rate of convergence is exponential while in the other cases it is polynomial.

math.PR↗
arXiv2021

Stochastic approximation of quasi-stationary distributions for diffusion processes in a bounded domain

We study a random process with reinforcement, which evolves following the dynamics of a given diffusion process in a bounded domain and is resampled according to its occupation measure when it reaches the boundary. We show that its occupation measure converges to the unique quasi-stationary distribution of the diffusion process absorbed at the boundary of the domain. Our proofs use recent results in the theory of quasi-stationary distributions and stochastic approximation techniques.

math.PR↗
arXiv2019

Analysis of an Adaptive Biasing Force method based on self-interacting dynamics

This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the bias in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the bias depends on the past of the trajectory and is designed to approximate the free energy. The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the bias as time goes to infinity, and that the limit is close to the ideal bias, as an auxiliary parameter of the algorithm goes to $0$. The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.

math.PR↗
arXiv2019

Persistence and extinction for stochastic ecological models with internal and external variables

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. We prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight challenges for future research.

q-bio.PE↗
arXiv2018

Random Switching between Vector Fields Having a Common Zero

Let $E$ be a finite set, $\{F^i\}_{i \in E}$ a family of vector fields on $\mathbb{R}^d$ leaving positively invariant a compact set $M$ and having a common zero $p \in M.$ We consider a piecewise deterministic Markov process $(X,I)$ on $M \times E$ defined by $\dot{X}_t = F^{I_t}(X_t)$ where $I$ is a jump process controlled by $X:$ $\Pr(I_{t+s} = j | (X_u, I_u)_{u \leq t}) = a_{i j}(X_t) s + o(s)$ for $i \neq j$ on $\{I_t = i \}.$ We show that the behavior of $(X,I)$ is mainly determined by the behavior of the linearized process $(Y,J)$ where $\dot{Y}_t = A^{J_t} Y_t,$ $A^i$ is the Jacobian matrix of $F^i$ at $p$ and $J$ is the jump process with rates $(a_{ij}(p)).$ We introduce two quantities $Λ^-$ and $Λ^+$ respectively %called the {\em minimal} and {\em maximal average growth rate.} $Λ^-$ (respectively $Λ^+$) is defined as the {\em minimal} (respectively {\em maximal}) {\em growth rate} of $\|Y_t\|,$ where the minimum (respectively maximum) is taken over all the ergodic measures of the angular process $(Θ, J)$ with $Θ_t = \frac{Y_t}{\|Y_t\|}.$ It is shown that $Λ^+$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions $Λ^- = Λ^+.$ We then prove that, under certain irreducibility conditions, $X_t \to p$ exponentially fast when $Λ^+ < 0$ and $(X,I)$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $M \setminus \{p\} \times E$ when $Λ^- > 0.$ Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.

math.PR↗
arXiv2018

A user-friendly condition for exponential ergodicity in randomly switched environments

We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hörmander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof of Li, Liu and Cui and using results of Benaïm, Le Borgne, Malrieu and Zitt, the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.

math.PR↗
arXiv2018

Convergence analysis of Adaptive Biasing Potential methods for diffusion processes

This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages. We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results , we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context.

math.PR↗