Two level natural selection with a quasi-stationarity approach

In a view for a simple model where natural selection at the individual level is confronted to selection effects at the group level, we consider some individual-based models of some large population subdivided into a large number of groups. We then obtain the convergence to the law of a stochastic process with some Feynman-Kac penalization. To analyze the limiting behavior of this law, we exploit a recent approach, designed for the convergence to quasi-stationary distributions. We are able to deal with the fixation of the stochastic process and relate the convergence to equilibrium to the one where fixation implies extinction. We notably establish different regimes of convergence. Besides the case of an exponential rate (the rate being uniform over the initial condition), critical regimes with convergence in 1/t are also to notice. We finally address the relevance of such limiting behaviors to predict the long-time behavior of the individual-based model and describe more specifically the cases of weak selection. Consequences in term of evolutive dynamics are also derived, where such competition is assumed to occur repeatedly at each de novo mutation.Comment: 38 pages, 13 figure

Exponential quasi-ergodicity for processes with discontinuous trajectories

This paper establishes exponential convergence to a unique quasi-stationary distribution in the total variation norm for a very general class of strong Markov processes. Specifically, we can treat nonreversible processes with discontinuous trajectories, which seems to be a substantial breakthrough. Considering jumps driven by Poisson Point Processes in two different applications, we intend to illustrate the potential of these results and motivate our criteria. Our set of conditions is expected to be much easier to verify than an implied property which is crucial in our proof, namely a comparison of asymptotic extinction rates between different initial conditions. Keywords : continuous-time and continuous-space Markov process , jumps , quasi-stationary distribution , survival capacity , Q-process , Harris recurrenc

Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned to never be absorbed. The technique relies on a coupling procedure that is related to Doeblin's type conditions. The main novelty is that we modulate each coupling step depending both on a final horizon of time --for survival-- and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment

Individual-based models under various time-scales

This article is a presentation of specific recent results describing scaling limits of individual-based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the individual-based models. Although these results are by no means exhaustive, both on the mathematical and the biological level, they complement each other. Indeed, they provide a viewpoint for many classical time-scales. Namely, they encompass the timescale typical of the life-expectancy of a single individual, the longer one wherein a population can be characterized through its demographic dynamics, and at least four interconnected ones wherein selection occurs. The limiting behavior is generally deterministic. Yet, since there are selective effects on randomness in the history of lineages, probability theory is shown to be a key factor in understanding the results. Besides, randomness can be maintained in the limiting dynamics, for instance to model rare mutations fixing in the population.Comment: Review on several papers based on the presentations given during the session Stochastic Processes and Biology at the conference Journ\'{e}es MAS 2018 (Mod\'{e}lisation Al\'{e}atoire et Stochastiques). arXiv admin note: text overlap with arXiv:1310.6274, arXiv:1711.10732, arXiv:1505.02421, arXiv:1507.00397 by other authors. additional note of the author : also arXiv:1903.1016

Individual based SIS models on (not so) dense large random networks

Starting from a stochastic individual-based description of an SIS epidemic spreading on a random network, we study the dynamics when the size n of the network tends to infinity. We recover in the limit an infinite-dimensional integro-differential equation studied by Delmas, Dronnier and Zitt (2022) for an SIS epidemic propagating on a graphon. Our work covers the case of dense and sparse graphs, provided that the number of edges grows faster than n, but not the case of very sparse graphs with O(n) edges. In order to establish our limit theorem, we have to deal with both the convergence of the random graphs to the graphon and the convergence of the stochastic process spreading on top of these random structures: in particular, we propose a coupling between the process of interest and an epidemic that spreads on the complete graph but with a modified infection rate. Keywords: random graph, mathematical models of epidemics, measure-valued process, large network limit, limit theorem, graphon.Comment: Acknowledgments: This work was financed by the Labex B\'ezout (ANR-10-LABX-58) and the COCOON grant (ANR-22-CE48-0011), and by the platform MODCOV19 of the National Institute of Mathematical Sciences and their Interactions of CNRS. 32 pages, including an appendix of 6 page

Exponential quasi-ergodicity for processes with discontinuous trajectories

This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack’s inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property that we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on ℝd that do not satisfy the classical Harnack’s inequality, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps

Individual-based models under various time-scales

This article is a presentation of specific recent results describing scaling limits of individual- based models. Thanks to them, we wish to relate the time-scales typical of demographic dynamics and natural selection to the parameters of the individual-based models. Although these results are by no means exhaustive, both on the mathematical and the biological level, they complement each other. Indeed, they provide a viewpoint for many classical time-scales. Namely, they encompass the timescale typical of the life-expectancy of a single individual, the longer one wherein a population can be characterized through its demographic dynamics, and at least four interconnected ones wherein selection occurs. The limiting behavior is generally deterministic. Yet, since there are selective effects on randomness in the history of lineages, probability theory is shown to be a key factor in understanding the results. Besides, randomness can be maintained in the limiting dynamics, for instance to model rare mutations fixing in the population

Unique quasi-stationary distribution, with a possibly stabilizing extinction

International audienceWe establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditionned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.Nous établissons des conditions suffisantes pour une convergence exponentielle vers une unique distribution quasi-stationnaire en norme de variation totale. Ces conditions assurent également l'existence et l'ergodicité exponentielle du Q-processus, le processus conditionné par le fait de ne jamais être absorbé. La technique repose sur une procédure de couplage qui est liée à la récurrence de Harris (pour les chaînes de Markov). Elle s'applique aux processus généraux de Markov à temps continu et à espace continu. La principale nouveauté est que nous ajustons chaque étape de couplage en fonction à la fois d'un horizon de temps final (pour la survie) et de la distribution initiale. De cette façon, nous pouvons notamment inclure dans la convergence une dépendance à la condition initiale. A titre d'illustration, nous considérons un processus de naissance-mort en temps continu avec catastrophes et un processus de diffusion décrivant une population (localisée) s'adaptant à son environnement