191 research outputs found

### 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

### Branching Feller diffusion for cell division with parasite infection

We describe the evolution of the quantity of parasites in a population of
cells which divide in continuous-time. The quantity of parasites in a cell
follows a Feller diffusion, which is splitted randomly between the two daughter
cells when a division occurs. The cell division rate may depend on the quantity
of parasites inside the cell and we are interested in the cases of constant or
monotone division rate. We first determine the asymptotic behavior of the
quantity of parasites in a cell line, which follows a Feller diffusion with
multiplicative jumps. We then consider the evolution of the infection of the
cell population and give criteria to determine whether the proportion of
infected cells goes to zero (recovery) or if a positive proportion of cells
becomes largely infected (proliferation of parasites inside the cells)

### 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.Comment: 29 page

### Slow and fast scales for superprocess limits of age-structured populations

International audienceA superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence

### The 2d-Directed Spanning Forest is almost surely a tree

6 figuresInternational audienceWe consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF

### Nonlinear historical superprocess approximations for population models with past dependence

We are interested in the evolving genealogy of a birth and death process with
trait structure and ecological interactions. Traits are hereditarily
transmitted from a parent to its offspring unless a mutation occurs. The
dynamics may depend on the trait of the ancestors and on its past and allows
interactions between individuals through their lineages. We define an
interacting historical particle process describing the genealogies of the
living individuals; it takes values in the space of point measures on an
infinite dimensional c\`adl\`ag path space. This individual-based process can
be approximated by a nonlinear historical superprocess, under the assumptions
of large populations, small individuals and allometric demographies. Because of
the interactions, the branching property fails and we use martingale problems
and fine couplings between our population and independent branching particles.
Our convergence theorem is illustrated by two examples of current interest in
biology. The first one relates the biodiversity history of a population and its
phylogeny, while the second treats a spatial model with competition between
individuals through their past trajectories.Comment: 31 page

### The 2d-directed spanning forest converges to the Brownian web

The two-dimensional directed spanning forest (DSF) introduced by Baccelli and
Bordenave is a planar directed forest whose vertex set is given by a
homogeneous Poisson point process $\mathcal{N}$ on $\mathbb{R}^2$. If the DSF
has direction $-e_y$, the ancestor $h(u)$ of a vertex $u \in \mathcal{N}$ is
the nearest Poisson point (in the $L_2$ distance) having strictly larger
$y$-coordinate. This construction induces complex geometrical dependencies. In
this paper we show that the collection of DSF paths, properly scaled, converges
in distribution to the Brownian web (BW). This verifies a conjecture made by
Baccelli and Bordenave in 2007

### Large graph limit for an SIR process in random network with heterogeneous connectivity

We consider an SIR epidemic model propagating on a configuration model
network, where the degree distribution of the vertices is given and where the
edges are randomly matched. The evolution of the epidemic is summed up into
three measure-valued equations that describe the degrees of the susceptible
individuals and the number of edges from an infectious or removed individual to
the set of susceptibles. These three degree distributions are sufficient to
describe the course of the disease. The limit in large population is
investigated. As a corollary, this provides a rigorous proof of the equations
obtained by Volz [Mathematical Biology 56 (2008) 293--310].Comment: Published in at http://dx.doi.org/10.1214/11-AAP773 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

We study the evolution of a particle system whose genealogy is given by a
supercritical continuous time Galton--Watson tree. The particles move
independently according to a Markov process and when a branching event occurs,
the offspring locations depend on the position of the mother and the number of
offspring. We prove a law of large numbers for the empirical measure of
individuals alive at time t. This relies on a probabilistic interpretation of
its intensity by mean of an auxiliary process. The latter has the same
generator as the Markov process along the branches plus additional jumps,
associated with branching events of accelerated rate and biased distribution.
This comes from the fact that choosing an individual uniformly at time t favors
lineages with more branching events and larger offspring number. The central
limit theorem is considered on a special case. Several examples are developed,
including applications to splitting diffusions, cellular aging, branching
L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP757 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

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