147 research outputs found
A microscopic interpretation for adaptive dynamics trait substitution sequence models
We consider an interacting particle Markov process for Darwinian evolution in
an asexual population with non-constant population size, involving a linear
birth rate, a density-dependent logistic death rate, and a probability of
mutation at each birth event. We introduce a renormalization parameter
scaling the size of the population, which leads, when , to a
deterministic dynamics for the density of individuals holding a given trait. By
combining in a non-standard way the limits of large population ()
and of small mutations (), we prove that a time scales separation
between the birth and death events and the mutation events occurs and that the
interacting particle microscopic process converges for finite dimensional
distributions to the biological model of evolution known as the ``monomorphic
trait substitution sequence'' model of adaptive dynamics, which describes the
Darwinian evolution in an asexual population as a Markov jump process in the
trait space
Large deviations for singular and degenerate diffusion models in adaptive evolution
In the course of Darwinian evolution of a population, punctualism is an
important phenomenon whereby long periods of genetic stasis alternate with
short periods of rapid evolutionary change. This paper provides a mathematical
interpretation of punctualism as a sequence of change of basin of attraction
for a diffusion model of the theory of adaptive dynamics. Such results rely on
large deviation estimates for the diffusion process. The main difficulty lies
in the fact that this diffusion process has degenerate and non-Lipschitz
diffusion part at isolated points of the space and non-continuous drift part at
the same points. Nevertheless, we are able to prove strong existence and the
strong Markov property for these diffusions, and to give conditions under which
pathwise uniqueness holds. Next, we prove a large deviation principle involving
a rate function which has not the standard form of diffusions with small noise,
due to the specific singularities of the model. Finally, this result is used to
obtain asymptotic estimates for the time needed to exit an attracting domain,
and to identify the points where this exit is more likely to occur
Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates
This article studies the quasi-stationary behaviour of multidimensional birth
and death processes, modeling the interaction between several species, absorbed
when one of the coordinates hits 0. We study models where the absorption rate
is not uniformly bounded, contrary to most of the previous works. To handle
this natural situation, we develop original Lyapunov function arguments that
might apply in other situations with unbounded killing rates. We obtain the
exponential convergence in total variation of the conditional distributions to
a unique stationary distribution, uniformly with respect to the initial
distribution. Our results cover general birth and death models with stronger
intra-specific than inter-specific competition, and cases with neutral
competition with explicit conditions on the dimension of the process.Comment: 18 page
Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing
This article studies the quasi-stationary behaviour of absorbed
one-dimensional diffusion processes with killing on . We obtain
criteria for the exponential convergence to a unique quasi-stationary
distribution in total variation, uniformly with respect to the initial
distribution. Our approach is based on probabilistic and coupling methods,
contrary to the classical approach based on spectral theory results. Our
general criteria apply in the case where is entrance and 0 either
regular or exit, and are proved to be satisfied under several explicit
assumptions expressed only in terms of the speed and killing measures. We also
obtain exponential ergodicity results on the -process. We provide several
examples and extensions, including diffusions with singular speed and killing
measures, general models of population dynamics, drifted Brownian motions and
some one-dimensional processes with jumps.Comment: arXiv admin note: text overlap with arXiv:1506.0238
Adaptive dynamics in logistic branching populations
We consider a trait-structured population subject to mutation, birth and
competition of logistic type, where the number of coexisting types may
fluctuate. Applying a limit of rare mutations to this population while keeping
the population size finite leads to a jump process, the so-called `trait
substitution sequence', where evolution proceeds by successive invasions and
fixations of mutant types. The probability of fixation of a mutant is
interpreted as a fitness landscape that depends on the current state of the
population. It was in adaptive dynamics that this kind of model was first
invented and studied, under the additional assumption of large population.
Assuming also small mutation steps, adaptive dynamics' theory provides a
deterministic ODE approximating the evolutionary dynamics of the dominant trait
of the population, called `canonical equation of adaptive dynamics'. In this
work, we want to include genetic drift in this models by keeping the population
finite. Rescaling mutation steps (weak selection) yields in this case a
diffusion on the trait space that we call `canonical diffusion of adaptive
dynamics', in which genetic drift (diffusive term) is combined with directional
selection (deterministic term) driven by the fitness gradient. Finally, in
order to compute the coefficients of this diffusion, we seek explicit
first-order formulae for the probability of fixation of a nearly neutral mutant
appearing in a resident population. These formulae are expressed in terms of
`invasibility coefficients' associated with fertility, defense, aggressiveness
and isolation, which measure the robustness (stability w.r.t. selective
strengths) of the resident type. Some numerical results on the canonical
diffusion are also given
Splitting trees with neutral Poissonian mutations I: Small families
We consider a neutral dynamical model of biological diversity, where
individuals live and reproduce independently. They have i.i.d. lifetime
durations (which are not necessarily exponentially distributed) and give birth
(singly) at constant rate b. Such a genealogical tree is usually called a
splitting tree, and the population counting process (N_t;t\ge 0) is a
homogeneous, binary Crump--Mode--Jagers process. We assume that individuals
independently experience mutations at constant rate \theta during their
lifetimes, under the infinite-alleles assumption: each mutation instantaneously
confers a brand new type, called allele, to its carrier. We are interested in
the allele frequency spectrum at time t, i.e., the number A(t) of distinct
alleles represented in the population at time t, and more specifically, the
numbers A(k,t) of alleles represented by k individuals at time t,
k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes
and branching processes counted by random characteristics. We provide explicit
formulae for the expectation of A(k,t) in a coalescent point process
conditional on population size, which apply to the special case of splitting
trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t
thanks to random characteristics. Last, we separately compute the expected
homozygosity by applying a method characterizing the dynamics of the tree
distribution as the origination time of the tree moves back in time, in the
spirit of backward Kolmogorov equations.Comment: 32 pages, 2 figures. Companion paper in preparation "Splitting trees
with neutral Poissonian mutations II: Large or old families
Evolution of discrete populations and the canonical diffusion of adaptive dynamics
The biological theory of adaptive dynamics proposes a description of the
long-term evolution of a structured asexual population. It is based on the
assumptions of large population, rare mutations and small mutation steps, that
lead to a deterministic ODE describing the evolution of the dominant type,
called the ``canonical equation of adaptive dynamics.'' Here, in order to
include the effect of stochasticity (genetic drift), we consider self-regulated
randomly fluctuating populations subject to mutation, so that the number of
coexisting types may fluctuate. We apply a limit of rare mutations to these
populations, while keeping the population size finite. This leads to a jump
process, the so-called ``trait substitution sequence,'' where evolution
proceeds by successive invasions and fixations of mutant types. Then we apply a
limit of small mutation steps (weak selection) to this jump process, that leads
to a diffusion process that we call the ``canonical diffusion of adaptive
dynamics,'' in which genetic drift is combined with directional selection
driven by the gradient of the fixation probability, also interpreted as an
invasion fitness. Finally, we study in detail the particular case of multitype
logistic branching populations and seek explicit formulae for the invasion
fitness of a mutant deviating slightly from the resident type. In particular,
second-order terms of the fixation probability are products of functions of the
initial mutant frequency, times functions of the initial total population size,
called the invasibility coefficients of the resident by increased fertility,
defence, aggressiveness, isolation or survival.Comment: Published at http://dx.doi.org/10.1214/105051606000000628 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions
A multitype Dawson-Watanabe process is conditioned, in subcritical and
critical cases, on non-extinction in the remote future. On every finite time
interval, its distribution is absolutely continuous with respect to the law of
the unconditioned process. A martingale problem characterization is also given.
Several results on the long time behavior of the conditioned mass process|the
conditioned multitype Feller branching diffusion are then proved. The general
case is first considered, where the mutation matrix which models the
interaction between the types, is irreducible. Several two-type models with
decomposable mutation matrices are analyzed too
- …