@article{cattiaux_quasi-stationary_2009,
Author = {Cattiaux, Patrick and Collet, Pierre and Lambert,
Amaury and Martínez, Servet and Méléard,
Sylvie and Martín, Jaime San},
Title = {Quasi-stationary distributions and diffusion models in
population dynamics},
Journal = {The Annals of Probability},
Volume = {37},
Pages = {1926--1969},
abstract = {In this paper we study quasi-stationarity for a large
class of Kolmogorov diffusions. The main novelty here
is that we allow the drift to go to
\$-\backslash{}infty\$ at the origin, and the diffusion
to have an entrance boundary at \$+\backslash{}infty\$.
These diffusions arise as images, by a deterministic
map, of generalized Feller diffusions, which themselves
are obtained as limits of rescaled birth-death
processes. Generalized Feller diffusions take
nonnegative values and are absorbed at zero in finite
time with probability 1. An important example is the
logistic Feller diffusion. We give sufficient
conditions on the drift near 0 and near
\$+\backslash{}infty\$ for the existence of
quasi-stationary distributions, as well as rate of
convergence in the Yaglom limit and existence of the
Q-process. We also show that, under these conditions,
there is exactly one quasi-stationary distribution, and
that this distribution attracts all initial
distributions under the conditional evolution, if and
only if \$+\backslash{}infty\$ is an entrance boundary.
In particular, this gives a sufficient condition for
the uniqueness of quasi-stationary distributions. In
the proofs spectral theory plays an important role on
\$L^2\$ of the reference measure for the killed
process. },
doi = {10.1214/09-AOP451},
year = 2009
}

@InProceedings{champagnat_adaptive_2008,
Author = {Champagnat, Nicolas and Lambert, Amaury},
Title = {Adaptive dynamics in logistic branching populations},
BookTitle = {Banach Center Publications},
Volume = {80},
Pages = {235--235},
Publisher = {Institute of Mathematics Polish Academy of Sciences},
abstract = {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. },
doi = {10.4064/bc80-0-14},
language = {en},
url = {http://journals.impan.pl/cgi-bin/doi?bc80-0-14},
urldate = {2014-08-31},
year = 2008
}