Abstract:
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.

Abstract:
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 $\mu$ of mutation at each birth event. We introduce a renormalization parameter $K$ scaling the size of the population, which leads, when $K\to+\infty$, 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 ($K\to+\infty$) and of small mutations ($\mu\to 0$), 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.

Abstract:
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.

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.

Abstract:
We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate $\theta$ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn. We are interested in the sizes and ages at time $t$ of the clonal families carrying the most abundant alleles or the oldest ones, as $t\to\infty$, on the survival event. Intuitively, the results must depend on how the mutation rate $\theta$ and the Malthusian parameter $\alpha>0$ compare. Hereafter, $N\equiv N_t$ is the population size at time $t$, constants $a,c$ are scaling constants, whereas $k,k'$ are explicit positive constants which depend on the parameters of the model. When $\alpha>\theta$, the most abundant families are also the oldest ones, they have size $cN^{1-\theta/\alpha}$ and age $t-a$. When $\alpha<\theta$, the oldest families have age $(\alpha /\theta)t+a$ and tight sizes; the most abundant families have sizes $k\log(N)-k'\log\log(N)+c$ and all have age $(\theta-\alpha)^{-1}\log(t)$. When $\alpha=\theta$, the oldest families have age $kt-k'\log(t)+a$ and tight sizes; the most abundant families have sizes $(k\log(N)-k'\log\log(N)+c)^2$ and all have age $t/2$. Those informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes, we are also able, when $\alpha\leq\theta$, to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes.

Abstract:
This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusion processes with killing on $[0,\infty)$. 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 $\infty$ 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 $Q$-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.

Abstract:
For general, almost surely absorbed Markov processes, we obtain necessary and 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). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.

Abstract:
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.

Abstract:
This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusion processes. We obtain a necessary and sufficient condition 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. We prove that this criterion is satisfied in most practical cases and is equivalent to an original strong property of strict local martingale for one-dimensional diffusion processes on natural scale. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps. We also give exponential ergodicity results on the $Q$-process.

Abstract:
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.