Dynamics of lineages in adaptation to a gradual environmental change

Abstract

We investigate a simple quantitative genetics model subjet to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past traits of their ancestors are shaped by the adaptation to the varying environment. The individuals are characterized by a one-dimensional trait. The dynamics -births and deaths-depend on a time-changing mortality rate that shifts the optimal trait to the right at constant speed. The population size is regulated by a nonlinear non-local logistic competition term. The macroscopic behaviour can be described by a PDE that admits a unique positive stationary solution. In the stationary regime, the population can persist, but with a lag in the trait distribution due to the environmental change. For the microscopic (individual-based) stochastic process, the evolution of the lineages can be traced back using the historical process, that is, a measure-valued process on the set of continuous real functions of time. Assuming stationarity of the trait distribution, we describe the limiting distribution, in large populations, of the path of an individual drawn at random at a given time T . Freezing the non-linearity due to competition allows the use of a many-to-one identity together with Feynman-Kac's formula. This path, in reversed time, remains close to a simple Ornstein-Uhlenbeck process. It shows how the lagged bulk of the present population stems from ancestors once optimal in trait but still in the tail of the trait distribution in which they lived.

Evolving genealogies for branching populations under selection and competition

“…By projectivity, this shows that (68) has a unique pathwise solution up to the time at which its mass process ζ goes to extinction or explodes. In view of ( 9) and ( 10), the solution of ( 11), (7) coincides with that of (68) up to that time at which ζ exceeds C c or C b . Again by projectivity, this implies the assertion of Theorem 2.2.…”

“…The above construction and Theorem 2.2 show that the process (ζ, X) is the pathwise unique solution the system of SDEs given by ( 7), ( 11) and (24), which is driven by (L ij ), (K i ), W and extends (7), (11) to include also the genealogical distances.…”

“…Recent work on evolving genealogies in the neutral case, with a focus on heavy-tailed offspring distributions, has been reviewed in [24]. Evolving ancestral path configurations under competition are studied in [33,25] or [7], building on the framework of historical processes which in the non-interactive case was pioneered in [8,14]. Inference methods in the presence of selection, varying population size and evolving population structure are described in [32], extending results of [5]; in the latter models, a time-scale separation allows to treat separately the type structure and population size on the one hand, and the genealogies on the other hand.…”

Time reversal of spinal processes for linear and non-linear branching processes near stationarity

“…This trait can change through time (by mutations occurring continuously in time). The case where it is driven by a Brownian motion has been considered in a previous paper by the authors [4]. Here, we are interested in a non-local mutation kernel.…”

“…it is proportional to the size N K t and does not account for the whole trait distribution. During their life, the trait of an individual mutates according to a kernel γm(x, y)dy and experiences a linear drift with environmental velocity ρ ∈ R due to environmental changes (see [4] for details). We assume that γ > 0 is the jump rate and that m(x, y)dy is the probability measure describing the jumps (assumed to be absolutely continuous with respect to the Lebesgue measure, for the sake of simplicity).…”