Jimmy Garnier

Allee effect promotes diversity in traveling waves of colonization

Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves.

Accelerating Solutions in Integro-Differential Equations

In this paper, we study the spreading properties of the solutions of an integro-differential equation of the form

Modeling the Spatio-temporal Dynamics of the Pine Processionary Moth

u_t=J\ast u-u+f(u)

Thin front limit of an integro–differential Fisher–KPP equation with fat–tailed kernels

. We focus on equations with slowly decaying dispersal kernels

Beneficial mutation-selection dynamics in finite asexual populations: a free boundary approach

J(x)

Isolation and sequence of a cDNA encoding the Jerusalem artichoke cinnamate 4-hydroxylase, a major plant cytochrome P450 involved in the general phenylpropanoid pathway.

which correspond to models of population dynamics with long-distance dispersal events. We prove that for kernels J, which decrease to 0 slower than any exponentially decaying function, the level sets of the solution u propagate with an infinite asymptotic speed. Moreover, we obtain lower and upper bounds for the position of any level set of u. These bounds allow us to estimate how the solution accelerates, depending on the kernel J: the slower the kernel decays, the faster the level sets propagate. Our results are in sharp contrast with most results on this type of equation, where the dispersal kernels are generally assumed to decrease exponentially fast, leading to finite propagation speeds.

Inside dynamics of delayed travelling waves

“This chapter summarizes several modeling studies conducted on the pine processionary moth range expansion in a spatio-temporally heterogeneous environment. These studies provide new approaches for analyzing and modeling range expansions and contribute to a better understanding of the effects of a wide variety of factors on the spatio-temporal dynamics of the pine processionary moth. These dynamics mostly depend on the dispersal, survival and reproduction characteristics of the species, and these characteristics fluctuate in time and space, depending on environmental and biological factors.”

Uniqueness from pointwise observations in a multi-parameter inverse problem

We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton-Jacobi limit.

The spatio-temporal dynamics of neutral genetic diversity

Using a free boundary approach based on an analogy with ice melting models, we propose a deterministic PDE framework to describe the dynamics of fitness distributions in the presence of beneficial mutations with non-epistatic effects on fitness. Contrarily to most approaches based on deterministic models, our framework does not rely on an infinite population size assumption, and successfully captures the transient as well as the long time dynamics of fitness distributions. In particular, consistently with stochastic individual-based approaches or stochastic PDE approaches, it leads to a constant asymptotic rate of adaptation at large times, that most deterministic approaches failed to describe. We derive analytic formulas for the asymptotic rate of adaptation and the full asymptotic distribution of fitness. These formulas depend explicitly on the population size, and are shown to be accurate for a wide range of population sizes and mutation rates, compared to individual-based simulations. Although we were not able to derive an analytic description for the transient dynamics, numerical computations lead to accurate predictions and are computationally efficient compared to stochastic simulations. These computations show that the fitness distribution converges towards a travelling wave with constant speed, and whose profile can be computed analytically.