arXiv subjects

Christopher Henderson

Publications and source records attributed to Christopher Henderson.

44 records · Page 3Linked to original sources

arXiv2016

The Bramson logarithmic delay in the cane toads equations

We study a nonlocal reaction-diffusion-mutation equation modeling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits traveling wave solutions [7]. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the traveling waves grows as (3/(2$λ$ *)) log t. This result relies on a present-time Harnack inequality which allows to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.

math.AP↗

arXiv2016

Super-linear spreading in local bistable cane toads equations

In this paper, we study the influence of an Allee effect on the spreading rate in a local reaction-diffusion-mutation equation modelling the invasion of cane toads in Australia. We are, in particular, concerned with the case when the diffusivity can take unbounded values. We show that the acceleration feature that arises in this model with a Fisher-KPP, or monostable, non-linearity still occurs when this non-linearity is instead bistable, despite the fact that this kills the small populations. This is in stark contrast to the work of Alfaro, Gui-Huan, and Mellet-Roquejoffre-Sire in related models, where the change to a bistable non-linearity prevents acceleration.

math.AP↗

arXiv2016

Equivalence of a mixing condition and the LSI in spin systems with infinite range interaction

We investigate unbounded continuous spin-systems with infinite-range interactions. We develop a new technique for deducing decay of correlations from a uniform Poincaré inequality based on a directional Poincaré inequality, which we derive through an averaging procedure. We show that this decay of correlations is equivalent to the Dobrushin-Shlosman mixing condition. With this, we also state and provide a partial answer to a conjecture regarding the relationship between the relaxation rates of non-ferromagnetic and ferromagnetic systems. Finally, we show that for a symmetric, ferromagnetic system with zero boundary conditions, a weaker decay of correlations can be bootstrapped.

math.PR↗

arXiv2016

Ricci curvature bounds for weakly interacting Markov chains

We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of spin systems including the Ising and Curie--Weiss models, a class of hard-core models and random walks on groups induced by a conjugacy invariant set of generators.

math.PR↗

arXiv2015

Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. Building on the results of Hamel and Roques '10, in this paper we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the level sets. In addition, we characterize the profile of the inhomogeneous problem for large times.

math.AP↗

arXiv2015

Stability of Vortex Solutions to an Extended Navier-Stokes System

We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.

math.AP↗

arXiv2014

Population Stabilization in Branching Brownian Motion With Absorption

We consider, through PDE methods, branching Brownian motion with drift and absorption. It is well know that there exists a critical drift which separates those processes which die out almost surely and those which survive with positive probability. In this work, we consider lower order terms to the critical drift which ensures a non-negative, bounded expected number of particles and convergence of this expectation to a limiting number, $α_0\geq 0$, which is positive for some initial data. In particular, we show that, in an average sense, the expected number of particles stabilizes to $α_0$ like $O(\log(t)/t)$ if and only if the multiplicative factor of the $O(t^{-1/2})$ correction term is $3\sqrtπ t^{-1/2}$. Otherwise, the convergence is like $O(1/\sqrt{t})$. We point out some connections between this work and recent work investigating the expansion of the front location for the initial value problem in Fisher-KPP.

math.AP↗

arXiv2013

Pulsating Fronts in a 2D Reactive Boussinesq System

We consider a reactive Boussinesq system with no stress boundary conditions in a periodic domain which is unbounded in one direction. Specifically, we couple the reaction-advection-diffusion equation for the temperature, $T$, and the linearized Navier-Stokes equation with the Boussinesq approximation for the fluid flow, $u$. We show that this system admits smooth pulsating front solutions that propagate with a positive, fixed speed.

math.AP↗