Related papers: Proliferating L\'evy Walkers and Front Propagation

Predicting evolution of expanding populations is critical to control biological threats such as invasive species and cancer metastasis. Expansion is primarily driven by reproduction and dispersal, but nature abounds with examples of…

This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v*, the asymptotic rate with which initially…

We study continuous-time (variable speed) random walks in random environments on $\mathbb{Z}^d$, $d\ge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary…

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the…

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a L\'evy process and can reach resources in a…

We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant…

The L\'evy walk process for a lower interval of an excursion times distribution ($\alpha<1$) is discussed. The particle rests between the jumps and the waiting time is position-dependent. Two cases are considered: a rising and diminishing…

We study a hydrodynamic limit approach to move-to-front rules, namely, a scaling limit as the number of items tends to infinity, of the joint distribution of jump rate and position of items. As an application of the limit formula, we…

In this article, the continuous time random walk on the circle is studied. We derive the corresponding generalized master equation and discuss the effects of topology, especially important when Levy flights are allowed. Then, we work out…

We employed the method of virial expansion in order to compute the retarded density correlation function (generalized diffusion propagator) in the critical random matrix ensemble in the limit of strong multifractality. We found that the…

Reaction-diffusion equations describe various spatially extended processes that unfold as traveling fronts moving at constant velocity. We introduce and solve analytically a model that, besides such fronts, supports solutions advancing as…

We consider the diffusion scaling limit of the one-dimensional vicious walker model of Fisher and derive a system of nonintersecting Brownian motions. The spatial distribution of $N$ particles is studied and it is described by use of the…

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical…

In this paper, we first focus on the speed selection problem for the reaction-diffusion equation of the monostable type. By investigating the decay rates of the minimal traveling wave front, we propose a sufficient and necessary condition…

L\'evy-type walks with correlated jumps, induced by the topology of the medium, are studied on a class of one-dimensional deterministic graphs built from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a standard…

A Levy walk is a non-Markovian stochastic process in which the elementary steps of the walker consist of motion with constant speed in randomly chosen directions and for a random period of time. The time of flight is chosen from a…

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes…

Many studies on animal and human movement patterns report the existence of scaling laws and power-law distributions. Whereas a number of random walk models have been proposed to explain observations, in many situations individuals actually…

We consider one-dimensional hyperbolic PDEs, linear and nonlinear, with random initial data. Our focus is the {\em pointwise statistics,} i.e., the probability measure of the solution at any fixed point in space and time. For linear…

This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by n converges to a constant as n goes to infinity. Furthermore, a…