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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Effects of fluctuations on propagating fronts
 Physics ReportsReview Section of Physics Letters
, 2004
"... Propagating fronts are seen in varieties of nonequilibrium pattern forming systems in Physics, Chemistry and Biology. In the last two decades, many researchers have contributed to the understanding of the underlying dynamics of the propagating fronts. Of these, the deterministic and meanfield dyna ..."
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Propagating fronts are seen in varieties of nonequilibrium pattern forming systems in Physics, Chemistry and Biology. In the last two decades, many researchers have contributed to the understanding of the underlying dynamics of the propagating fronts. Of these, the deterministic and meanfield dynamics of the fronts were mostly understood in late 1980s and 1990s. On the other hand, although the earliest work on the effect of fluctuations on propagating fronts dates back to early 1980s, the subject of fluctuating fronts did not reach its adolescence until the mid 1990s. From there onwards the last few years witnessed a surge in activities in the effect of fluctuations on propagating fronts. Scores of papers have been written on this subject since then, contributing to a significant maturity of our understanding, and only recently a full picture of fluctuating fronts has started to emerge. This review is an attempt to collect all the works on fluctuating (propagating) fronts in a coherent and cogent manner in proper perspective. It is based on the idea of making our knowledge in this field available to a broader audience, and it is also expected to help to collect
BrunetDerrida particle systems, free boundary problems and WienerHopf equations
, 2009
"... We consider a branchingselection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in di ..."
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We consider a branchingselection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measurevalued process whose densities solve a free boundary integrodifferential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c ≥ a or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of WienerHopf equations. 1 Introduction and statement of the results We will consider the following branchingselection particle system. At any time t we have N particles on the real line with positions η N t (1) ≥ · · · ≥ η N t (N). Each one of the N particles gives birth at rate 1 to a new particle whose position is chosen, relative to the
RANDOM WALKS & TREES
"... These notes provide an elementary and selfcontained introduction to branching random walks. Chapter 1 gives a brief overview of Galton–Watson trees, whereas Chapter 2 presents the classical law of large numbers for branching random walks. These two short chapters are not exactly indispensable, but ..."
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These notes provide an elementary and selfcontained introduction to branching random walks. Chapter 1 gives a brief overview of Galton–Watson trees, whereas Chapter 2 presents the classical law of large numbers for branching random walks. These two short chapters are not exactly indispensable, but they introduce the idea of using sizebiased trees, thus giving motivations and an avantgoût to the main part, Chapter 3, where branching random walks are studied from a deeper point of view, and are connected to the model of directed polymers on a tree. Treerelated random processes form a rich and exciting research subject. These notes cover only special topics. For a general account, we refer to the StFlour lecture notes of Peres [47] and to the forthcoming book of Lyons and Peres [42], as well as to Duquesne and Le Gall [23] and Le Gall [37] for continuous random trees. I am grateful to the organizers of the Symposium for the kind invitation, and to my coauthors for sharing the pleasure of random climbs. Contents 1 Galton–Watson trees 1
A stochastic model for wound healing
, 2008
"... We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front spee ..."
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We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front speed as a power series expansion in a parameter, p, that gives the relative size of proliferation and diffusion processes for the invading cells. In two dimensions the model becomes the Eden model for p≈1. In both one and two dimensions for small p, front propagation for this model should approach that of the FisherKolmogorov equation. However, as in other cases, this discrete model approaches FisherKolmogorov behavior slowly. 1
Kevrekidis, Equationfree modelling of evolving diseases: coarsegrained computations with individualbased models, Proc Roy Soc
"... We demonstrate how direct simulation of stochastic, individualbased models can be combined with continuum numerical analysis techniques to study the dynamics of evolving diseases. Sidestepping the necessity of obtaining explicit populationlevel models, the approach analyzes the (unavailable in clo ..."
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We demonstrate how direct simulation of stochastic, individualbased models can be combined with continuum numerical analysis techniques to study the dynamics of evolving diseases. Sidestepping the necessity of obtaining explicit populationlevel models, the approach analyzes the (unavailable in closed form) ‘coarse ’ macroscopic equations, estimating the necessary quantities through appropriately initialized, short ‘bursts ’ of individualbased dynamic simulation. We illustrate this approach by analyzing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts. Building up from classical SIR and SIRS models, our example uses a onedimensional lattice for variant space, and assumes a finite number of individuals. Macroscopic computational tasks enabled through this approach include stationary state computation, coarse projective integration, parametric continuation and stability analysis. 1
Evolution in predatorprey systems
, 2010
"... We study the adaptive dynamics of predator prey systems modeled by a dynamical system in which the characteristics are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexiste ..."
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We study the adaptive dynamics of predator prey systems modeled by a dynamical system in which the characteristics are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the parameters of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied we see (i) the number of coexisting predators remains tight and the differences of the parameters from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we conjecture that the differences converge to a deterministic limit. Key words: predatorprey, adaptive dynamics, branching random walk, branchingselection particle system 2000 MSC: primary 92D15, 92D25; secondary 60J60, 60K35 1.
A short proof of the logarithmic Bramson correction
 in FisherKPP equations, Networks and Heterogeneous Media
"... In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions u(t, x) of FisherKPP reactiondiffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison ..."
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In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions u(t, x) of FisherKPP reactiondiffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of u to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the largetime convergence of the solutions u along their level sets to the profile of the minimal travelling front. 1
A geometric approach to bistable front propagation in scalar reactiondiffusion equations with cutoff
 Physica D
, 1999
"... Abstract. ‘Cutoffs ’ were introduced to model front propagation in reactiondiffusion systems in which the reaction is effectively deactivated at points where the concentration lies below some threshold. In this article, we investigate the effects of a cutoff on fronts propagating into metastabl ..."
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Abstract. ‘Cutoffs ’ were introduced to model front propagation in reactiondiffusion systems in which the reaction is effectively deactivated at points where the concentration lies below some threshold. In this article, we investigate the effects of a cutoff on fronts propagating into metastable states in a class of bistable scalar equations. We apply the method of geometric desingularization from dynamical systems theory to calculate explicitly the change in front propagation speed that is induced by the cutoff. We prove that the asymptotics of this correction scales with fractional powers of the cutoff parameter, and we identify the source of these exponents, thus explaining the structure of the resulting expansion. In particular, we show geometrically that the speed of bistable fronts increases in the presence of a cutoff, in agreement with results obtained previously via a variational principle. We first discuss the classical Nagumo equation as a prototypical example of bistable front propagation. Then, we present corresponding results for the (equivalent) cutoff Schlögl equation. Finally, we extend our analysis to a general family of reactiondiffusion equations that support bistable fronts, and we show that knowledge of an explicit front solution to the associated problem without cutoff is necessary for the correction induced by the cutoff to be computable in closed form. 1.