Results 1  10
of
50
Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities
, 2010
"... Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the selforganization principles underneath the complexity of these systems. A ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
(Show Context)
Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the selforganization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social ” behavior. A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial
Effect of noise on front propagation in reactiondiffusion equations of KPP type
, 2009
"... We consider reactiondiffusion equations of KPP type in one spatial dimension, perturbed by a FisherWright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed FisherKPP equations and ∂tu = ∂ 2 xu + u(1 − u) + ǫ p u(1 − u) ˙ W, (0.1) ∂tu = ∂ 2 ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
We consider reactiondiffusion equations of KPP type in one spatial dimension, perturbed by a FisherWright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed FisherKPP equations and ∂tu = ∂ 2 xu + u(1 − u) + ǫ p u(1 − u) ˙ W, (0.1) ∂tu = ∂ 2 xu + u(1 − u) + ǫ √ u ˙ W, (0.2) where ˙ W = ˙ W(t, x) is a spacetime white noise.
QuasiStationary Regime of a Branching Random Walk in Presence of an Absorbing Wall
 J STAT PHYS
, 2008
"... A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as the velocity v of the wall varies. Below the critical velocity vc, the population has a nonzero survival probability and when the population survives its size grows exponential ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as the velocity v of the wall varies. Below the critical velocity vc, the population has a nonzero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time T. We study the quasistationary regime for v < vc when T is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time T. We then use this construction to show that the properties of the quasistationary regime are universal when v → vc. We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasistationary regime can be reduced to the analysis of a single onedimensional map.
Spreading speeds in slowly oscillating environments
"... In this paper, we derive exact asymptotic estimates of the spreading speeds of solutions of some reactiondiffusion models in periodic environments with very large periods. Contrarily to the other limiting case of rapidly oscillating environments, there was previously no explicit formula in the case ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
In this paper, we derive exact asymptotic estimates of the spreading speeds of solutions of some reactiondiffusion models in periodic environments with very large periods. Contrarily to the other limiting case of rapidly oscillating environments, there was previously no explicit formula in the case of slowly oscillating environments. The knowledge of these two extremes permits to quantify the effect of environmental fragmentation on the spreading speeds. On the one hand, our analytical estimates and numerical simulations reveal speeds which are higher than expected for ShigesadaKawasakiTeramoto models with FisherKPP reaction terms in slowly oscillating environments. On the other hand, spreading speeds in very slowly oscillating environments are proved to be 0 in the case of models with strong Allee effects; such an unfavorable effect of aggregation is merely seen in reactiondiffusion models.
Selforganization of mobile populations in cyclic competition, 2008. Manuscript submitted
"... The formation of outofequilibrium patterns is a characteristic feature of spatiallyextended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the ‘rockpaperscissors’ game. Both experimentally and theoretically, such ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
The formation of outofequilibrium patterns is a characteristic feature of spatiallyextended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the ‘rockpaperscissors’ game. Both experimentally and theoretically, such nontransitive interactions have been found to induce selforganization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals ’ mobility on the spatial structures emerging in rockpaperscissors games. We devise a quantitative approach to analyze the spatial patterns selfforming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system’s behavior in the proper continuum limit is aptly captured by a set of stochastic partial differential equations. The system’s stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals ’ wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex GinzburgLandau equation, their entanglement results from the inherent stochastic nature of the system. These findings and our methods have important applications for understanding the formation of noisy patterns, e.g., in ecological and evolutionary contexts, and are also of relevance for the kinetics of (bio)chemical reactions. 1
Existence of traveling waves of invasion for GinzburgLandautype problems in infinite cylinders
, 2008
"... We study a class of systems of reaction–diffusion equations in infinite cylinders which arise within the context of Ginzburg–Landau theories and describe the kinetics of phase transformation in secondorder or weakly firstorder phase transitions with nonconserved order parameters. We use a variati ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
We study a class of systems of reaction–diffusion equations in infinite cylinders which arise within the context of Ginzburg–Landau theories and describe the kinetics of phase transformation in secondorder or weakly firstorder phase transitions with nonconserved order parameters. We use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of nonlinearities. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or nonexistence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computerassisted approach. 1.
A slow pushed front in a LotkaVolterra competition model
, 2012
"... We study invasion speeds in the LotkaVolterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric sin ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
(Show Context)
We study invasion speeds in the LotkaVolterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric singular perturbation theory and geometric desingularization. The main challenge arises from the slow passage through a saddlenode bifurcation. From a perspective of linear versus nonlinear speed selection, this front provides an interesting example as the propagation speed is slower than the linear spreading speed. However, our front shares many characteristics with pushed fronts that arise when the influence of nonlinearity leads to faster than linear speeds of propagation. We show that this is a result of the linear spreading speed arising as a simple pole of the resolvent instead of as a branch pole. Using the pointwise Green’s function, we show that this pole poses no a priori obstacle to marginal stability of the nonlinear traveling front, thus explaining how nonlinear systems can exhibit slower spreading that their linearization in a robust fashion. MSC numbers: 35C07, 35K57, 34E99 Keywords: invasion fronts, pushed and pulled fronts, LotkaVolterra systems, pointwise Green’s function
Coarsening fronts
, 2006
"... We characterize the spatial spreading of the coarsening process in the AllenCahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the AllenCahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneo ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
We characterize the spatial spreading of the coarsening process in the AllenCahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the AllenCahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneous, stable equilibrium. During one cycle of the oscillatory propagation, two layers of the periodic pattern are annihilated. Galerkin approximations and Conley index for illposed spatial dynamics are used to show existence of modulated fronts for all parameter values. In the limit of small amplitude patterns or large wave speeds, we establish uniqueness and asymptotic stability of the modulated fronts. We show that the minimal speed of propagation can be characterized by a dichotomy depending on the existence of pulled fronts. Main tools here are an Evans function type construction for the infinitedimensional illposed dynamics and an analysis of the complex dispersion relation based on SturmLiouville theory.