A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion

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Abstract

The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman-Rach ford operator splitting scheme. Numerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.
Lingua originaleEnglish
pagine (da-a)1059-1074
Numero di pagine16
RivistaApplied Numerical Mathematics
Volume59
Stato di pubblicazionePublished - 2009

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Cross-diffusion
Self-diffusion
Lotka-Volterra System
Travelling Fronts
Linear Diffusion
Operator Splitting
Lotka-Volterra
Particle Method
Time Discretization
Segregation
Term
Pattern Formation
Reaction-diffusion Equations
Diffusion equation
Numerical Experiment
Fick's laws
Approximation

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Numerical Analysis
  • Computational Mathematics

Cita questo

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title = "A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion",
abstract = "The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman-Rach ford operator splitting scheme. Numerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.",
keywords = "Particle methods; Nonlinear diffusion; Reaction-diffusion; ADI schemes; Pattern formation; Travelling fronts",
author = "Sammartino, {Marco Maria Luigi} and Lombardo, {Maria Carmela} and Gaetana Gambino",
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language = "English",
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journal = "Applied Numerical Mathematics",
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T1 - A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion

AU - Sammartino, Marco Maria Luigi

AU - Lombardo, Maria Carmela

AU - Gambino, Gaetana

PY - 2009

Y1 - 2009

N2 - The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman-Rach ford operator splitting scheme. Numerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.

AB - The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman-Rach ford operator splitting scheme. Numerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.

KW - Particle methods; Nonlinear diffusion; Reaction-diffusion; ADI schemes; Pattern formation; Travelling fronts

UR - http://hdl.handle.net/10447/39992

M3 - Article

VL - 59

SP - 1059

EP - 1074

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

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