Deriving reaction-diffusion models in ecology from interacting particle systems

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We use a scaling procedure based on averaging Poisson distributed random variables to derive population level models from local models of interactions between individuals. The procedure is suggested by using the idea of hydrodynamic limits to derive reaction-diffusion models for population interactions from interacting particle systems. The scaling procedure is formal in the sense that we do not address the issue of proving that it converges; instead we focus on methods for computing the results of the scaling or deriving properties of rescaled systems. To that end we treat the scaling procedure as a transform, in analogy with the Laplace or Fourier transform, and derive operational formulas to aid in the computation of rescaled systems or the derivation of their properties. Since the limiting procedure is adapted from work by Durrett and Levin, we refer to the transform as the Durrett-Levin transform. We examine the effects of rescaling in various standard models, including Lotka-Volterra models, Holling type predator-prey models, and ratio-dependent models. The effects of scaling are mostly quantitative in models with smooth interaction terms, but ratio-dependent models are profoundly affected by the scaling. The scaling transforms ratio-dependent terms that are singular at the origin into smooth terms. Removing the singularity at the origin eliminates some of the unique dynamics that can arise in ratio-dependent models.

Original languageEnglish (US)
Pages (from-to)187-217
Number of pages31
JournalJournal of Mathematical Biology
Volume48
Issue number2
DOIs
StatePublished - Feb 2004

Fingerprint

Interacting Particle Systems
Reaction-diffusion Model
Ecology
Ratio-dependent
Scaling
ecology
Transform
Term
Interaction
Model
Hydrodynamic Limit
Lotka-Volterra Model
Fourier Analysis
Hydrodynamics
Predator-prey Model
Rescaling
Population
Laplace transform
Averaging
Analogy

Keywords

  • Competition
  • Hawk-Dove game
  • Hydrodynamic limits
  • Interacting particle systems
  • Lotka-Volterra
  • Population dynamics
  • Predator-prey
  • Ratio-dependence
  • Reaction-diffusion
  • Scaling
  • Spatial models

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Mathematics (miscellaneous)

Cite this

Deriving reaction-diffusion models in ecology from interacting particle systems. / Cantrell, Robert; Cosner, George.

In: Journal of Mathematical Biology, Vol. 48, No. 2, 02.2004, p. 187-217.

Research output: Contribution to journalArticle

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