The importance of being discrete (and spatial)
type: Item
instance of: scholarly article
title: The importance of being discrete (and spatial)
zbMATH Open document ID: 0846.92027
published in: Theoretical Population Biology
publication date: 1996-09-30T00:00:00Z
review text: In recent years the importance of spatial phenomena has been recognized increasingly in discussions of the ecology and evolution of species in terms of modeling approaches. Each approach has its advantages and disadvantages and leads to unique conclusions. We select three prototypical spatial approaches from among this bestiary and compare the predictions with the null model of no spatial detail. Specifically, our focus is on the comparison of four different approaches to modeling the dynamics of spatially distributed systems: 1. mean field approaches (described by ordinary differential equations) in which every individual is considered to have equal probability of interacting with every other individual; 2. patch models that group discrete individuals into patches without other levels of spatial structure (i.e., individuals interact equally with all individuals in the same patch and there is between-patch migration that treats all other patches equally); 3. reaction-diffusion equations, in which infinitesimal individuals diffuse in space and undergo purely local interactions; 4. interacting particle systems, in which individuals are discrete and space is treated explicitly. To compare and contrast these four approaches we examined three examples of interactions in spatially distributed populations. The differences we have observed in the four approaches are of interest beyond the models and point to the consequences of particular influences such as localization of interactions or discreteness of individuals. In the first case where the fitness of one type of individual is enhanced by the presence of the other, all four approaches reach the same conclusion: there is a unique equilibrium that is the limit starting from any initial state in which both species have positive density. More generally, we expect this consensus of opinion to occur in most cases in which the ordinary differential equation has a globally attracting fixed point. In the second case where two in…
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 718696
zbMATH Keywords: mean field approaches
zbMATH Keywords: patch models
zbMATH Keywords: reaction-diffusion equations
zbMATH Keywords: interacting particle systems
zbMATH Keywords: spatially distributed populations
zbMATH Keywords: localization of interactions
zbMATH Keywords: discreteness
zbMATH Keywords: fitness
zbMATH Keywords: spatial models
zbMATH Keywords: stable equilibria
zbMATH Keywords: traveling wave
author: Rick Durrett
author: Simon Levin
MaRDI profile type: Publication
full work available at URL: tpbi.1994.1032
OpenAlex ID: W2016469924
Wikidata QID: Q112839304
DOI: 10.1006/TPBI.1994.1032
description: scientific article; zbMATH DE number 718696