Stochastic spatial model of producer-consumer systems on the lattice

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The objective of this paper is to give a rigorous analysis of a stochastic spatial model of producer-consumer systems that has been recently introduced by Kang and the author to understand the role of space in ecological communities in which individuals compete for resources. Each point of the square lattice is occupied by an individual which is characterized by one of two possible types, and updates its type in continuous time at rate 1. Each individual being thought of as a producer and consumer of resources, the new type at each update is chosen at random from a certain interaction neighborhood according to probabilities proportional to the ability of the neighbors to consume the resource produced by the individual to be updated. In addition to giving a complete qualitative picture of the phase diagram of the spatial model, our results indicate that the nonspatial deterministic mean-field approximation of the stochastic process fails to describe the behavior of the system in the presence of local interactions. In particular, we prove that, in the parameter region where the nonspatial model displays bistability, there is a dominant type that wins regardless of its initial density in the spatial model, and that the inclusion of space also translates into a significant reduction of the parameter region where both types coexist.

Original languageEnglish (US)
Pages (from-to)1157-1181
Number of pages25
JournalAdvances in Applied Probability
Volume45
Issue number4
DOIs
StatePublished - Dec 2013

Fingerprint

Spatial Model
Stochastic Model
Resources
Update
Local Interaction
Bistability
Mean-field Approximation
Square Lattice
Phase Diagram
Continuous Time
Stochastic Processes
Display
Inclusion
Directly proportional
Random processes
Ecosystems
Phase diagrams
Display devices
Interaction
Model

Keywords

  • Interacting particle system
  • Richardson model
  • Threshold contact process
  • Voter model

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistics and Probability

Cite this

Stochastic spatial model of producer-consumer systems on the lattice. / Lanchier, Nicolas.

In: Advances in Applied Probability, Vol. 45, No. 4, 12.2013, p. 1157-1181.

Research output: Contribution to journalArticle

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