Survival and extinction results for a patch model with sexual reproduction

Eric Foxall, Nicolas Lanchier

Research output: Contribution to journalArticle

Abstract

We consider a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents’ patch or are sent to a nearby patch with some fixed probabilities. As the patch size tends to infinity, we identify a mean-field limit consisting of an infinite set of coupled differential equations. For the mean-field equations, we find explicit conditions for survival and extinction that we call expansion and retreat. Using duality techniques to compare the stochastic model to its mean-field limit, we find that expansion and retreat are also precisely the conditions needed to ensure survival and extinction of the stochastic model when the patch size is large. In addition, we study the dependence of survival on the dispersal range. We find that, with probability close to one and for a certain set of parameters, the metapopulation survives in the presence of nearest neighbor interactions while it dies out in the presence of long range interactions, suggesting that the best strategy for the population to spread in space is to use intermediate dispersal ranges.

Original languageEnglish (US)
Pages (from-to)1-51
Number of pages51
JournalJournal of Mathematical Biology
DOIs
StateAccepted/In press - Sep 19 2016

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Stochastic models
sexual reproduction
Extinction
Reproduction
Patch
extinction
Population
Differential equations
Mean-field Limit
Metapopulation
Contact Process
Model
Stochastic Model
Mean Field Equation
Long-range Interactions
Interaction
Range of data
Nearest Neighbor
Duality
Die

Keywords

  • Allee effect
  • Block construction
  • Duality
  • Interacting particle system
  • Metapopulation

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

Cite this

Survival and extinction results for a patch model with sexual reproduction. / Foxall, Eric; Lanchier, Nicolas.

In: Journal of Mathematical Biology, 19.09.2016, p. 1-51.

Research output: Contribution to journalArticle

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