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Persistence and extinction for stochastic ecological models with internal and external variables.

Michel Benaïm1, Sebastian J Schreiber2

  • 1Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand, 2000, Neuchâtel, Switzerland.

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|May 5, 2019
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Stochastic ecological models reveal conditions for species persistence or extinction. Environmental fluctuations can promote coexistence in some games but hinder it in others, impacting population dynamics.

Keywords:
CoexistenceExtinctionStochastic difference equationsStochastic persistence

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Population Dynamics

Background:

  • Species density is influenced by internal factors (phenotypes, spatial distribution) and external factors (environment, other species).
  • Environmental fluctuations introduce stochasticity, impacting population persistence and extinction dynamics.

Purpose of the Study:

  • To develop mathematical conditions for stochastic persistence and exclusion in ecological models.
  • To quantify extinction probabilities and persistence times under environmental stochasticity.

Main Methods:

  • Proving theorems for stochastic persistence and exclusion using a stochastic analog of average Lyapunov functions.
  • Analyzing stochastic difference equations incorporating internal and external variables.

Main Results:

  • Established sufficient and necessary conditions for stochastic persistence and asymptotic extinction.
  • Provided quantitative estimates for time near extinction and probability of extinction.
  • Demonstrated environmental stochasticity's varied effects on coexistence in evolutionary games and predator-prey models.

Conclusions:

  • Environmental stochasticity can promote persistence in evolving populations and diseases in patchy landscapes.
  • Results bridge the gap between deterministic and stochastic persistence theories in ecology.
  • Further research is needed to address remaining challenges in stochastic ecological modeling.