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A random walk with catastrophes.

Iddo Ben-Ari1, Alexander Roitershtein2, Rinaldo B Schinazi3

  • 1Department of Mathematics, University of Connecticut, Storrs, CT 06269-1009, USA.

Electronic Journal of Probability
|August 20, 2019
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Summary
This summary is machine-generated.

This study analyzes random population dynamics with linear growth and binomial catastrophes. We demonstrate a sharp cutoff phenomenon in population models using a coupling construction for convergence analysis.

Keywords:
catastrophescutoffpersistencepopulation modelsspectral gap

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Area of Science:

  • Mathematical Biology
  • Probability Theory
  • Stochastic Processes

Background:

  • Population dynamics models are crucial for understanding ecological systems.
  • Catastrophes significantly impact population stability and long-term survival.
  • Ergodic models provide a framework for analyzing systems that return to equilibrium.

Purpose of the Study:

  • To investigate an ergodic model of random population dynamics incorporating linear growth and binomial catastrophes.
  • To derive precise bounds on the rate of convergence to stationarity for this population model.
  • To establish the presence of a cutoff phenomenon in the studied population dynamics.

Main Methods:

  • Utilizing a coupling construction to analyze the convergence properties of the population model.
  • Developing sharp two-sided bounds for the rate of convergence to stationarity.
  • Applying these bounds to identify and characterize the cutoff phenomenon.

Main Results:

  • Established sharp two-sided bounds for the rate of convergence to stationarity in the population model.
  • Demonstrated that the model exhibits a cutoff phenomenon, indicating abrupt changes in population behavior.
  • Provided a rigorous mathematical analysis of population dynamics under catastrophic events.

Conclusions:

  • The studied ergodic model with linear growth and binomial catastrophes exhibits a distinct cutoff phenomenon.
  • The coupling construction provides an effective method for analyzing convergence rates and phenomena in population dynamics.
  • This research contributes to a deeper mathematical understanding of population stability and resilience in the face of random events.