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Noise-induced switching and extinction in systems with delay.

Ira B Schwartz1, Lora Billings2, Thomas W Carr3

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This study investigates noise-induced switching and extinction rates in dissipative dynamical systems with hard delay. Analytical and numerical methods reveal exponentially small rates and acausal most probable paths for population dynamics.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Theoretical Ecology
  • Stochastic Processes

Background:

  • Dissipative dynamical systems with delay are crucial for modeling phenomena like population dynamics.
  • Understanding noise-induced transitions between stable states and extinction events is vital in these systems.
  • The presence of 'hard delay' introduces unique challenges in analyzing system behavior.

Purpose of the Study:

  • To investigate the rates of noise-induced switching between stable states and extinction in dissipative dynamical systems with hard delay.
  • To develop analytical methods for calculating these rates, particularly for weak noise.
  • To identify and characterize the most probable paths associated with switching and extinction events.

Main Methods:

  • Formulation of the problem in terms of variational problems for logarithmic accuracy.
  • Derivation and analysis of acausal equations for the most probable paths.
  • Development of a direct variational method for rate calculation.
  • Comparison of analytical results with numerical simulations.

Main Results:

  • Noise-induced switching and extinction rates are exponentially small for weak noise.
  • The most probable paths governing these transitions are acausal, requiring specific boundary conditions.
  • Explicit results were derived for systems with small delay relative to the relaxation rate.
  • Analytical predictions show strong agreement with numerical simulations.

Conclusions:

  • The study provides a robust framework for analyzing noise-induced transitions in delayed dynamical systems.
  • The findings offer insights into the mechanisms of population extinction and state switching under stochastic perturbations.
  • The developed methods are applicable to a range of dissipative systems with time delays.