Escape from a metastable state in non-Markovian population dynamics

Area of Science:

• Theoretical Physics
• Stochastic Processes
• Mathematical Biology

Background:

• Non-Markovian models capture systems with memory, deviating from standard Markovian assumptions.
• Fat-tailed waiting time distributions introduce long-term memory effects in reaction dynamics.
• Understanding long-time behavior is crucial for biological and physical systems.

Purpose of the Study:

• Investigate the long-time dynamics of single-population stochastic models with nonexponential waiting times.
• Analyze the emergence of memory-induced states and scaling regimes.
• Develop theoretical frameworks for non-Markovian systems.

Main Methods:

• Utilized a Wentzel-Kramers-Brillouin (WKB) approach for non-Markovian systems.
• Analyzed systems with finite and infinite mean waiting time distributions.
• Examined prototypical examples: genetic switching, population establishment, and extinction.

Main Results:

• In the finite mean regime, a memory-induced (meta)stable state can emerge with strong memory.
• Derived explicit results for mean population size and escape times.
• In the infinite mean regime, bistable systems exhibit two distinct scaling regimes separated by long times.

Conclusions:

• Non-Markovian effects, particularly long-term memory, significantly alter system dynamics.
• The system's behavior depends critically on the properties of the waiting time distribution (finite vs. infinite mean).
• The developed WKB approach provides a powerful tool for analyzing complex non-Markovian phenomena.