Reaction fronts in persistent random walks with demographic stochasticity.

Area of Science:

• Mathematical Biology
• Theoretical Ecology
• Statistical Physics

Background:

• Standard reaction-diffusion models assume infinite speeds and no movement persistence, which are unrealistic for biological systems.
• Living organisms exhibit finite movement speeds and persistent behaviors, necessitating more complex modeling approaches.

Purpose of the Study:

• To investigate how persistent random walks and logistic growth in discrete particle models affect population dynamics.
• To bridge the gap between individual-based models and continuous reactive Cattaneo equations (RCE).
• To analyze finite-population effects and fluctuations on propagation speed.

Main Methods:

• Developed a discrete particle model with persistent random walks and logistic growth.
• Analyzed the model's behavior for large populations, comparing it to the continuous reactive Cattaneo equation (RCE).
• Investigated fluctuation effects using RCE with a reaction term cutoff and numerical simulations of the individual-based model.
• Examined Lévy walk processes for individual transport.

Main Results:

• The individual-based model aligns with the reactive Cattaneo equation for large populations.
• Significant finite-population effects emerge at lower carrying capacities.
• Fluctuations impact propagation speed, as evidenced by RCE with cutoff and simulations.
• A proposed expression for the front speed of traveling waves resulting from Lévy walks.

Conclusions:

• Persistent random walks and finite population sizes introduce crucial deviations from standard reaction-diffusion theory.
• The reactive Cattaneo equation provides a valid continuous approximation for large populations, but finite-population effects require careful consideration.
• Fluctuations and generalized transport processes like Lévy walks are important for accurately describing biological pattern formation and spread.