Velocity-jump models with crowding effects.

Area of Science:

• Mathematical Biology
• Collective Motion Modeling
• Agent-Based Modeling

Background:

• Velocity-jump processes model collective motion, often simulating bacterial 'run and tumble' behavior.
• Traditional models neglect agent-to-agent interactions, limiting their use to dilute biological systems.
• Dense cell populations in tissues are crucial for processes like wound healing and cancer invasion.

Purpose of the Study:

• To develop novel velocity-jump processes that incorporate crowding and agent-to-agent interactions.
• To enable accurate modeling of collective cell behavior in high-density biological tissues.
• To bridge the gap between discrete stochastic models and continuum descriptions for crowded systems.

Main Methods:

• Introduction of three distinct classes of crowding interactions into a one-dimensional velocity-jump model.
• Utilizing simulation data and averaging arguments to derive continuum descriptions.
• Developing systems of hyperbolic partial differential equations to represent the interacting processes.

Main Results:

• Successfully derived continuum descriptions for interacting velocity-jump processes.
• Demonstrated that the derived hyperbolic partial differential equations accurately predict the mean behavior of stochastic simulations.
• Validated the model's applicability to high-cell-density scenarios.

Conclusions:

• The developed interacting velocity-jump processes provide a robust framework for modeling collective motion in crowded biological environments.
• The resulting continuum models offer accurate predictions for dense cellular systems, overcoming limitations of traditional approaches.
• This work advances the understanding of cell-cell interactions in tissue dynamics.