Individual based and mean-field modeling of direct aggregation

Area of Science:

• Mathematical Biology
• Statistical Physics
• Dynamical Systems

Background:

• Biological aggregation is crucial for species survival and ecosystem function.
• Existing models often rely on explicit attractive forces between individuals.
• Understanding emergent aggregation from local interactions is a key challenge.

Purpose of the Study:

• To introduce and analyze novel models of biological aggregation.
• To investigate aggregation driven solely by density-dependent stochasticity reduction.
• To explore the mathematical properties and simulation outcomes of these models.

Main Methods:

• Development of two models: first-order (position-based) and second-order (velocity-based) density-dependent random walks.
• Formal derivation of mean-field limits yielding nonlocal degenerate diffusions.
• Mathematical analysis including existence of weak solutions, steady states, and linear stability analysis.
• Numerical simulations on both individual-based and continuum levels.

Main Results:

• Aggregation emerges without explicit attractive forces, solely through reduced individual stochasticity.
• The models yield nonlocal degenerate diffusion equations.
• Existence of weak solutions and measure-valued steady states for the first-order model.
• Identification of conditions for pattern formation through linear stability analysis.
• Numerical simulations confirm emergent aggregation and pattern formation.

Conclusions:

• Individual stochasticity reduction in response to local density is a sufficient mechanism for biological aggregation.
• The derived mathematical framework (nonlocal degenerate diffusions) accurately describes emergent collective behavior.
• These models offer a new perspective on understanding self-organization in biological systems.