Aggregation-diffusion in heterogeneous environments.

Area of Science:

• Mathematical Biology
• Ecology
• Statistical Mechanics

Background:

• Aggregation-diffusion equations model collective organism movement and space use.
• Existing models often neglect environmental influences on movement patterns.
• Environmental heterogeneity is crucial for understanding emergent spatial distributions.

Purpose of the Study:

• To investigate aggregation-diffusion equations in one-dimensional heterogeneous environments.
• To develop a method for predicting emergent space use patterns by minimizing an energy functional.
• To analyze the combined effects of self-attraction and environmental resource attraction on aggregation.

Main Methods:

• Derivation of analytic expressions for steady-state solutions with quadratic diffusion.
• Minimization of an energy functional to predict emergent space use patterns.
• Numerical simulations to verify analytical predictions in a resource clump scenario.

Main Results:

• Identified counter-intuitive findings: non-monotonic clump width dependence and positive correlation between self-attraction and aggregation width under strong resource attraction.
• Demonstrated the combined influence of self-attraction and resource distribution on aggregation patterns.
• Validated analytical predictions through numerical simulations.

Conclusions:

• Rigorous demonstration of how environmental factors and collective behavior interact to shape organism space use.
• Highlights the potential for unexpected emergent patterns in biological aggregations.
• Provides a mathematical framework for studying spatial ecology in complex environments.