Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

Area of Science:

• Mathematical Biology
• Computational Science
• Biophysics

Background:

• Nonlocal models describe phenomena like cell adhesion and chemotaxis.
• Local models represent similar processes using haptotaxis and classical chemotaxis.
• Existing mathematical frameworks have limitations in unifying these model types.

Purpose of the Study:

• To develop a rigorous limit procedure linking nonlocal and local models of cell behavior.
• To unify the mathematical treatment of adhesion/nonlocal chemotaxis with haptotaxis/classical chemotaxis.
• To extend the existing mathematical framework to handle general solution-dependent coefficients.

Main Methods:

• Reformulation of nonlocalities using integral operators applied to gradients.
• Development of a unified mathematical approach for diverse model types.
• Comparison of novel nonlocal operators with previous forms.

Main Results:

• A rigorous procedure is established to connect nonlocal and local models.
• Novel integral operators are introduced for a unified treatment.
• The extended framework accommodates general solution-dependent coefficients.
• Numerical simulations illustrate theoretical findings.

Conclusions:

• The proposed approach provides a unified mathematical link between different classes of biological models.
• The novel integral operators offer advantages over previous methods.
• This work advances the mathematical understanding of cell migration and pattern formation.