Stabilization and pattern formation in chemotaxis models with acceleration and logistic source

Area of Science:

• Mathematical Biology
• Partial Differential Equations
• Chemotaxis Modeling

Background:

• Classical chemotaxis models often exhibit blow-up solutions, limiting their predictive power for long-term biological phenomena.
• Understanding the dynamics of cell aggregation and pattern formation is crucial in developmental biology and disease modeling.

Purpose of the Study:

• To investigate a modified chemotaxis-growth system with an acceleration assumption.
• To establish conditions for global boundedness of solutions and analyze pattern formation.
• To compare the behavior of this modified system with classical chemotaxis models.

Main Methods:

• Analysis of a coupled system of partial differential equations governing cell density, chemoattractant, and collective cell motion.
• Application of mathematical techniques including stability analysis and perturbation expansions.
• Numerical simulations to visualize emergent patterns and validate theoretical findings.

Main Results:

• Global bounded solutions are proven to exist under specific parameter regimes, contrasting with potential blow-up in classical models.
• Exponential convergence to a steady state is demonstrated for small chemotactic sensitivity parameters.
• The system exhibits rich pattern formation, including stationary, merging, chaotic, and periodic behaviors, with pitchfork bifurcations observed.

Conclusions:

• The modified chemotaxis-growth system offers a more robust framework for modeling biological aggregation, preventing blow-up.
• The study reveals complex pattern dynamics and bifurcations, highlighting the influence of the acceleration term.
• Further research is suggested to explore open questions regarding pattern selection and stability in extended parameter ranges.