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

Chunlai Mu1, Weirun Tao1,2

  • 1College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China.

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|March 11, 2023
PubMed
Summary
This summary is machine-generated.

This study proves global bounded solutions for a modified chemotaxis-growth system, unlike classical models that can exhibit blow-up. The research explores pattern formation and stability in biological aggregation models.

Keywords:
accelerationamplitude equationchemotaxispattern formationstabilization

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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.