Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling

Area of Science:

• Mathematical Biology
• Partial Differential Equations
• Chemotaxis Modeling

Background:

• Investigates a three-component indirect signaling chemotaxis system with diffusion and reaction terms.
• Addresses the biological relevance of signaling processes occurring on faster timescales than species diffusion and interactions.

Purpose of the Study:

• To analyze singular limits of the chemotaxis system under parabolic-elliptic simplification (PES) and indirect-direct simplification (IDS).
• To rigorously establish convergence in critical dimensions for both simplification regimes.
• To determine convergence rates and analyze initial layer effects and convergence to the critical manifold.

Main Methods:

• For PES (ε→0+, τ fixed): Utilizes entropy functions, Adam-type inequalities, slow evolution regularization, and energy equation methods.
• For IDS ((ε,τ)→(0+,0+)): Employs a bootstrap argument on the L^p-energy function to derive uniform bounds.
• Analysis is performed up to critical dimensions N=4 for PES and N=2 for IDS.

Main Results:

• Achieved strong convergence in representative spaces for the PES regime.
• Obtained suitable uniform bounds for singular limits in the IDS regime.
• Provided convergence rates for both PES and IDS, highlighting initial layer effects and convergence to the critical manifold.

Conclusions:

• Successfully demonstrated singular limits in critical dimensions for the indirect signaling chemotaxis system.
• The methods developed provide a robust framework for analyzing similar systems in mathematical biology.
• The results offer insights into the dynamics of biological systems where signaling and diffusion operate at different rates.