Gaussian approximations for chemostat models in finite and infinite dimensions

Area of Science:

• Mathematical Biology
• Stochastic Processes

Background:

• Chemostat dynamics are modeled using mass-structured individual-based models.
• Previous work proved convergence to a classic partial differential equation.

Purpose of the Study:

• To investigate the convergence of fluctuation processes in chemostat models.
• To derive a Gaussian approximation for the Crump-Young model and analyze its long-time behavior.

Main Methods:

• Utilizing central limit theorems on Hilbert space.
• Analyzing processes within Sobolev spaces.
• Deriving invariant distributions and convergence properties.

Main Results:

• Convergence in law of the fluctuation process to an infinite-dimensional Gaussian process.
• A two-dimensional Gaussian approximation for the Crump-Young model.
• Derivation of the invariant distribution and its convergence.

Conclusions:

• The study provides a rigorous mathematical framework for understanding fluctuations in chemostat bacterial populations.
• The derived Gaussian approximation offers new insights into the long-term dynamics of the Crump-Young model.
• Numerical simulations support the theoretical findings.