A general class of bounded diffusion processes with applications in the biosciences.

Area of Science:

• Mathematical Biology
• Stochastic Processes
• Dynamical Systems

Background:

• Bounded biological processes are often modeled using diffusion processes.
• Existing models, like stochastic logistic growth, have limitations in scope and flexibility.

Purpose of the Study:

• To develop and analyze a unified class of bounded continuous-time diffusion processes.
• To extend existing models to include polynomial, logistic, and Gompertz dynamics.
• To provide a flexible framework for empirical modeling of bounded biological processes.

Main Methods:

• Development of stochastic differential equations for bounded diffusion.
• Analysis of polynomial drift, generalized logistic, and Gompertz dynamics.
• Classification of boundary behavior and conditions for stationary distributions.

Main Results:

• A unified class of bounded diffusion processes is established.
• Explicit expressions for stationary distributions are derived when they exist.
• The framework unifies and extends several widely used bioscience diffusion models.

Conclusions:

• The developed framework significantly expands the repertoire of tractable bounded diffusions.
• This offers enhanced capabilities for modeling diverse bounded biological phenomena.
• The study provides transparent conditions for stationary distributions and boundary behavior.