Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with...

Area of Science:

• Environmental Science
• Mathematical Biology
• Stochastic Systems

Background:

• Microcystins pose environmental and health risks.
• Understanding their degradation dynamics is crucial.
• Stochastic models offer a robust framework for complex biological systems.

Purpose of the Study:

• To analyze a stochastic Microcystins degradation model with distributed delay.
• To establish the existence and uniqueness of solutions.
• To derive the probability density function (PDF) of the model.

Main Methods:

• Demonstration of global positive solutions.
• Derivation of a stochastic critical value ($R_0^s$).
• Construction of Lyapunov functions to prove ergodic stationary distribution.
• Application of Fokker-Planck equation methods for PDF derivation.

Main Results:

• Existence and uniqueness of a global positive solution confirmed.
• A stochastic critical value ($R_0^s$) related to the basic reproduction number ($R_0$) was derived.
• Ergodic stationary distribution exists when $R_0^s > 1$.
• The exact expression for the PDF around the quasi-endemic equilibrium was obtained.

Conclusions:

• The derived PDF fully characterizes the stochastic model's dynamics.
• Theoretical findings are validated through numerical simulations.
• The model provides insights into Microcystins degradation processes.