Transit-time and age distributions for nonlinear time-dependent compartmental systems.

Area of Science:

• Environmental Science
• Mathematical Modeling
• Systems Ecology

Background:

• Compartmental systems are widely used to model natural processes, often as first-order differential equations.
• Key diagnostics include the age of matter and transit times, crucial for diverse scientific applications.
• Explicit formulas for nonlinear, time-dependent systems were previously unavailable.

Purpose of the Study:

• To derive explicit formulas for transit-time and age distributions in nonlinear, time-dependent compartmental systems.
• To generalize existing density formulas for linear, time-independent systems.
• To provide a method for calculating these distributions even when only numerical solutions are available.

Main Methods:

• Assumed well-mixed compartments.
• Constructed a linear time-dependent system from a nonlinear system's numerical solution.
• Derived equations for the time evolution of quantiles and moments of age distributions.
• Applied formulas to a global carbon cycle model.

Main Results:

• Developed novel explicit formulas for transit-time and age distributions in nonlinear, time-dependent compartmental systems.
• Showcased how to compute these distributions using numerical solutions of the original nonlinear system.
• Derived time-dependent age distributions for a global carbon cycle model, estimating fossil carbon removal times.

Conclusions:

• The derived formulas extend existing methods for linear systems.
• This work provides a powerful tool for analyzing complex environmental systems with time-varying dynamics.
• Enables quantitative assessment of pollutant or carbon residence times in dynamic systems.