Decay, turnover, and residence time of pools of stuff

Mathmatical models describing the decay or turnover of stuff. I use these to describe the docomposition of organic matter, the size of organic matter pools as a function of their inputs and outputs, but the same concepts and mathematical models are applicable to the decay of radionuclides, hydrologic turnover, and other phenomena.

Decay functions (no inputs)

The general function for decay of leaf litter (for example) is:

$$L_t = L_0e\^{-kt} $$ or, $$ ln \frac{L_t}{L_0} = -kt $$

where $L_0$ is the mass at time 0, $L_t$ is the mass at time $t$, and $k$ is the decomposition constant. The mean residence time, or time required for $L_0$ to decompose under steady state conditions equals $\frac{1}{k}$.

Dual pool decay

Turnover functions (inputs and outputs)

When a pool of stuff has both inputs and outputs, the change in the pool with time is defined as

$$ \frac{\partial S}{\partial t} = I - kS$$

where $I$ is the input to the pool, $k$ is the decomposition rate, and $S$ is the size of the pool.