# 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 $\frac{1}{k}$.

## 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.