River flow: how a stock responds to its flows

A river fills a reservoir and the dam releases a fixed share of whatever is stored each month. Change the flows and watch the water level head toward the point where inflow equals outflow.

View as:

Inflow

Stock

Outflow

Time t

0 months

Stock S

Inflow I(t)

Outflow O(t)

Net change dS/dt = I − O

Steady state S* = Ī/k

Time constant 1/k

0 months

Half-life ln2/k

0 months

Average inflow Ī Mean amount that flows in each month.

Seasonal swing a Inflow swings ± this % around its mean over a 12-month cycle.

Outflow rate k Share of the current stock that leaves each month.

Simulation speed Simulated months per real second.

Reservoir capacity 1,000 (excess spills)

Stock S (left axis) Steady state S* (left axis) Inflow I (right axis) Outflow O (right axis)

dS/dt = I(t) − k·S · I(t) = Ī(1 + a·sin(2πt/12)) · S* = Ī/k