System Dynamics

Stocks and flows

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The image below represents a simple model of a bathtub. You can control the flows in and out. If more water flows is, the bath tub eventually empties. if more water flows out it will reach the maximun ( set at 1000liters) and an error message will be triggered. If thetwo flows are identical, the content of the bathtub remains constant at the inital value of 100 liters

Mathematical representation

Integral and differential equation of stock and flows (Sterman, 2000, p. 195)

S t o c k (t) = S t o c k (t_{0}) \int_{t_{o}}^{t} [I n (s) - O u t (s)] d s

Differential equation

d (S t o c k) / d t = I n (t) - O u t (t)

Feedback loops

Things get interesting when system have, and they usually do, feedback loops. A Feedback loop exists when a downstream velue has an impact on an upstream one.
In the model below , the value of the inflow is also controlled by the current value of the stock through a feedback that can be positive ( the inflow increses with the value of th bathtub) or negative ( the inflow decreases with the value of the bath tub). In the seond case, the result is that the bathtub reaches gradually a stable value (balancing loop)