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Feedback Loops

Initial Publication Date: December 21, 2006
Feedback Loops can enhance or buffer changes that occur in a system.

Positive feedback loops enhance or amplify changes; this tends to move a system away from its equilibrium state and make it more unstable.
Negative feedbacks tend to dampen or buffer changes; this tends to hold a system to some equilibrium state making it more stable.

Several examples are given here to help clarify feed back loops and to introduce loop diagrams.

Understanding negative and positive connections is helpful for understanding loop structure.

Hot coffee means large cooling rate
A positive connection is one in which a change (increase or decrease) in some variable results in the same type of change (increase or decrease)in a second variable. The positive connection in the figure below for a cooling coffee cup implies that the hotter the coffee is the faster it cools. The variables Tc and Tr are coffee and room temperature respectively.

Large cooling rate means drop in coffee temp
A negative connection is one in which a change (increase or decrease) in some variable results in the opposite change (decrease/increase) in a second variable. The negative connection in the figure below for a cooling coffee cup implies a positive cooling rate makes the coffee temperature drop.

Negative feedback example using a coffee cup
When these two connections are combined we get a negative feedback loop as shown at left in which the coffee temperature approaches the stable equilibrium of the room temperature. Going around the loop the positive connection times the negative connection gives a negative loop feedback effect. This same trick of multiplying the signs of the connections around a loop together to find out whether it is a positive or negative feedback loop works for more complicated loop structures with many more connections.

An example of positive feedback is world population with a fixed percentage birth rate. Positive feedbacks will result in unlimited growth (until checked) and are sometimes referred to as vicious cycles. In the figure below connecting population to births, large populations cause large numbers of births and large numbers of births result in larger population. This idea can be modeled nicely with the differential equation dP/dt=+rP, where P is population and r is the percent birth rate. The solution to this is P(t)=Po(exp[rt]) or exponential growth. Not all positive feedbacks give exponential growth but all, left unchecked, will result in unlimited (or unstable) growth. In the graph below we show the world population predicted for a fixed 2% growth rate from 1950 to 2050. Also shown is an estimate of future world population which is close to the mid-range United Nation Environmental Program (UNEP) best guess for future population to stress that exponential growth is not realistic for world population although it works fairly well for the time between 1950 and 1990. Logistic (S-shaped) growth would be a better choice for modeling world population for this 100 year time interval shown. UNEP Population

1950to2050WorldPopwithfixed2 growth and with more realistic UNEP projections for positive feedback discussion
Positive feedback example using world population