Feedback Loops: The Math of Systems That Change Themselves
Put $100 in a savings account at 5% interest per year and leave it alone. After year one you have $105. But year two's interest is calculated on $105, not $100, so you earn $5.25. The next year, even more. Your money is feeding on its own growth. This is a feedback loop, and it is the beating heart of systems thinking. A feedback loop happens whenever the OUTPUT of a system loops back and becomes part of its own INPUT. Learn to spot loops and you can predict the behavior of everything from bank accounts to animal populations to a thermostat in your house.
What You'll Learn
- The difference between a reinforcing loop and a balancing loop - Why reinforcing loops produce exponential (not straight-line) growth - How balancing loops create stability and hold a value near a target - How to model a real loop with a simple recursive rule and a table
Reinforcing Loops: More Leads to More
A reinforcing loop amplifies itself. More causes more; less causes less. Compound interest is the classic case: the recursive rule is A(next) = A(now) x 1.05. Run it forward: 100, 105, 110.25, 115.76, 121.55... Notice the gaps between terms keep growing. That is the signature of exponential growth: the change each step is proportional to the current size. Rabbit populations, viral rumors, and a snowball rolling downhill all follow reinforcing loops until something stops them.
Balancing Loops: Pushing Back Toward a Target
A balancing loop resists change and seeks a target. A home thermostat is the model: if the room is colder than 68 degrees F, the heater turns on; once it passes 68, the heater turns off. The output (temperature) feeds back to control the input (heating), holding the room near a set point. Mathematically, the correction each step is proportional to the GAP from the target: change = k x (target - current). The bigger the gap, the harder the system pushes; as the gap shrinks toward zero, the system settles. This is why your coffee cools quickly at first and then slowly levels off near room temperature.
A reinforcing loop has no opinion about good or bad. At 5% it grows your savings; at 22% credit-card interest it grows your debt just as relentlessly. The math is identical; only the sign of the outcome differs.
Real loops often have a time lag between action and effect. A delay in a balancing loop causes overshoot and oscillation, which is why a shower with slow-reacting pipes swings from freezing to scalding as you overcorrect.
Match each real-world system to the type of feedback loop that best describes it.
Terms
Definitions
Drag terms onto their definitions, or click a term then click a definition to match.
A population follows the rule P(next) = P(now) x 1.08. What kind of growth is this, and why?
In a balancing loop with the rule change = k x (target - current), what happens as 'current' approaches 'target'?
Model Two Loops in a Table
On paper or a spreadsheet, build two 10-step tables. TABLE 1 (reinforcing): start at 100 and apply A(next) = A(now) x 1.10 each row. TABLE 2 (balancing): start at 40, target 70, and apply next = current + 0.5 x (70 - current) each row. Plot both as line graphs. Write a short paragraph explaining which curve keeps steepening and which one flattens toward a value, and name one real system that behaves like each.
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