264. Feedback Loops
Closed causal chains where an effect comes back to influence its cause. The engine of dynamic behavior in system dynamics.
264.1. Two types
Reinforcing loops (R, positive feedback): the loop amplifies changes. Math: with . Solution: exponential growth .
Balancing loops (B, negative feedback): the loop resists changes. Math: with . Solution: exponential approach to target with time constant .
264.2. Behavior patterns
| Loop structure | Behavior | Example |
|---|---|---|
| Pure reinforcing | Exponential growth/collapse | Compound interest, viral spread |
| Pure balancing | Goal-seeking to steady state | Thermostat, predator-prey equilibrium |
| R + B (limit) | S-curve (logistic) | Market penetration, learning curves |
| Two B with delay | Oscillation (damped or sustained) | Inventory cycles, beer game |
| R + B unstable | Overshoot and collapse | Resource depletion |
264.3. First-order behavior
Reinforcing ():
- Doubling time:
- Halving time (if ): same formula, opposite sign
Balancing ():
- Time constant : time to reach of the way to
- Settling time (within 5%):
- Settling time (within 1%):
264.4. Combined loops: limits to growth
Reinforcing growth () plus balancing loop tied to capacity :
Initially R dominates → exponential growth. As , the balancing factor shrinks . Asymptote: . S-curve.
See Logistic Growth.
264.5. Second-order: oscillation
A loop with delays produces oscillation. Simple example: (damped harmonic oscillator).
| Damping | Behavior | Pattern |
|---|---|---|
| Sustained oscillation | sine wave | |
| Damped oscillation | decaying sine | |
| Critical damping | smoothest approach | |
| Over-damped | no oscillation, slow approach |
264.6. Why feedback loops matter
Most “surprising” system behavior comes from feedback:
- Bullwhip in supply chains: balancing loops with delays amplify variance
- Boom-and-bust cycles: capacity adjustment delays
- Population dynamics: birth/death feedbacks
- Climate: ice-albedo, methane release reinforcing loops; weathering balancing loops
Without feedback, a system can’t generate dynamic patterns; it’s just open-loop input → output.
264.7. Identifying loops in a system
- List variables and how they affect each other
- Trace cycles in the diagram
- Compute polarity of each cycle (R or B)
- Identify dominant loops in each behavior phase
The same model can have R-dominant phase (early growth) and B-dominant phase (saturation).