1Demonstration 1 of 4
An early limit is easy to dismiss
How much of the unconstrained growth is a limit already taking away at low load?
The pack's logistic step adds rate x state x (1 - state / capacity). Dividing by the unconstrained growth, rate x state, leaves 1 - state / capacity, a straight line from 1 at no load to 0 at full capacity.
load is the state as a share of the capacity of 100. rate is the unconstrained growth rate per step. The multiplier 1 - load / capacity is the share of the unconstrained growth that remains.
Predict first. At 10% of capacity, what share of the unconstrained growth is left?
Choose an example
Constructed example: the chapter's logistic engine with load shares and rates chosen for this reader.
Calculated values
- Load (units of 100)
- 2.0
- Share of unconstrained rate
- 0.98
- Growth with no limit (per step)
- 0.60
- Growth with the limit (per step)
- 0.59
- Shortfall (per step)
- 0.01
Multiplier = 1 - 2.0 / 100 = 0.98. With no limit one step adds 0.3 x 2.0 = 0.60; with the limit it adds 0.60 x 0.98 = 0.59. The shortfall is 0.01 per step, 2% of the unconstrained growth.
Use the idea
A modest shortfall at low load is exactly the size that reads as noise, so name the limit before it is measurable.
Where the conclusion applies
A capacity of 100 held fixed and one step of dt = 1. The logistic form assumes the shortfall is linear in load; a limit that bites only near the top would not look like this.
Check your understanding: At 50% of capacity with rate 0.4, how many units does one step add?
Chapter 9 source: section "Four templates and what each one warns about". Demonstration C09-D01.
2Demonstration 2 of 4
Same engine, two boundaries
If the limit sits outside the model, then inside it, do the two runs agree?
A approaches its fixed ceiling and stays. In B the growing state consumes the capacity that supports it, so growth stalls, then the state follows the shrinking capacity down.
A holds capacity at 100. B starts at 100 and loses 0.02 x state of capacity each step. Both start at 1 and use the same growth rate.
Predict first. With rate 0.4 over 200 steps, does B end higher or lower than A?
Choose an example
Constructed example: the chapter's own calls to the pack, with the rate and horizon varied for this reader.
Calculated values
- A: value at the end
- 100.0
- B: peak
- 87.6
- B: step of the peak
- 20
- B: value at the end
- 2.0
- B: capacity at the end
- 1.9
Both start with 1 + 0.4 x 1 x (1 - 1 / 100) = 1.396 after one step. Then B's capacity falls by 0.02 x state each step: 100 - 0.02 x 1 = 99.98. After 200 steps A is at 100.0, while B peaked at 87.6 (step 20), ended at 2.0, and left a capacity of 1.9.
Use the idea
Before accepting an archetype name, ask whether the limit is a fixed property of the environment or something the growth consumes.
Where the conclusion applies
One erosion rate, 0.02, and a state that cannot go below zero. If the growing thing also replenished its limit, B would not collapse.
Check your understanding: With rate 0.3 and 120 steps, what value does A end at?
Chapter 9 source: section "The boundary decides the behavior". Demonstration C09-D02.
3Demonstration 3 of 4
The window decides what the record can show
How long must a record be before it can separate an eroding limit from slowed growth?
Both observables are properties of the record, not of the structure. A window that ends before the state has fallen 5% below its peak fires neither test.
The record is B's state path over the chosen number of steps. Peak is its highest value. The peak-then-fall test asks whether the last value is below 95% of the peak. The settle test asks whether the last step moved by less than 0.01.
Predict first. With erosion 0.005, can a 30 step record show the fall?
Choose an example
Constructed example: paths from the pack's eroding_limit with erosion and window chosen for this reader.
Calculated values
- Peak
- 87.6
- End of record
- 39.1
- 95% of peak
- 83.2
- Last step change
- (-0.84)
- Settles
- no
- Peaks then falls
- yes
Threshold = 0.95 x 87.6 = 83.2. The record ends at 39.1, below it. Last step change = 39.06 - 39.90 = (-0.84). Peaks then falls: the end is more than 5% below the peak, so the record separates B from A.
Use the idea
Choose the window for a distinguishing observation from the time the mechanism needs to act, and write it beside the claim.
Where the conclusion applies
Rate 0.4, capacity 100 and a 5% margin. A different margin moves the window at which the fall is detected; a noisy real record would need a wider one.
Check your understanding: A peak of 87.6 and an end of 73.9: is the end below 95% of the peak?
Chapter 9 source: section "The instantiation worksheet". Demonstration C09-D03.
4Demonstration 4 of 4
One parameter turns B into A
What happens to the eroding-limit model when the erosion rate is zero?
With no erosion B's capacity stays at 100 and its update is the same as A's, so the paths coincide. Each positive erosion rate opens a gap that grows with the rate.
\[erosion_rate\]
erosion_rate is the capacity lost per unit of state per step. Everything else, the capacity of 100, the start of 1 and the growth rate, is shared by A and B.
Predict first. At erosion rate 0, how large is the largest gap between B and A?
Choose an example
Constructed example: the chapter's zero-erosion test, run for several erosion rates chosen for this reader.
Calculated values
- Erosion rate
- 0.000
- Largest gap between B and A
- 0.00
- B: value at the end
- 100.0
- B: capacity at the end
- 100.0
- Same path as A
- yes (tie)
After the first step the capacity is 100 - 0.000 x 1 = 100.000. Over 200 steps the largest gap between B and A is 0.00 units and B ends at 100.0 against A's 100.0. The two paths are the same, so B has become A.
Use the idea
When two accounts differ in outcome, find the single quantity whose value makes one reduce to the other; the archetype name rarely mentions it.
Where the conclusion applies
A 200 step horizon and a shared growth rate. The exact tie holds for this update rule only; a different erosion mechanism would need its own zero case.
Check your understanding: At erosion rate 0.02, what is B's capacity after the first step from a state of 1?
Chapter 9 source: section "The boundary decides the behavior". Demonstration C09-D04.