Logistic growth, its limits, and what the numbers tell us about a real pond in the west of Ireland.
These are real tadpoles from a real clutch, spawned on 22 February 2026. Everything below was calculated from weekly observations, and the survival counts were submitted to the IPCC Hop To It survey. Before the maths, three things worth knowing about the animal you are modelling.
The chart plots the tadpole's actual measured length each week against the length the model predicts. They agree beautifully from week 0 to about week 11. Then, in the shaded weeks, they part company.
| Week | Actual (mm) | Model L(t) (mm) |
|---|---|---|
| 0 | 5 | 5.0 |
| 1 | 7 | 7.3 |
| 2 | 10 | 10.5 |
| 3 | 14 | 14.4 |
| 4 | 19 | 18.9 |
| 5 | 24 | 23.6 |
| 6 | 29 | 28.1 |
| 7 | 33 | 31.9 |
| 8 | 36 | 34.9 |
| 9 | 39 | 37.2 |
| 10 | 41 | 38.8 |
| 11 | 42 | 39.9 |
| 12 | 42 | 40.6 |
| 13 | 40 | 41.1 |
| 14 | 18 | 41.4 |
The photograph of the real tadpoles at week 3 showed them at about 9 mm, not 14.4. The model is not wrong, it is assuming the wrong pond. It expects water at about 18°C, but an Irish garden pond in March sits at 8 to 12°C, so the real tadpoles grow more slowly. The lesson is the quiet one: every published growth model carries hidden assumptions, and a real local place may not honour them.
That is a gap of more than half the length of the tadpole, and it is not measurement error. The tadpole has begun metamorphosis. A surge of thyroid hormone signals every tissue to remodel. The tail is consumed from the inside by programmed cell death, and the nutrients fuel the growing legs. The intestine shrinks by about 90% to rebuild for a carnivore's diet. The eyes migrate across the skull. The animal becomes shorter because it is becoming something else.
The logistic model has no term for this. It can describe growth, but not transformation.
(i) The model assumes the tadpole keeps doing one thing: growing longer toward a fixed ceiling, ever more slowly. (ii) That assumption breaks at week 12 because the animal stops simply growing and starts transforming, its body remodelled rather than enlarged. (iii) So the measured length falls while the model keeps rising, and the gap at week 14 is biology the equation was never built to hold.
Growth is only half the story. The other half is how many survive. Cumulative survival is each week's count divided by the original 200 eggs.
| Week | Date | Count | Survival | Notes |
|---|---|---|---|---|
| 0 | 22 Feb | 200 | 100% | Eggs in jelly |
| 1 | 1 Mar | 180 | 90% | Some eggs white (unfertilised) |
| 2 | 8 Mar | 120 | 60% | Hatchlings, yolk sac visible |
| 3 | 15 Mar | 95 | 47.5% | Free-swimming, feeding |
| 4 | 22 Mar | 72 | 36% | Growing, size variation |
| 5 | 29 Mar | 58 | 29% | Active, darker colour |
| 6 | 5 Apr | 45 | 22.5% | External gills gone |
| 7 | 12 Apr | 38 | 19% | Hind leg buds visible |
The single worst transition is week 1 to week 2, when 60 tadpoles are lost in seven days. This is the hatching bottleneck: the yolk sac runs out and the hatchlings must start feeding themselves. Many do not make it.
| Stage | Count (from 3,000 eggs) | % of original |
|---|---|---|
| Eggs spawned | 3,000 | 100% |
| Hatch | 2,790 | 93% |
| Froglets (metamorphosis) | 120 | 4% |
| Breeding adults (year 3) | 5 | 0.17% |
The tank runs at roughly five times the wild survival rate, because it removes the pressures that kill most wild tadpoles: predators, frost, drying out, food insecurity. But the tank is not neutral either. High density, restricted volume and no natural food web bring stresses the wild animal never meets. Five breeding adults from three thousand eggs is a 99.83% loss. Every garden pond matters.
Same maths, new medium, real contribution. The next step turns this spreadsheet into a camera: a few lines of Python and a small sensor at the pond, capturing growth and survival every day, and feeding the same national survey. The numbers a student calculates by hand become a living data stream.