INTERACTIVE COMPANION · FIELD NOTE · MARINE PHYSIOLOGY

The Oxygen Budget Bench

A live model accompanying “Clownfish Shrink to Survive Heatwaves”

During a marine heatwave in 2023, a hundred wild clown anemonefish out of 134 measurably lost body length, and the ones that shrank were more likely to be alive at the end. That is the observation. The two benches below let you take the usual explanation apart and see how much of the work it can actually do.

Both models are the club's own simplified calculation rather than the paper's analysis. Every constant in them comes from the linked Python script, and the numbers they print at their default settings are the same numbers the article quotes.

Model 1. The oxygen budget

Warming does two things to a fish at once, and they pull in opposite directions. Warm water holds less dissolved oxygen, so supply drops. A warm ectotherm burns more energy, so demand rises. Everything in this bench is the arithmetic of those two lines crossing.

supply = mass^d_g × O₂(T) demand = mass^0.75 × Q₁₀^{(T−28.5)/10}
ratio = supply ÷ demand, normalised to 1.000 for a 67.50 mm fish at 28.5 °C

The only place body size enters is through the difference between the two exponents. Gill surface area scales with mass as mass^d_g, metabolism scales as mass^0.75, and mass itself scales as length cubed. So the size term is length raised to 3(d_g − 0.75). Set d_g below 0.75 and a shorter fish gets a better ratio. Set it above 0.75 and a shorter fish gets a worse one. Nobody has measured it for this species.

Water temperature 32.5 °C 32.5 is the club's reference of 28.5 plus the 4 °C anomaly the study anemones actually recorded. Drag down to 28.5 and watch the ratio return to exactly 1.000. Drag up to 34 and watch how much faster the demand line climbs than the oxygen line falls. Body length 67.50 mm 67.50 mm is the mean starting length of the dominant females. Try shrinking by 1.4 percent, to 66.55 mm, which is the typical real shrink, and see how little the ratio moves. Then try to find the length that pushes the ratio back to 1.000 at this temperature. You will run out of slider. Gill-area exponent d_g 0.70 This is the softest joint in the whole model. At 0.75 body size drops out of the ratio completely and the length slider stops doing anything at all. Above 0.75 the sign flips and shrinking hurts. Published intraspecific values straddle 0.75. Q₁₀ for metabolic demand 2.0 How much faster the fish burns energy per 10 °C. 2.0 is conventional for teleost resting metabolism. Push it to 3.0 for a picture of an animal near its thermal limit, where the squeeze gets much worse and shrinking still barely helps.

dissolved O₂: – supply: – demand: – ratio: – length that restores balance: –

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At the default settings the ratio reads 0.7121, which is the number in Figure 2 of the article, and the restoring length reads 7.02 mm. A 67.50 mm fish would have to lose 89.60 percent of its length to balance its books this way. The fish lost one or two percent.

Why the ratio is the wrong ledger

The bench above asks whether a fish can improve its supply per unit of demand. That is the gill-limitation question, and the answer it gives is dismal: a real shrink buys about 0.0248 °C of relief.

There is a second ledger. A smaller fish needs less oxygen and less food in absolute terms, because metabolic rate goes as mass^0.75 and mass goes as length cubed. A 1.4 percent shrink cuts the total bill by 3.12 percent, which is worth 0.4577 °C, eighteen times more. Watch the demand bar in the stage above, rather than the ratio, and you are reading that second ledger instead of the first.

Model 2. The survival experiment

The paper reports that shrinking raised survival probability by as much as 78 percent, an odds ratio of about 1.78, from a Cox proportional-hazards model on 134 fish. This bench runs a much cruder experiment: a simulated population of 6,000 fish in 3,000 breeding pairs, through one heatwave, with a binary survived-or-died outcome.

Each fish shrinks or does not, at the rank-specific rates the paper reports, 71 percent of dominant females and 79 percent of breeding males. Shrinkers all shrink by the amount you choose. Survival comes from the logistic coefficients our Python fit returned, which are printed under the stage, and a fish whose partner also shrank carries an extra bonus.

Shrink magnitude 1.4 % of total length 1.4 percent is the club's assumed typical shrink, just under a millimetre on a dominant female. Slide to 0 and the odds ratio collapses towards 1, because nothing distinguishes a shrinker from a non-shrinker any more. Slide to 5 and watch how far past the paper's figure the model runs. Population 6000 fish Drag this down to 134, the real study size, and press Run a few times. The odds ratio bounces around wildly and its confidence interval usually includes 1. That is the whole reason the real study used repeated measures and time-to-event models instead of a single yes-or-no at the end. Force tandem shrinking within pairs Off, each fish decides independently and pairs end up matched by chance about three quarters of the time. On, every pair either shrinks together or not at all, which pushes far more fish into the tandem bonus and lifts the whole population's survival. Compare the two bar groups on the right of the stage. Each run redraws the population from a fresh seed. At 6,000 fish the answer barely moves. At 134 it moves a great deal.

survived: – no shrink: – shrank alone: – shrank in tandem: – fitted odds ratio: –

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Model survival, no shrink: 71.59%
Model survival, 1.4% solo: 76.14%
Model survival, 1.4% tandem: 83.52%
Expected odds ratio at defaults: 1.7666
The paper's figure: 1.78 (a 78% survival gain)
Our article's fitted value: 1.7310 [1.5121, 1.9816]
This bench, first run: 1.6088 [1.4037, 1.8438]

What would break all of this

Three things, and we would like you to try all of them on the benches above rather than take our word for it.

First, set d_g to 0.80 in Model 1. The length slider now runs backwards: shrinking makes the oxygen ratio worse, and the whole gill-limitation explanation reverses sign on an input nobody has measured for this species.

Second, set the shrink magnitude in Model 2 to zero and then to five. The odds ratio tracks the magnitude smoothly, because we built it that way. The model cannot tell you whether shrinking causes survival or merely accompanies it, and neither can an observational study in the wild.

Third, drop the population in Model 2 to 134 and run it ten times. Note how often the confidence interval covers 1. A study of that size with a binary endpoint has about a 15 percent chance of detecting a true odds ratio of 1.78, which tells you how much work the real paper's repeated measurements are doing.