A Neighbor's Calm as a Safety Signal express
A lone fish sees a shadow overhead. Should it flee? If it's just a leaf, it wastes energy and gives away its position to a predator. If it's a heron, hesitation is fatal. That's the eternal trade-off of the solitary individual: quick reactions mean many false alarms, while caution means risking being eaten.
In a school, it's different. A neighbor that calmly keeps nibbling on algae is living proof that there's no danger. Scientists built a model where each fish accumulates ambiguous signals like drops in a glass and darts off when the level overflows. It turns out that a neighbor's flight is strong evidence of a threat, but its stillness is evidence against one. A naive fish reacts only to others' flights and, as the school grows, starts darting at every little thing. A smart fish weighs both: it's as if it discounts the threat while all is quiet around it.
This discount has a precise measure. From the school's behavior, one can calculate how much an animal devalues an alarm signal when neighbors are calm. And this number has a limit: if a fish watches too many neighbors, a single discount is no longer enough to suppress false alarms. Observations of wild mollies attacked by birds showed that their discount is much larger than what simple individual accumulation of signals could produce. That's why false alarms in the school don't increase even as the number of fish grows.
Thus, the school turns the solitary individual's weakness, inevitable errors, into strength. A neighbor's calm becomes as important a signal as its flight.
Original abstract
as written by the authors arXiv:2608.14965v1
Solitary animals face a tradeoff when detecting threats: faster detection means accepting more false alarms. We show that groups can manage this tradeoff better by treating an undisturbed neighbor as evidence against a threat, becoming both faster and more accurate than lone individuals. Modeling each animal as a noisy evidence-accumulator that flees when its belief crosses a threshold, we find that a neighbor's flight signals danger while its stillness signals safety. A naive responder reacts only to flights and inflates false alarms as the group grows; a Bayesian responder weighs both, approximated by a single social discounting rate that interpolates between these limits. This yields closed-form expressions for group performance, including cascade branching ratios that stay strongly subcritical in safety and turn supercritical under threat, so the rate at which an animal discounts a threat while its neighbors stay still can be inferred from behavior alone, and it sets a ceiling on how many neighbors an animal can attend before discounting alone can no longer hold its false-alarm rate. Wild sulphur molly shoals under bird attack are best described by discounting rates well above what individually Bayesian updating supplies over any neighborhood they could plausibly attend, and the same model, at the inferred value, predicts a false-alarm rate that stays constant as shoals grow.
In mollies, false alarms don't become more frequent as the school grows, even though it might seem that the more fish there are, the higher the chance that one of them makes a mistake.