# Fertilization in strong flows and high egg density.

Previous measurements of fertilization probability *P*_{fert}(*ϵ*) for sea urchin *S. purpuratus* at strong turbulence, characterized by density-normalized dissipation rate *ϵ* (filled gray triangles) [25, 38] and our corresponding simulations *P*_{fert}(*α*) as function of shear rate *α* (open blue circles, mean ± SD) match well, using a single fit parameter *a* = 0.075 that relates dissipation rate *ϵ* and typical shear rate *α* (using the known relationship [24, 26]). Both simulation and experiment are well captured by a minimal theory of a ballistic swimmer in simple shear flow (red), see Sec A in S1 Text. Fertilization probability *P*_{fert} rapidly drops above a characteristic flow strength *α* > 100 s^{−1}, which is consistent with a scale estimate *α* = 2*πv*_{h}/(0.1*r*_{egg}) (vertical dotted line). At these high shear rates, active swimming becomes negligible compared to convection. The case of low shear rates is well described by the limit case of a ballistic swimmer in the absence of flow *α* = 0 s^{−1} (dotted horizontal line, Eq (5) with *P*_{sperm:egg}(*t*) = 1 − exp(−*qt*) and rate ). The fertilizability *p*_{f} = 10% is obtained from an independent experiment [25], see Fig B in S1 Text. From the experimental protocol, we estimate a high background concentration *c*_{bg} = 500 − 4000 nM of chemoattractant, which renders sperm chemotaxis ineffective. Corresponding results for simulations with co-rotation are shown in Fig C in S1 Text.