Extended Data Fig. 1: Shape of interaction rate functions is preserved under continuous variation in the degree of directed motion.

(a) Rate of drift toward target as a function of distance from the target for the two-state model described in the main text with l_*=100 body lengths (black line), and for three alternative models in which drift rate toward the target is a saturating function of a signal that decays exponentially (purple), like a Gaussian (orange) or like an inverse-square power law (green) with distance from the target. Signal decay parameters were chosen so that the mean drift speed at distance 100 body lengths are equal for all models. Drift rate, H(l), is calculated as a saturating function of the signal to constrain searchers to a maximum drift speed: H(l)=−v_maxtanh(S(l)), where S(l) is the signal value at a distance l, from a target (see Supplementary Discussion for details). The hyperbolic tangent form is motivated by past work on signal-dependent taxis responses^15,16; other saturating functions yield similar results. (b) Interaction rate computed by solving Eq. (1) using the drift rates shown in panel (a). (c) Per-capita interaction rates corresponding to interaction rates shown in panel (b). Note non-monotonic form similar to that shown in Fig. 2c of the Main Text. (d) Range of drift functions with different parameter values. Each curve is one parameterization of the exponential signal function shown in panel (b) with drift rate again given by H(l)=−v_maxtanh(S(l)). Dark colored curves indicate functions in which change in drift rate with distance from target is relatively slow, whereas lighter colors are drift rate functions with more abrupt transition from high to low drift rate as distance from target increases. (e) Per-capita encounter rate as a function of target density for the same curves shown in panel (d). Note that location and height of peak changes but qualitative shape of curve is preserved.