Abstract
Catch of randomly walking organisms by a trap can occur only after an intersection with the trap or the attractive plume from a trap. The proportion of a population of ballistic (straight-line) movers caught after embarking on random headings from a point at distance r from a trap is given by the size of the trap (or reach of its plume) divided by the circumference of a circle centered on the movers’ origin and bisecting the trap (or 2πr). Since r resides in the denominator of this equation rather than numerator, the proportion of ballistic movers caught vs. distance of origin is a simple inverse function, not a negative logarithmic function for which it has previously been mistaken. Therefore, plotting distance of ballistic mover origin against the inverse of proportion caught (Miller–Adams–McGhee, MAG plot) yields a straight line. Dividing 2π by the MAG plot slope conveniently returns trap length for ballistic movers. Meandering in the tracks of random walkers alters this relationship only slightly by imparting an additional increment to the above measure of trap length that we call gain. Gain is directly correlated with foraging efficiency of random walkers and is maximized under particular regimes of resource size and spatial distribution by the use of particular c.s.d. values. For example, c.s.d. values of ca. 10–30° yield the highest gains for simulated insects responding to typical pheromone plumes of their calling mates or monitoring traps. Optimal c.s.d. values will be smaller for organisms foraging for larger resources and vice versa. Knowledge of these relationships combined with simulation tools led to the discovery of how to translate data from single-trap, multiple-release experiments into estimates of the reach of attractive plumes from traps.
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Notes
- 1.
Intersection probabilities have been extensively studied for Brownian motion (Wiener process) and uncorrelated random walks. In particular, Spitzer (1964) proved an asymptotic formula that can be used to compute the behavior of the expected catch C( t) at time \(t\rightarrow\infty\) in the limit t , for a trap positioned in an infinite plane populated with a density D of Brownian movers:
$$C(t)\sim 2\Pi Dt \left[\frac{1}{{\rm {1n}}\,t}+\frac{1}{({\rm{1n}}\,t)^{2}}[rc(T)+1+\gamma -1n 2]\right]$$where γ ≈ 0.5772… is Euler’s constant and rc( T) is “conformal radius,” or “logarithmic capacity,” of the trapping radius (see Kuz’mina for the definition). This result is not particularly useful for our analysis because we study correlated walks over time scales too short for this universal Brownian behavior to emerge. However, the above relation does suggest that the conformal radius of the trap might be the correct geometric parameter to correlate with capture probability, not the perimeter as considered in Sect. 4.5. In this regard, it is significant that each of the traps considered in Fig. 4.9 is a convex body (for which conformal radius and perimeter are comparable). We do not expect the relationship between perimeter and catch to extend to nonconvex traps, as such traps may have deep fjords in their boundary which greatly increase perimeter without increasing catch. Indeed, a trap with a fractal boundary could have infinite perimeter and finite catch.
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Miller, J., Adams, C., Weston, P., Schenker, J. (2015). Intersections of Movers with Traps. In: Trapping of Small Organisms Moving Randomly. SpringerBriefs in Ecology. Springer, Cham. https://doi.org/10.1007/978-3-319-12994-5_4
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DOI: https://doi.org/10.1007/978-3-319-12994-5_4
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