Hierarchical random walks in trace fossils and the origin of optimal search behavior

Abstract

Efficient searching is crucial for timely location of food and other resources. Recent studies show that diverse living animals use a theoretically optimal scale-free random search for sparse resources known as a Lévy walk, but little is known of the origins and evolution of foraging behavior and the search strategies of extinct organisms. Here, using simulations of self-avoiding trace fossil trails, we show that randomly introduced strophotaxis (U-turns)--initiated by obstructions such as self-trail avoidance or innate cueing--leads to random looping patterns with clustering across increasing scales that is consistent with the presence of Lévy walks. This predicts that optimal Lévy searches may emerge from simple behaviors observed in fossil trails. We then analyzed fossilized trails of benthic marine organisms by using a novel path analysis technique and find the first evidence, to our knowledge, of Lévy-like search strategies in extinct animals. Our results show that simple search behaviors of extinct animals in heterogeneous environments give rise to hierarchically nested Brownian walk clusters that converge to optimal Lévy patterns. Primary productivity collapse and large-scale food scarcity characterizing mass extinctions evident in the fossil record may have triggered adaptation of optimal Lévy-like searches. The findings suggest that Lévy-like behavior has been used by foragers since at least the Eocene but may have a more ancient origin, which might explain recent widespread observations of such patterns among modern taxa.

Keywords: Brownian motion; climate change; scale invariance; superdiffusion.

Fig. 1.

Simple fossil behavior may give rise to Lévy patterns. (A) Simulated self-avoiding, trail-following random walk and (B) an example of a self-avoiding fossil trail, Cosmorhaphe, from the Lower Eocene (Beloveža Beds, sample UJTF77; Table S1). (C) The same simulation as in A but with introduced strophotaxis (U-turns) in response to randomly distributed obstructions or innate cues. Note the similarity in trail form between B and C. (D) Cumulative frequency distribution of move step lengths between turns (○) for the simulation exemplified in C shows a model best fit to a truncated power law with µ = 2.14 (red line) compared with the exponential (blue line). In A and C, track intersections are evident, as occasionally a self-avoiding walker traps itself and is forced to cross (intersect) itself. The trajectories in A and C are random, although this becomes clear only for very long tracks.

Fig. 2.

Brownian walks in trails of Eocene worm-like animals. Trace fossils of (A and B) Helminthorhaphe flexuosa and (E and F) Cosmorhaphe tremens (samples D1L2 and D2L6, respectively; Table S1 and Fig. S1) and the digitized trails. For scale: coin diameter, 19.75 mm. MLE with wAIC shows model fits for H. flexuosa to a (C) two-exponential CB distribution on the x axis (red line, CB best fit; blue line, truncated power law) and (D) a truncated power law fitting only the tail of the data (red line) on the y axis vs. an exponential (blue line). For C. tremens, model best fits (G and H) to a CB with two and three exponentials on the x and y axes, respectively (red lines), compared with a truncated power law.

Fig. 3.

Hierarchically nested random walks in fossil echinoid trails. (A) Scolicia specimen (MBA-3d) from the Lower Eocene Higuer–Getaria Formation (coin diameter, 20 mm) and (B) the digitized trail. (C and D) Model best fits to truncated Pareto–Lévy power laws (red line) for the move step-length distributions in both x and y axes compared with exponential fits (blue line). (E and F) Marginally better fits to four-exponential CB distributions (red line) in both x and y axes vs. truncated power laws (blue line) indicate very similar fits to empirical move step data and, therefore, that a multimodal or CB walk of this Scolicia specimen is finely tuned to a Lévy walk.

Fig. 4.

Scolicia trails reflect sparse and abundant resource landscapes. (A and B) A spatially extensive Scolicia trail (MBA-1) 6.2 meters long shows (C) similar model fits of a truncated Lévy power law (blue line) and a CB walk (red line), with many long, ballistic move steps characteristic of movement responses to resource scarcity. Gray box in B denotes photographed area in A. (D and E) Scolicia specimen (MBA-4a) from a different location within the Lower Eocene Higuer–Getaria Formation shows more intensive walk clusters at several different scales. (Scale: acetate sheet length, 29.7 cm.) (F) A model best fit to a CB distribution with three exponentials (red line; y-axis steps shown) and strong fits to truncated Lévy distributions in both axes also were found (blue line, y-axis steps shown; Table S3), indicating fine-tuning of multimodal Brownian walks to a Lévy walk that is optimal in a patchy resource environment. (G) Putative assemblage of many individual self-avoiding Scolicia trails from the same formation may represent aggregation in an abundant food patch. Area shown, 1.85 × 1.16 m.