Hierarchical Random Walks in Trace Fossils and the Origin of Optimal Search Behavior
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Hierarchical random walks in trace fossils and the origin of optimal search behavior David W. Simsa,b,c,1, Andrew M. Reynoldsd, Nicolas E. Humphriesa, Emily J. Southalla, Victoria J. Wearmoutha, Brett Metcalfee, and Richard J. Twitchettf aMarine Biological Association of the United Kingdom, Plymouth PL1 2PB, United Kingdom; bOcean and Earth Science, National Oceanography Centre Southampton, University of Southampton, Waterfront Campus, Southampton SO14 3ZH, United Kingdom; cCentre for Biological Sciences, University of Southampton, Highfield Campus, Southampton SO17 1BJ, United Kingdom; dRothamsted Research, Harpenden, Hertfordshire AL5 2JQ, United Kingdom; eEarth and Climate Cluster, Faculty of Earth and Life Sciences, VU University Amsterdam, 1081 HV, Amsterdam, The Netherlands; and fDepartment of Earth Sciences, The Natural History Museum, London SW7 5BD, United Kingdom Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved June 19, 2014 (received for review April 1, 2014) −μ Efficient searching is crucial for timely location of food and other “walk clusters” with no characteristic scale, such that P(l) ∼ l , resources. Recent studies show that diverse living animals use a with 1 < μ ≤ 3, where l is the move step length between turns and theoretically optimal scale-free random search for sparse resources μ the power-law exponent. Over many iterations, a Lévy walk will known as a Lévy walk, but little is known of the origins and evo- be distributed much further from its starting position than a lution of foraging behavior and the search strategies of extinct Brownian walk of the same length [hence is termed super- organisms. Here, using simulations of self-avoiding trace fossil trails, diffusive (8)], because small-step walk clusters are interspersed we show that randomly introduced strophotaxis (U-turns)—initiated by long “steps” to new locations, with this pattern repeating by obstructions such as self-trail avoidance or innate cueing—leads across all scales. It has been demonstrated that a Lévy walk with to random looping patterns with clustering across increasing scales exponent μ ∼2 is optimal when the search targets are not de- that is consistent with the presence of Lévy walks. This predicts that pleted or rejected once visited but instead may be revisited prof- optimal Lévy searches may emerge from simple behaviors observed itably, either because they replenish overtime or because targets in fossil trails. We then analyzed fossilized trails of benthic marine are distributed patchily (1, 2). Importantly, Lévy searches with ECOLOGY organisms by using a novel path analysis technique and find the first μ ∼2 are optimal for a very broad range of target densities and evidence, to our knowledge, of Lévy-like search strategies in extinct distributions (9). In the very low-density regime, Lévy strategies animals. Our results show that simple search behaviors of extinct remain the optimal solution with the optimal exponent 1 < μopt ≤ 2 animals in heterogeneous environments give rise to hierarchically dependent on specific environmental properties, such as the nested Brownian walk clusters that converge to optimal Lévy pat- degree of spatial landscape heterogeneity or temporal target terns. Primary productivity collapse and large-scale food scarcity revisitability (10, 11). Consequently, optimal Lévy searches result characterizing mass extinctions evident in the fossil record may in more predictable target encounters during foraging in other- have triggered adaptation of optimal Lévy-like searches. The find- wise unpredictable environments (9). Because Lévy walks can SCIENCES ings suggest that Lévy-like behavior has been used by foragers optimize search efficiencies in this way, it is proposed that natural APPLIED PHYSICAL since at least the Eocene but may have a more ancient origin, which selection should have led to adaptations for Lévy walk foraging might explain recent widespread observations of such patterns [the Lévy flight foraging (LFF) hypothesis] (1–3). The apparent among modern taxa. Brownian motion | superdiffusion | scale invariance | climate change Significance How best to search for food in heterogeneous landscapes is he specific pattern of searching movements used by an or- a universal problem facing mobile organisms. Diverse modern ganism to locate food relative to the food’s distribution closely T animals use a random search strategy called a Lévy walk, determines the number of successful encounters (1–3). The evo- composed of many small move steps interspersed by rare long lution of optimal search patterns is predicted because natural steps, which theoretically is optimal for locating sparse resour- selection favors individuals that are best able to find resources ces. Here, we find the first evidence, to our knowledge, that critical to survival (4). It is recognized, however, that the natural extinct animals, in this case 50 My-old sea urchins, used a Lévy- environment is too complex for evolution to produce a behavior like search strategy. Our results are important because they pattern that is optimal across all scales and contexts (5). Rather, indicate Lévy walks likely have an ancient origin and may arise simple rules probably will evolve that, on average, perform well from simple behaviors observed in much older fossil trails. This in their natural environment (5). Such rules are exemplified in foraging strategy may have adapted in response to decreased the different movement modes that tend to characterize behav- food availability after productivity collapse associated with past ior across different spatiotemporal scales. For instance, simple climate change and mass extinctions. deterministic foraging searches, such as Archimedean spirals (6) or area-restricted searching (7), that are driven by sensory and Author contributions: D.W.S., A.M.R., and R.J.T. designed research; D.W.S. and A.M.R. cognitive abilities are efficient where food distributions are known, conceived the research independently and brought their results together in the closing easily detected, or predictable. However, these patterns are in- stages of their work; D.W.S. and R.J.T. planned and led the empirical research; D.W.S., efficient when food resources are sparsely or patchily distributed A.M.R., N.E.H., B.M., and R.J.T. performed research; B.M. and R.J.T. undertook field re- and the forager has incomplete information on resource location; search and specimen tracings; D.W.S., N.E.H., E.J.S., and V.J.W. conducted track and sta- tistical analyses; A.M.R. undertook simulations and MLE modeling; N.E.H. wrote the under these conditions, probabilistic searches such as Lévy walks software for track digitization and for MLE and model selection for track analysis; become advantageous (1–3). N.E.H. contributed new reagents/analytic tools; D.W.S., A.M.R., N.E.H., E.J.S., V.J.W., and Theory predicts that Lévy walk search strategies should be R.J.T. analyzed data; and D.W.S. wrote the paper with contributions from all authors. optimal where food is sparse and distributed unpredictably (1), The authors declare no conflict of interest. whereas Brownian walks are sufficiently efficient for locating This article is a PNAS Direct Submission. abundant prey (2). A Lévy walk search pattern comprises dis- 1To whom correspondence should be addressed. Email: [email protected]. placements (move steps) drawn from a probability distribution This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. with a heavy power-law tail that results in a fractal pattern of 1073/pnas.1405966111/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1405966111 PNAS Early Edition | 1of6 Downloaded by guest on September 27, 2021 ubiquity of Lévy patterns among extant organisms, including Simulation outputs show that before encountering an obstruc- humans (1–3, 12–18), suggests that searches that approximate tion, the model organism spirals outward, forming an Archimedes them have evolved naturally (3). It has been hypothesized that spiral (Fig. 1A) characteristic of some early fossil traces (6). After behavioral adaptations to changes in environmental resources encountering obstructions, random looping patterns appear (Fig. cue the switching between localized Brownian and Lévy ran- 1A). The presence of starting spirals and, in particular, clusters of dom searching (2, 3, 13) or that sensory interactions with localized “walks” are common in meandering trace fossils, such heterogeneous environments may give rise to Lévy movement as in Cosmorhaphe (Fig. 1B). Our simulation shows that initial patterns (an emergent phenomena) (19); however, the origins spiraling becomes less evident as the number of obstructions of such potential mechanisms remain elusive. increases (Fig. 1C). In the presence of sufficiently numerous The fossilized records of animal movements preserved as trails obstructions, model parameter estimation using maximum likeli- and burrows (trace fossils) are the only direct record of extinct hood estimation (MLE) and Akaike information criteria weights organisms’ behavior and may provide a means to understand the (wAIC) for best-fit model selection (Materials and Methods)favors evolution of search strategies in ancient landscapes, including a power-law over an exponential (Brownian) model as being the during dramatic environmental changes (6, 20, 21). At intervals better model of the step-length distribution, i.e., the distribution of throughout evolutionary history, organisms have faced large-scale distances traveled