Mussels realize Weierstrassian Lévy walks as composite correlated random walks

OPEN SUBJECT AREAS: EVOLUTIONARY THEORY BEHAVIOURAL METHODS Mussels realize Weierstrassian Lévy walks as composite correlated random walks Andy M. Reynolds Received 4 November 2013 Accepted 3 March 2014 Published 18 March 2014 Correspondence and requests for materials should be addressed to Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK. Composite correlated random walks (CCRW) have been posited as a potential replacement for Lévy walks and it has also been suggested that CCRWs have been mistaken for Lévy walks. Here I test an alternative, emerging hypothesis: namely that some organisms approximate Lévy walks as an innate CCRW. It is shown that the tri-modal CCRW found to describe accurately the movement patterns of mussels (Mytilus edulis) during spatial pattern formation in mussel beds can be regarded as being the first three levels in a hierarchy of nested movement patterns which if extended indefinitely would correspond to a Lévy walk whose characteristic (power-law) exponent is tuned to nearly minimize the time required to form patterned beds. The mussels realise this Lévy walk to good approximation across a biologically meaningful range of scales. This demonstrates that the CCRW not only describes mussel movement patterns, it explains them. A.M.R. (andy. reynolds@rothamsted. ac.uk) L évy walks are a popular model of organism movement pattern data1. In a Lévy walk the mean-squared steplength diverges over time and this implies the absence of a characteristic scale and so fractal scaling. It has long been recognised that these superdiffusive and fractal properties of Lévy walks can be advantageous when searching and as a consequence may be selected for2,3. This expectation is now amply supported by empirical observations. Many organisms including E-coli, T-cells, honeybees, the wandering albatross and some marine predators have been reported to have movement patterns that can be approximated by Lévy walks4–12. Nonetheless, Lévy walks have not been accepted in some quarters. This is partly because many earlier studies had wrongly ascribed Lévy walks to some species through the use of inappropriate statistical analyses and through misinterpreting data13,14. This has cast a long shadow over Lévy walk research15–17. It is also because ‘composite correlated random walks’ (CCRW) appear to be a strong alternative model of movement pattern data resembling Lévy walks18. In these models, organisms are assumed to switch between two or more kinds of simple walk pattern. CCRW can resemble Lévy walks when frequently occurring movements with relatively short steps are interspersed with more rarely occurring longer steps. This leaves open the question of why CCRW can come to resemble Lévy walks. Close resemblance requires fine tuning of the parameters in a CCRW that determine the step-lengths for each mode and the rates of switching between the different modes. The issue was first articulated by de Jager et al.19 who hypothesised that organisms can approximate a Lévy walk by adopting an intrinsic CCRW in which switching between different modes is internally triggered rather than externally triggered, by, for instance, the detection or depletion of food, as in the original model of Benhamou1. The hypothesis of de Jager et al.19 stemmed from analyses of the movement patterns of mussels (Mytilus edulis) made during the formation of regularly patterned beds when individuals are searching for conspecifics20. The mussels aggregate with some conspecifics to minimize wave forces from the water, but also keep their distance from other clusters of mussels to avoid high competition. The mussel movement patterns resemble Lévy walks20 but are, in fact, better represented by CCRW21. This finding led Jansen et al.21 to suggest that Lévy walks have been wrongly identified in mussels and to conclude that one has to be cautious in inferring the presence of Lévy walks in biological systems, implying that the concept is not applicable to organisms. Nonetheless, repeated switching between movement strategies induced by changing environmental conditions, as in the model of Benhamou18, does not provide a plausible explanation for the observed composite walk, as the mussels were placed in a bare, homogeneous environment19. de Jager et al.19 also precluded the possibility that variation in individual walking behaviours – for example, multiple different Brownian (exponential) walks - together make up the observed composite walk as Brownian walks fitted individual movement pattern data very poorly. The observed CCRW therefore appears to be intrinsic. de Jager et al.22 subsequently showed that the intrinsic pattern is Levy-like in a bare tank (sparse conditions) but emerges as SCIENTIFIC REPORTS | 4 : 4409 | DOI: 10.1038/srep04409 1 www.nature.com/scientificreports Brownian when encounters with conspecifics are frequent. A similar resemblance between intrinsic CCRW and LW to that seen in the mussels has subsequently been found in the movement patterns of the Australian desert ants Melophorus bagoti23. In these desert ants a bi-modal walk is utilized when searching in visually unfamiliar surroundings, a setting which favours Lévy walk searching. When searching in visually familiar surroundings, the ants adopt a Brownian walk. The hypothesis of de Jager et al.19 finds support in the theoretical analysis of Reynolds24 who showed how selection pressures can give intrinsic CCRW Lévy walk characteristics. In this note I show explicitly that the CCRW seen in mussels approximates to a Lévy walk that is optimized for the formulation of patterned beds, i.e., for searching for conspecifics. The approach taken draws heavily upon the work of Hughes et al.25 who constructed a family of random walks – now sometimes called Weierstrass random walks or Weierstrass Lévy flights because of their association with the Weierstrass function - having a hierarchy of self-similar clusters that coincides with a Lévy walk. The tri-modal CCRW is shown to correspond to the first three hierarchy levels of a Weierstrass Lévy flight. Weierstrass random walks are one of the simplest random walks which do not satisfy the Central Limit Theorem. In the continuum limit they are governed by Lévy stable distributions and not by Gaussians. Weierstrass random walks have thus become the paradigmatic Markov process giving rise to Lévy walks and have come epitomize scale-invariance27. The new finding explains why the CCRW so closely resembles a Lévy walk and accounts naturally for the optimization, as Weierstrass Lévy flights can have similitude with self-avoiding random walks26. Self-avoidance is advantageous when randomly searching because it avoids needlessly revisiting previously-searched locations. Results The step-length distribution found to describe accurately the movements of mussels is the tri-exponential pðlÞ~ 3 X pi l{1 expð{l=li Þ i occurrence probability for this mode is predicted (...truncated)