Lévy Walks Suboptimal under Predation Risk

Virtual prey with Lévy motion are preferentially attacked by predatory fish

Of widespread interest in animal behavior and ecology is how animals search their environment for resources, and whether these search strategies are optimal. However, movement also affects predation risk through effects on encounter rates, the conspicuousness of prey, and the success of attacks. Here, we use predatory fish attacking a simulation of virtual prey to test whether predation risk is associated with movement behavior. Despite often being demonstrated to be a more efficient strategy for finding resources such as food, we find that prey displaying Lévy motion are twice as likely to be targeted by predators than prey utilizing Brownian motion. This can be explained by the predators, at the moment of the attack, preferentially targeting prey that were moving with straighter trajectories rather than prey that were turning more. Our results emphasize that costs of predation risk need to be considered alongside the foraging benefits when comparing different movement strategies.

Sediment availability provokes a shift from Brownian to Lévy-like clonal expansion in a dune building grass

In biogeomorphic landscapes, plant traits can steer landscape development through plant-mediated feedback interactions. Interspecific differences in clonal expansion strategy can therefore lead to the emergence of different landscape organisations. Yet, whether landscape-forming plants adopt different clonal expansion strategies depending on their physical environment remains to be tested. Here, we use a field survey and a complementary mesocosm approach to investigate whether sediment deposition affects the clonal expansion strategy employed by dune-building marram grass individuals. Our results reveal a consistent shift in expansion pattern from more clumped, Brownian-like, movement in sediment-poor conditions, to patchier, Lévy-like, movement under high sediment supply rates. Additional model simulations illustrate that the sediment-dependent shift in movement strategies induces a shift in optimisation of the cost-benefit relation between landscape engineering (i.e. dune formation) and expansion. Plasticity in expansion strategy may therefore allow landscape-forming plants to optimise their engineering ability depending on their physical landscape.

Termite males enhance mating encounters by changing speed according to density

Search theory predicts that animals evolve efficient movement patterns to enhance encounter rates with specific targets. The optimal movements vary with the surrounding environments, which may explain the observation that animals often switch their movement patterns depending on conditions. However, the effectiveness of behavioural change during search is rarely evaluated because it is difficult to examine the actual encounter dynamics. Here we studied how partner-seeking termites update their search strategies depending on the local densities of potential mates. After a dispersal flight, termites drop their wings and walk to search for a mate; when a female and a male meet, they form a female-led tandem pair and search for a favourable nesting site. If a pair is separated, they have two search options-reunite with their stray partner, or seek a new partner. We hypothesized that the density of individuals affects separation-reunion dynamics and thus the optimal search strategy. We observed the searching process across different densities and found that termite pairs were often separated but obtained a new partner quickly at high mate density. After separation, while females consistently slowed down, males increased their speed according to the density. Under high mate density, separated males obtained a partner earlier than females, who do not change movement with density. Our data-based simulations confirmed that the observed behavioural change by males contributes to enhancing encounters. Males at very low mate densities did best to move slowly and thereby reduce the risk of missing their stray partner, who is the only available mate. On the other hand, males that experienced high mate densities did better in mating encounters by moving fast because the risk of isolation is low, and they must compete with other males to find a partner. These results demonstrate that termite males adaptively update their search strategy depending on conditions. Understanding the encounter dynamics experienced by animals is key to connecting the empirical work to the idealized search processes of theoretical studies.

Functional advantages of Lévy walks emerging near a critical point

A special class of random walks, so-called Lévy walks, has been observed in a variety of organisms ranging from cells, insects, fishes, and birds to mammals, including humans. Although their prevalence is considered to be a consequence of natural selection for higher search efficiency, some findings suggest that Lévy walks might also be epiphenomena that arise from interactions with the environment. Therefore, why they are common in biological movements remains an open question. Based on some evidence that Lévy walks are spontaneously generated in the brain and the fact that power-law distributions in Lévy walks can emerge at a critical point, we hypothesized that the advantages of Lévy walks might be enhanced by criticality. However, the functional advantages of Lévy walks are poorly understood. Here, we modeled nonlinear systems for the generation of locomotion and showed that Lévy walks emerging near a critical point had optimal dynamic ranges for coding information. This discovery suggested that Lévy walks could change movement trajectories based on the magnitude of environmental stimuli. We then showed that the high flexibility of Lévy walks enabled switching exploitation/exploration based on the nature of external cues. Finally, we analyzed the movement trajectories of freely moving Drosophila larvae and showed empirically that the Lévy walks may emerge near a critical point and have large dynamic range and high flexibility. Our results suggest that the commonly observed Lévy walks emerge near a critical point and could be explained on the basis of these functional advantages.

Adaptive switch to sexually dimorphic movements by partner-seeking termites

How should females and males move to search for partners whose exact location is unknown? Theory predicts that the answer depends on what they know about where targets can be found, raising the question of how actual animals update their mate search patterns to increase encounter probability when conditions change. Here, we show that termites adaptively alternate between sexually monomorphic and dimorphic movements during mate search. When the location of potential mates was completely unpredictable, both sexes moved in straight lines to explore widely. In contrast, when the stray partner was at least nearby, males moved while females paused. Data-based simulations confirmed that these movements increase the rate of successful encounters. The context-dependent switch of search modes is a key to enhance random encounters.

Lévy like patterns in the small-scale movements of marsupials in an unfamiliar and risky environment

We investigate the movement patterns of three different Neotropical marsupials in an unfamiliar and risky environment. Animals are released in a matrix from which they try to reach a patch of forest. Their movements, performed on a small spacial scale, are best approximated by Lévy flights. Patterns of oriented and non-oriented individuals - with forest patches within or beyond their perceptual range - differ only slightly in the value of their exponents. These facts suggest that, for these species, the appearance of Lévy flights is the product of animals innate behaviour that emerges spontaneously, as a neutral characteristic proper of a default movement mode for alerted animals.

Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo

Metastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long "flights". Such movements are characterized by heavy-tailed, truncated power law distributions of persistence times and are consistent with the Lévy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for "reprogramming" the Lévy walks into either diffusive or ballistic movements.

The optimal movement patterns for mating encounters with sexually asymmetric detection ranges

Animals have evolved various sex-specific characteristics to improve the efficiency of mating encounters. One is the sex-specific attracting signal. Signal receivers perform a combination of random search and navigation before and after signal detections. On the other hand, signal senders can also modify their movement patterns to optimize their encounter rates, which invokes a reverse side of random search problems that asks for the most efficient movement patterns of signal senders to be found by signal receivers. In this study, we focused on visual and auditory signals in particular, and quantified the efficiency of mating encounters of individual animals performing a Lévy walk, a special class of random walk, with a variety of speeds before signal detection. We found that signal senders should move more slowly and/or less diffusively than receivers to improve mating encounters. The optimal movement patterns of senders ranged from relatively slow to stationary ones depending on the density of individuals, the effective range of signals, and the ability of receivers to locate senders. By focusing on the optimal movement patterns of individuals that are often assumed to be given targets, the present study provides insights into strategies of effective attraction beyond the case of mate search.

Optimizing mating encounters by sexually dimorphic movements

All organisms with sexual reproduction undergo a process of mating, which essentially involves the encounter of two individuals belonging to different sexes. During mate search, both sexes should mutually optimize their encounters, thus raising a question of how they achieve this. Here, we show that a population with sexually dimorphic movement patterns achieves the highest individual mating success under a limited lifespan. Extensive simulations found and analytical approximations corroborated the existence of conditions under which sexual dimorphism in the movement patterns (i.e. how diffusively they move) is advantageous over sexual monomorphism. Mutual searchers with limited lifespans need to balance the speed and accuracy of finding their mates, and dimorphic movements can solve this trade-off. We further demonstrate that the sexual dimorphism can evolve from an initial sexually monomorphic population. Our results emphasize the importance of considering mutual optimization in problems of random search.

The evolutionary origins of Lévy walk foraging

We study through a reaction-diffusion algorithm the influence of landscape diversity on the efficiency of search dynamics. Remarkably, the identical optimal search strategy arises in a wide variety of environments, provided the target density is sparse and the searcher’s information is restricted to its close vicinity. Our results strongly impact the current debate on the emergentist vs. evolutionary origins of animal foraging. The inherent character of the optimal solution (i.e., independent on the landscape for the broad scenarios assumed here) suggests an interpretation favoring the evolutionary view, as originally implied by the Lévy flight foraging hypothesis. The latter states that, under conditions of scarcity of information and sparse resources, some organisms must have evolved to exploit optimal strategies characterized by heavy-tailed truncated power-law distributions of move lengths. These results strongly suggest that Lévy strategies—and hence the selection pressure for the relevant adaptations—are robust with respect to large changes in habitat. In contrast, the usual emergentist explanation seems not able to explain how very similar Lévy walks can emerge from all the distinct non-Lévy foraging strategies that are needed for the observed large variety of specific environments. We also report that deviations from Lévy can take place in plentiful ecosystems, where locomotion truncation is very frequent due to high encounter rates. So, in this case normal diffusion strategies—performing as effectively as the optimal one—can naturally emerge from Lévy. Our results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

How organisms improve the search for food, mates, etc., is a key factor to their survival. Mathematically, the best strategy to look for randomly distributed re-visitable resources—under scarce information and sparse conditions—results from Lévy distributions of move lengths (the probability of taking a step ℓ is proportional to 1/ℓ²). Today it is well established that many animal species in different habitats do perform Lévy foraging. This fact has raised a heated debate, viz., the emergent versus evolutionary hypotheses. For the former, a Lévy foraging is an emergent property, a consequence of searcher-environment interactions: certain landscapes induce Lévy patterns, but others not. In this view, the optimal strategy depends on the particular habitat. The evolutionary explanation, in contrast, is that Lévy foraging strategies are adaptations that evolved via natural selection. In this article, through simulations we exhaustively analyze the influence of distinct environments on the foraging efficiency. We find that the optimal procedure is the same in all situations, provided density is low and landscape information is scarce. So, the best search strategy is remarkably independent of details. These results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

Online games: a novel approach to explore how partial information influences human random searches

Many natural processes rely on optimizing the success ratio of a search process. We use an experimental setup consisting of a simple online game in which players have to find a target hidden on a board, to investigate how the rounds are influenced by the detection of cues. We focus on the search duration and the statistics of the trajectories traced on the board. The experimental data are explained by a family of random-walk-based models and probabilistic analytical approximations. If no initial information is given to the players, the search is optimized for cues that cover an intermediate spatial scale. In addition, initial information about the extension of the cues results, in general, in faster searches. Finally, strategies used by informed players turn into non-stationary processes in which the length of e ach displacement evolves to show a well-defined characteristic scale that is not found in non-informed searches.