Andrew M. Edwards

Revisiting Levy flight search patterns of wandering albatrosses, bumblebees and deer

The study of animal foraging behaviour is of practical ecological importance, and exemplifies the wider scientific problem of optimizing search strategies. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical and chemical systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface. This well known finding was followed by similar inferences about the search strategies of deer and bumblebees. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer, microzooplankton, grey seals, spider monkeys and fishing boats. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood and Akaike weights. We apply this to the four original deer and bumblebee data sets, finding that none exhibits evidence of Lévy flights, and that the original graphical approach is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool. Our results question the strength of the empirical evidence for biological Lévy flights.

Using likelihood to test for Lévy flight search patterns and for general power‐law distributions in nature

1. Ecologists are obtaining ever-increasing amounts of data concerning animal movement. A movement strategy that has been concluded for a broad variety of animals is that of Lévy flights, which are random walks whose step lengths come from probability distributions with heavy power-law tails. 2. The exponent that parameterizes the power-law tail, denoted micro, has repeatedly been found to be within the Lévy range of 1 < micro <or= 3. Here, we use Monte Carlo simulations to show that the methods used to infer the value of micro are inaccurate. 3. The widely used method of simply logarithmically transforming a standard histogram of movement lengths has been shown elsewhere to be problematic. Here, we further demonstrate how poor it is, and show that it actually biases estimates of micro towards the Lévy range of 1 < micro <or= 3, and can bias estimates towards the value of micro = 2. Thus, previous reports of animals undergoing Lévy flights, or of micro being close to the reported optimal value of micro = 2, may simply be a consequence of the bias generated by this method. 4. A technique that has been recently recommended is to logarithmically bin the data and then normalize the resulting histogram. We show that this technique also produces biased results, and suffers from similar problems as those just outlined, although to a lesser extent. 5. The proposed solution is to use likelihood. We find that calculating the maximum likelihood estimate of micro gives the most accurate results (having also tested the rank/frequency method). Likelihood has the further advantages of being the easiest method to implement, and of yielding accurate confidence intervals. Results are applicable to power-law distributions in general, and so are not restricted to inference of Lévy flights. 6. We also re-analyse a data set of grey seal movements that was originally reported to demonstrate Lévy flight behaviour. Using Akaike weights, we test four models, and find no evidence for Lévy flights. Overall, our results suggest that Lévy flights might not be as common as previously thought.

Assessing Levy walks as models of animal foraging

The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.

Overturning conclusions of Lévy flight movement patterns by fishing boats and foraging animals

A surprisingly diverse variety of foragers have previously been concluded to exhibit movement patterns known as Lévy flights, a special type of random walk. These foragers range in size from microzooplankton in experiments to fishermen in the Pacific Ocean and the North Sea. The Lévy flight conclusion implies that all the foragers have similar scale-free movement patterns that can be described by a single dimensionless parameter, the exponent micro of a power-law (Pareto) distribution. However, the previous conclusions have been made using methods that have since been shown to be problematic: inaccurate techniques were used to estimate micro, and the power-law distribution was usually assumed to hold without testing any alternative hypotheses. Therefore, I address the open question of whether the previous data still support the Lévy flight hypothesis, and thus determine whether Lévy flights really are so ubiquitous in ecology. I present a comprehensive reanalysis of 17 data sets from seven previous studies for which Lévy flight behavior had been concluded, covering marine, terrestrial, and experimental systems from four continents. I use the modern likelihood and Akaike weights approach to test whether simple alternative models are more supported by the data than Lévy flights. The previously estimated values of the power-law exponent micro do not match those calculated here using the accurate likelihood approach, and almost all of them lie outside of the likelihood-based 95% confidence intervals. Furthermore, the original power-law Lévy flight model is overwhelmingly rejected for 16 out of the 17 data sets when tested against three other simple models. For one data set, the data are consistent with coming from a bounded power-law distribution (a truncated Lévy flight). For three other data sets, an exponential distribution corresponding to a simple Poisson process is suitable. Thus, Lévy flight movement patterns are not the common phenomena that was once thought, and are not suitable for use as ecosystem indicators for fisheries management, as has been proposed.

Incorrect Likelihood Methods Were Used to Infer Scaling Laws of Marine Predator Search Behaviour

Background Ecologists are collecting extensive data concerning movements of animals in marine ecosystems. Such data need to be analysed with valid statistical methods to yield meaningful conclusions. Principal Findings We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (Nature 451∶1098; Science 332∶1551). The first study analysed vertical movement data to conclude that diverse marine predators (Atlantic cod, basking sharks, bigeye tuna, leatherback turtles and Magellanic penguins) exhibited “Lévy-walk-like behaviour”, close to a hypothesised optimal foraging strategy. By reproducing the original results for the bigeye tuna data, we show that the likelihood of tested models was calculated from residuals of regression fits (an incorrect method), rather than from the likelihood equations of the actual probability distributions being tested. This resulted in erroneous Akaike Information Criteria, and the testing of models that do not correspond to valid probability distributions. We demonstrate how this led to overwhelming support for a model that has no biological justification and that is statistically spurious because its probability density function goes negative. Re-analysis of the bigeye tuna data, using standard likelihood methods, overturns the original result and conclusion for that data set. The second study observed Lévy walk movement patterns by mussels. We demonstrate several issues concerning the likelihood calculations (including the aforementioned residuals issue). Re-analysis of the data rejects the original Lévy walk conclusion. Conclusions We consequently question the claimed existence of scaling laws of the search behaviour of marine predators and mussels, since such conclusions were reached using incorrect methods. We discourage the suggested potential use of “Lévy-like walks” when modelling consequences of fishing and climate change, and caution that any resulting advice to managers of marine ecosystems would be problematic. For reproducibility and future work we provide R source code for all calculations.

Effects of ocean acidification on temperate coastal marine ecosystems and fisheries in the northeast Pacific.

As the oceans absorb anthropogenic CO2 they become more acidic, a problem termed ocean acidification (OA). Since this increase in CO2 is occurring rapidly, OA may have profound implications for marine ecosystems. In the temperate northeast Pacific, fisheries play key economic and cultural roles and provide significant employment, especially in rural areas. In British Columbia (BC), sport (recreational) fishing generates more income than commercial fishing (including the expanding aquaculture industry). Salmon (fished recreationally and farmed) and Pacific Halibut are responsible for the majority of fishery-related income. This region naturally has relatively acidic (low pH) waters due to ocean circulation, and so may be particularly vulnerable to OA. We have analyzed available data to provide a current description of the marine ecosystem, focusing on vertical distributions of commercially harvested groups in BC in the context of local carbon and pH conditions. We then evaluated the potential impact of OA on this temperate marine system using currently available studies. Our results highlight significant knowledge gaps. Above trophic levels 2–3 (where most local fishery-income is generated), little is known about the direct impact of OA, and more importantly about the combined impact of multi-stressors, like temperature, that are also changing as our climate changes. There is evidence that OA may have indirect negative impacts on finfish through changes at lower trophic levels and in habitats. In particular, OA may lead to increased fish-killing algal blooms that can affect the lucrative salmon aquaculture industry. On the other hand, some species of locally farmed shellfish have been well-studied and exhibit significant negative direct impacts associated with OA, especially at the larval stage. We summarize the direct and indirect impacts of OA on all groups of marine organisms in this region and provide conclusions, ordered by immediacy and certainty.

The invisible niche: Weakly density-dependent mortality and the coexistence of species

Weakly density-dependent effects, characterized by fractional scaling exponents close to one, are rarely studied in the ecological literature. Here, we consider the effect of an additional weakly density-dependent term on a simple competition model. Our investigation reveals that weak density-dependence opens up an invisible niche. This niche does not constitute a new mechanism for coexistence, but is a previously unexplored consequence of known mechanisms. In the invisible niche a weaker competitor can survive at very low density. Coexistence thus requires large habitat size. Such niches, if found in nature, would have a direct impact on species-area laws and species-abundance curves and should therefore receive more attention.

Hitting the moving target: modelling ontogenetic shifts with stable isotopes reveals the importance of isotopic turnover

Ontogenetic niche shifts are widely prevalent in nature and are important in shaping the structure and dynamics of ecosystems. Stable isotope analysis is a powerful tool to assess these shifts, with δ(15) N providing a measure of trophic level and δ(13) C a measure of energy source. Previous applications of stable isotopes to study ontogenetic niche shifts have not considered the appreciable time lag between diet and consumer tissue associated with isotopic turnover. These time lags introduce significant complexity into field studies of ontogenetic niche shifts. Juvenile Chinook salmon (Oncorhynchus tshawytscha) migrate from freshwater to marine ecosystems and shift their diet from feeding primarily on invertebrates to feeding primarily on fish. This dual ontogenetic habitat and diet shift, in addition to the long time lag associated with isotopic turnover, suggests that there is potential for a disconnect between the prey sources that juvenile salmon are consuming, and the inferred prey sources from stable isotopes. We developed a model that considered ontogenetic niche shifts and time lags associated with isotopic turnover, and compared this ontogeny model to one that considered only isotopic turnover. We used a Bayesian framework to explicitly account for parameter uncertainty. Data showed overwhelming support for the ontogeny model relative to the isotopic turnover model. Estimated variables from best model fits indicate that the ontogeny model predicts a much greater reliance on fish prey than does the stomach content data. Overall, we found that this method of quantifying ontogenetic niche shifts effectively accounted for both isotopic turnover and ontogenetic diet shifts; a finding that could be widely applicable to a variety of systems.

Fishing degrades size structure of coral reef fish communities

Fishing pressure on coral reef ecosystems has been frequently linked to reductions of large fishes and reef fish biomass. Associated impacts on overall community structure are, however, less clear. In size-structured aquatic ecosystems, fishing impacts are commonly quantified using size spectra, which describe the distribution of individual body sizes within a community. We examined the size spectra and biomass of coral reef fish communities at 38 US-affiliated Pacific islands that ranged in human presence from near pristine to human population centers. Size spectra steepened steadily with increasing human population and proximity to market due to a reduction in the relative biomass of large fishes and an increase in the dominance of small fishes. Reef fish biomass was substantially lower on inhabited islands than uninhabited ones, even at inhabited islands with the lowest levels of human presence. We found that on populated islands size spectra exponents decreased (analogous to size spectra steepening) linearly with declining biomass, whereas on uninhabited islands there was no relationship. Size spectra were steeper in regions of low sea surface temperature but were insensitive to variation in other environmental and geomorphic covariates. In contrast, reef fish biomass was highly sensitive to oceanographic conditions, being influenced by both oceanic productivity and sea surface temperature. Our results suggest that community size structure may be a more robust indicator than fish biomass to increasing human presence and that size spectra are reliable indicators of exploitation impacts across regions of different fish community compositions, environmental drivers, and fisheries types. Size-based approaches that link directly to functional properties of fish communities, and are relatively insensitive to abiotic variation across biogeographic regions, offer great potential for developing our understanding of fishing impacts in coral reef ecosystems.

Apparent power-law distributions in animal movements can arise from intraspecific interactions

Lévy flights have gained prominence for analysis of animal movement. In a Lévy flight, step-lengths are drawn from a heavy-tailed distribution such as a power law (PL), and a large number of empirical demonstrations have been published. Others, however, have suggested that animal movement is ill fit by PL distributions or contend a state-switching process better explains apparent Lévy flight movement patterns. We used a mix of direct behavioural observations and GPS tracking to understand step-length patterns in females of two related butterflies. We initially found movement in one species (Euphydryas editha taylori) was best fit by a bounded PL, evidence of a Lévy flight, while the other (Euphydryas phaeton) was best fit by an exponential distribution. Subsequent analyses introduced additional candidate models and used behavioural observations to sort steps based on intraspecific interactions (interactions were rare in E. phaeton but common in E. e. taylori). These analyses showed a mixed-exponential is favoured over the bounded PL for E. e. taylori and that when step-lengths were sorted into states based on the influence of harassing conspecific males, both states were best fit by simple exponential distributions. The direct behavioural observations allowed us to infer the underlying behavioural mechanism is a state-switching process driven by intraspecific interactions rather than a Lévy flight.