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.

As part of national biosecurity programs, cargo imports, passenger baggage, and international mail are inspected at ports of entry to verify compliance with phytosanitary regulations and to intercept potentially damaging nonnative species to prevent their introduction. Detection of organisms during inspections may also provide crucial information about the species composition and relative arrival rates in invasion pathways that can inform the implementation of other biosecurity practices such as quarantines and surveillance. In most regions, insects are the main taxonomic group encountered during inspections. We gathered insect interception data from nine world regions collected from 1995 to 2019 to compare the composition of species arriving at ports in these regions. Collectively, 8,716 insect species were intercepted in these regions over the last 25 yr, with the combined international data set comprising 1,899,573 interception events, of which 863,972 were identified to species level. Rarefaction analysis indicated that interceptions comprise only a small fraction of species present in invasion pathways. Despite differences in inspection methodologies, as well as differences in the composition of import source regions and imported commodities, we found strong positive correlations in species interception frequencies between regions, particularly within the Hemiptera and Thysanoptera. There were also significant differences in species frequencies among insects intercepted in different regions. Nevertheless, integrating interception data among multiple regions would be valuable for estimating invasion risks for insect species with high likelihoods of introduction as well as for identifying rare but potentially damaging species.

Many different species have been suggested to forage according to a Lévy walk in which the distribution of step lengths is heavy-tailed. Theoretical research has shown that a Lévy exponent of approximately 2 can provide a higher foraging efficiency than other exponents. In this paper, a composite search model is presented for non-destructive foraging behaviour based on Brownian (i.e. non-heavy-tailed) motion. The model consists of an intensive search phase, followed by an extensive phase, if no food is found in the intensive phase. Quantities commonly observed in the field, such as the distance travelled before finding food and the net displacement in a fixed time interval, are examined and compared with the results of a Lévy walk model. It is shown that it may be very difficult, in practice, to distinguish between the Brownian and the Lévy models on the basis of observed data. A mathematical expression for the optimal time to switch from intensive to extensive search mode is derived, and it is shown that the composite search model provides higher foraging efficiency than the Lévy model.