The Lévy Flight Foraging Hypothesis

Transcription

1 The Lévy Flight Foraging Hypothesis Viswanathan et al. (1999) proposed that predators may use Lévy flights when searching for prey that is sparsely and unpredictably distributed. Distances traveled when foraging will then follow a power law with tail index α = µ 1: p(x) x µ P(X > x) x 1 µ. Depending on whether the prey is exploited in a renewable or non-renewable fashion, the optimal exponent is predicted to be µ 2 or µ 1. At high prey densities, Brownian motion (µ 3) is expected. Jay Taylor (ASU) STP421 Spring / 11

2 Lévy flight search patterns of wandering albatrosses Source: Viswanathan et al. (1996) Jay Taylor (ASU) STP421 Spring / 11

3 The LFFH has been claimed to be consistent with movement in the following: chemotaxis in Acanthamoeba (Schuster & Levandowsky, 1996) dinoflagellates (Bartumeus et al., 2003); prey-dependent bumble-bees (Viswanathan, 1999); honeybees (Reynolds et al., 2007) sharks, bony fish, sea turtles, penguins (Sims et al., 2006) wandering albatross (Viswanathan et al., 1996) side-striped jackal (Atkinson et al., 2002) Geoffroy s spider monkey (Ramos-Fernandez et al., 2004) humans - Peruvian fishing boats (Bertrand et al., 2005) extinct sea urchins, using trace fossils (Sims et al., 2014) Jay Taylor (ASU) STP421 Spring / 11

4 Criticisms Super-diffusive motion can be generated by other mechanisms, including correlated random walks. Power law distributions could be generated by the distribution of the prey rather than the behavior of the predator. Reliable estimation of power laws is notoriously difficult, especially with small sample sizes. The results are sensitive to sampling interval. Few studies assessed goodness-of-fit of the LFF model to the data. Many studies failed to properly segment the data. Jay Taylor (ASU) STP421 Spring / 11

5 Humphries et al. (2010): Environmental context explains Lévy and Brownian movement patterns of marine predators Analyzed large data sets characterizing vertical movement by 55 individuals of 14 open-ocean predatory fish species: 8 shark spp., 2 tuna spp., 3 bill fish and ocean sunfish. Long time series were segmented using split moving-window analysis. AIC was used to fit power law, truncated power law and exponential distributions to each time series section. Goodness-of-fit tests were used to discard data with poor fit to any of these distributions. Foraging behavior was related to ecological context (prey density, water temperature). Jay Taylor (ASU) STP421 Spring / 11

6 Split moving window analysis 1 Each time series was divided into a series of overlapping windows each containing 6 time bins. 2 The time-at-depth distribution was estimated for each time bin using 10 m depth bins. 3 Euclidean distances were calculated between the time-at-depth distributions for each pair of windows. 4 Significance was assessed by randomly permuting the data. 5 This procedure was repeated with increasing width of the time bins (6-32). Jay Taylor (ASU) STP421 Spring / 11

7 Figure S1: Identification of movement pattern discontinuities in a blue shark (dissimilarity matrices) Red color shows areas of high dissimilarity. Time at depth (a); time step-length (b). Jay Taylor (ASU) STP421 Spring / 11

8 Results Of 129 depth sections analyzed, 1 was best fit by a pure power law; 60 were best fit by a truncated Pareto distribution, with µ (1, 3]; mean µ = were best fit by a truncated Pareto distribution with µ > 3 27 were best fit by an exponential distribution 35 did not show a good fit to any of these distributions Pareto distributions were concentrated in nutrient and prey-poor regions. Jay Taylor (ASU) STP421 Spring / 11

9 Figure 1: Examples of good fits to power-law and truncated power-law distributions (blue shark and sunfish). Jay Taylor (ASU) STP421 Spring / 11

10 Figure 2: Behavioral switching between Lévy and Brownian motion in relation to habitat type. Jay Taylor (ASU) STP421 Spring / 11

11 Figure 3: Spatial occurrence of Lévy and Brownian behaviour types. Jay Taylor (ASU) STP421 Spring / 11