Scaling laws of ambush predator 'waiting' behaviour are tuned to a common ecology

Abstract

The decisions animals make about how long to wait between activities can determine the success of diverse behaviours such as foraging, group formation or risk avoidance. Remarkably, for diverse animal species, including humans, spontaneous patterns of waiting times show random 'burstiness' that appears scale-invariant across a broad set of scales. However, a general theory linking this phenomenon across the animal kingdom currently lacks an ecological basis. Here, we demonstrate from tracking the activities of 15 sympatric predator species (cephalopods, sharks, skates and teleosts) under natural and controlled conditions that bursty waiting times are an intrinsic spontaneous behaviour well approximated by heavy-tailed (power-law) models over data ranges up to four orders of magnitude. Scaling exponents quantifying ratios of frequent short to rare very long waits are species-specific, being determined by traits such as foraging mode (active versus ambush predation), body size and prey preference. A stochastic-deterministic decision model reproduced the empirical waiting time scaling and species-specific exponents, indicating that apparently complex scaling can emerge from simple decisions. Results indicate temporal power-law scaling is a behavioural 'rule of thumb' that is tuned to species' ecological traits, implying a common pattern may have naturally evolved that optimizes move-wait decisions in less predictable natural environments.

Keywords: foraging strategy; human dynamics; intermittence; movement ecology; random walk; search.

Figure 1.

Temporal sequence of waiting times shows an intermittent pattern. Example patterns of waiting times for individual (a,b) European plaice P. platessa, (c,d) thornback ray R. clavata and (e,f) common sole S. solea in natural ((a) over 90 days duration; (c) 90 days; (e) 55.7 days) and in controlled environments ((b) 26.7 days; (d) 23.6 days; (f) 26 days), respectively. Note the burstiness of waiting times is exemplified by many short waits interspersed with some rare but very long waits.

Figure 2.

Power-law scaling of waiting time distributions among individuals of diverse species. Example log–log plots show model best fits to truncated power-law (red lines) and poor fits to an exponential (blue lines) for eight individuals from species tracked both (a–h) in the wild and (i–p) in captivity. (a,i) Common cuttlefish, (b,j) small-eyed ray, (c,k) thornback ray, (d,l) blonde ray, (e,m) European plaice, (f,n) turbot, (g,o) common sole and (h,p) anglerfish.

Figure 3.

Non-universality in scaling exponents predicted by traits linked to state. Relationships between species mean scaling exponent and (a) predator maximum body mass/maximum body length (a proxy for total energy reserves) and (b) predator's specialization on active prey (mean frequency of occurrence of fish prey in its diet). A multiple linear regression model with exponent as the response variable and predator body mass/body length and percentage fish in diet as predictor variables accounted for 63% of the variation in the response (r² = 0.63; ANOVA, F_2,8 = 6.81, p = 0.019). Considering only fish data improved the model fit, with predictors accounting for 77% of response variation (r² = 0.77; ANOVA, F_2,7 = 11.55, p = 0.006); see the electronic supplementary material.

Figure 4.

(a) Simulation data for the proportion of waiting times with duration more than T produced by the stochastic priority-list model (γ = 1) (open circles) together with the best fit inverse power-law (black line). The maximum-likelihood estimate for μ = 2.0 and predicts a predator waiting time pattern dominated by short waits. A straight line on this log–log plot is indicative of inverse power-law scaling similar to empirical waiting time distributions (figure 2). (b) The dependency of μ on the degree of determinism (γ) (filled circles), where μ = 2 predicts a waiting time pattern when prey has lower energy content and μ → 1 as the energy content of the prey increases. The line is added to guide the eye.