The effects of spatially heterogeneous prey distributions on detection patterns in foraging seabirds

Abstract

Many attempts to relate animal foraging patterns to landscape heterogeneity are focused on the analysis of foragers movements. Resource detection patterns in space and time are not commonly studied, yet they are tightly coupled to landscape properties and add relevant information on foraging behavior. By exploring simple foraging models in unpredictable environments we show that the distribution of intervals between detected prey (detection statistics) is mostly determined by the spatial structure of the prey field and essentially distinct from predator displacement statistics. Detections are expected to be Poissonian in uniform random environments for markedly different foraging movements (e.g. Lévy and ballistic). This prediction is supported by data on the time intervals between diving events on short-range foraging seabirds such as the thick-billed murre (Uria lomvia). However, Poissonian detection statistics is not observed in long-range seabirds such as the wandering albatross (Diomedea exulans) due to the fractal nature of the prey field, covering a wide range of spatial scales. For this scenario, models of fractal prey fields induce non-Poissonian patterns of detection in good agreement with two albatross data sets. We find that the specific shape of the distribution of time intervals between prey detection is mainly driven by meso and submeso-scale landscape structures and depends little on the forager strategy or behavioral responses.

Figure 1. Detections in random uniform prey landscapes.

(A) Albatross data and the model with destructive scenario. Green circles: accumulated distribution of flight lengths between successive detected prey of the model forager with perception radius r = 0.001 following a Lévy process with µ = 1.5 (from 20 simulations of 75 captures each). The foraging ground is represented by a square of area unity and contained 5000 prey. Continuous line: exponential fit. Red triangles: of the Bird Island albatross takeoff/landing data . Blue triangles: of the Crozet Islands albatross prey capture data , converted into flight durations assuming a constant flight velocity m/s. Inset: same curves represented in semi-log to better emphasize the non exponential nature of the observed albatross data versus the exponential form of the model forager detections. (B) Albatross data and model with non-destructive scenario. Violet circles: accumulated distribution for the model forager performing a Lévy process with µ = 2. Continuous line: exponential fit. Prey number: 3000; r = 0.0003. In A) and B), the lengths in the model with foraging arena of area unity are converted in hours (t) by using with the scaling factor v = 0.12. Inset: same curves represented in semi-log to better emphasize the non exponential nature of the observed albatross data versus the exponential form of the model forager detections. (C) Murre data and model with destructive scenario. Green circles: accumulated distribution of flight lengths between prey of the model forager with perception radius r = 0.001 following a Lévy process with µ = 1.5 (from 20 simulations of 75 captures each). Continuous line: exponential fit. Brown dots are the murre flight durations from . Inset: same data represented in semi-log in order to better emphasize the exponential nature of both the observed murres data and the model forager. (D) Murre data and model with non-destructive scenario. Violet circles: accumulated distribution for a forager performing a Lévy process with µ = 2. Continuous line: exponential fit. Prey number: 3000; r = 0.0003. Inset: same curves represented in semi-log. Similar close-to-exponential detections are obtained in all simulations with , in both destructive and non-destructive scenarii.

Figure 2. Three different theoretical patterns of spatial prey distribution in a unit box and their corresponding box-counting fractal dimension.

represent the size of the boxes and is the number of boxes of size in the box-counting algorithm. In the three cases, 5000 prey are distributed accordingly to a Lévy dust with fractal dimension (). (A) If the minimal distance between prey is large the Lévy process producing the fractal pattern bounces many times on the walls and the overall process tends to be space-filling. In this particular case, the minimal distance between prey was 1/7 and the process bounced around 2500 times which is equivalent to the superposition of 2500 separated fractals in the same domain. (B) As expected in this case, the fractal dimension measured by box-counting does not show a scaling region with exponent (red line) but approximates more the typical graph of a 2D random process with (blue line). (C) Pattern that corresponds to a prey distribution with a minimal distance of 1/700 between prey, leading to less than 150 bounces ( of the total prey number). (D) In this case a scaling region with is visible, followed by a two-dimensional behavior at larger length scales. (E) A very clumped and aggregated fractal pattern of prey is obtained when the minimal distance between prey is set to (F) In this case the fractal is nearly perfect with

Figure 3. Left panel.

Foraging arena composed of N = 5000 prey generated with a LD of exponent  = 1.5 (fractal dimension  = 0.5). Solid line: trajectory of a ballistic forager () with detection radius r = 0.001. The larger grey dots indicate detection events (destructive scenario). Right panels: Fractal Local Density (FLD) model. Upper figure: The medium is composed of patches of heterogeneous sizes R, drawn from a PDF Within a patch, prey are randomly and uniformly distributed. Lower figure: linear representation of the forager/medium system, which is solved here.

Figure 4. Detections in LD media.

(A)-(D): Accumulated histograms of prey detection times (grey circles) for a ballistic model predator (µ = 1.01) with r = 0.0003 foraging in LD environments (N = 5000) of varying fractal dimension at lower scales. Foraging is destructive in all cases. Bird Island data: red triangles, Crozet Islands: blue triangles. Recall that (A)  = 1.2 (  = 0.55), p-value of K-S test on Bird Island:  = 0.0045, Crozet Islands:  = 4.9e-08; (B)  = 1.5 (  = 0.50),  =  0.997,  = 0.248; (C)  = 1.8 (,  = 0.25),  =  0.997,  =  0.367 and (D)  = 2.2 (  = 0.20),  =  0.033,  =  0.033. (E)-(H): Same quantities for LD media with fixed  = 1.5 (N = 5000, ) and a model forager following processes with different step length distributions: (E) µ = 1.5 ( = 0.5),  =  1,  =  0.248; (F) µ = 2.0 ( = 0.67),  =  0.999,  =  0.0995; (G) µ = 2.5 ( = 0.67),  =  0.000955,  = 4.03e-09 and (H) µ = 3.0 ( = 0.67),  =  6.38e-05,  =  1.72e-13.

Figure 5. Cumulative distribution of prey detection times/distances obtained by fitting the FLD model to the albatross data (triangles), for two fixed fractal dimensions of the medium ( = 0.6 and 1.6).

Solid black line: responsive search; green dotted line: non-responsive search. Each curve is plotted with the MLE of the parameters, see Table 1. (A)-(B): Bird Island. (C)-(D): Crozet Islands. The best estimates of the patch size distribution parameters vary little in the different cases:  = 1.20 ± 0.05 and in the range of 160–240 km, independently of for the whole range considered. A more efficient strategy yields a lower dimensionless detection radius