The Effects of Spatially Heterogeneous Prey Distributions on Detection Patterns in Foraging Seabirds

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.

1 Landscape properties of the Fractal Local Density Model

Calculation of the fractal dimension
In this model, prey are distributed on a plane that is randomly subdivided into non-overlapping patches of heterogeneous diameters (see Materials and Methods, main article). The patch diameter probability distribution function (PDF), ψ(R), is given by the power-law and ψ(R) = 0 for R < R 0 , where R 0 is the minimum patch diameter. ( ∞ 0 ψ(R)dR = 1.) A patch of size R contains in average n p (R) prey uniformly and randomly distributed inside the patch. We assume n p (R) = kR ǫ with k a constant and ǫ a scaling exponent. Let N 0 (> 1) be the average number of prey that a patch of smallest size R 0 contains. Thus, If ǫ = 2, the number of prey per unit area, ρ(R) = n p (R)/R 2 = kR ǫ−2 , is independent of R and the medium is uniform. We show below that the prey system can be fractal if ǫ < 2, when ρ(R) → 0 for large patches, and calculate its fractal dimension D F using the standard box-counting method [1]. For a fractal medium, we expect a relation of the form where N (σ) is the average number (per patch) of non-overlapping square boxes of length σ that cover all the prey (σ > R 0 in the following). Two cases are distinguished below: 0 ≤ ǫ < 2 (big patches have more prey) and ǫ < 0 (big patches have fewer prey). Although fits to the albatross data suggest that the prey system has ǫ < 0 (see main article), we also present the results of the other case for completeness.
If ǫ > 0, there are always more than one prey per patch. As the prey density ρ(R) = kR ǫ−2 is uniform inside the patch, the typical distance between neighboring prey is d(R) = 1/ρ(R) 1/2 , which is < R. Consider square boxes of size σ, larger than the smallest patch diameter R 0 . If d(R) does not exceeds σ, the patch can be completely covered by (R/σ) 2 boxes on average [this number is < 1 if R < σ, meaning that one box covers more than one patch on average]. If on the other hand d(R) > σ, n p (R) = kR ǫ boxes are needed in average to cover all the prey inside the patch. For a fixed σ, there is a patch size R * such that d(R * ) = σ. Using notation (2), From the considerations above, one can write Several cases are to be distinguished. If the patch exponent ν is > 3, one finds Since Comparing with the definition (3), one concludes: Hence, if ν > 3, the prey system is bidimensional, although with a non-uniform density.
If ν < 1 + ǫ, the average prey number per patch is infinite and the medium can not be described as a fractal. This case will not be considered here. b) Case ǫ < 0.
The calculation above must be modified for negative values of ǫ. The average prey number is < 1 in patches larger than (11) One can figure these low density patches as containing one prey with probability kR ǫ and zero prey with probability 1 − kR ǫ . Hence, if R > R c , d(R) does not represent the distance between neighboring prey as in the previous case. Let us restrict our analysis to boxes of size σ > R c in the following. A patch with R < R c is completely covered (and therefore all its prey) by (R/σ) 2 (< 1) boxes. A patch with R c < R < σ is covered by (R/σ) 2 boxes, too, and has a probability kR ǫ (< 1) of containing one prey. A patch with R > σ has a probability kR ǫ of containing one prey and therefore requires kR ǫ boxes in average to be covered. Therefore: Performing the integrals, one finds (σ ≫ R 0 ): For a fixed patch exponent ν, the fractal dimension is given by the leading term(s) in eqs.(13)-(15), i.e., the term(s) with the slowest power-law decay as a function of σ/R 0 . One obtains:

Distribution of the local density
In this subsection, we show that in the Fractal Patch model with patch size distribution of the form ψ(R) ∼ R −ν and ǫ < 2, the probability distribution function f (ρ) of the local density ρ is also a power-law, f (ρ) ∼ ρ −αρ , with exponent: (See equation (4), main article). A motivation for calculating α ρ is that recent acoustic measurements have shown that the prey of some top marine predators have densities that are power-law distributed in space [2]. The probability that a small region of the plane has a density larger than a value ρ is equal to the fraction area occupied by patches smaller than R, with ρ = kR ǫ−2 . Namely, with ρ 0 = ρ(R 0 ) the largest density, that is found in the smallest patches.
Taking the derivative of eq.(19) with respect to ρ, one obtains Using the relation R ∼ ρ 1/(ǫ−2) yields to the above mentioned result. If ν < 3, then α ρ > 1 from eq.(18): The presence of very large patches of low densities produce a sharp increase of f (ρ) as ρ → 0. This is akin to the situation encountered in [2], where α ρ ≈ 1.7 was observed for krill densities. A realistic description of prey in the sea requires that the patch size distribution ψ(R) is exponentially cut-off at large scales, with few patch exceeding a value R m (see eq.(1), main article). This puts a lower bound on ρ and ensures that f (ρ) is integrable in the small ρ region.
If ν > 3, then α ρ < 1. The medium is more homogeneous than in the previous case due to the relative scarcity of large patches. Even if the patch size distribution has no large size cut-off (i.e. ψ(R) is a pure power-law), f (ρ) remains integrable in ρ = 0. At large densities, on the other hand, the distribution f (ρ) is very "flat". It drops to zero, however, once the highest density in the system, ρ 0 , is reached.