Collective foraging in heterogeneous landscapes

Abstract

Animals foraging alone are hypothesized to optimize the encounter rates with resources through Lévy walks. However, the issue of how the interactions between multiple foragers influence their search efficiency is still not completely understood. To address this, we consider a model to study the optimal strategy for a group of foragers searching for targets distributed heterogeneously. In our model, foragers move on a square lattice containing immobile but regenerative targets. At any instant, a forager is able to detect only those targets that happen to be in the same site. However, we allow the foragers to have information about the state of other foragers. A forager who has not detected any target walks towards the nearest location, where another forager has detected a target, with a probability exp(-αd), where d is the distance between the foragers and α is a parameter characterizing the propensity of the foragers to aggregate. The model reveals that neither overcrowding (α → 0) nor independent searching (α → ∞) is beneficial for the foragers. For a patchy distribution of targets, the efficiency is maximum for intermediate values of α. In addition, in the limit α → 0, the length of the walks can become scale-free.

Keywords: Lévy walks; heterogeneous landscapes; optimal foraging theory.

Figure 1.

The snapshots during evolution of the model for N = 128, L = 128, γ = 2.5 and α = 0 at times t = (a) 0, (b) 194, (c) 281, (d) 747, (e) 802 and (f) 979. The foragers are marked with blue circles and the targets are marked with green squares. The path of a typical forager is drawn. The red steps belong to random walks and the black steps belong to targeted walks. The fact that a single lattice site may be multiply occupied by foragers or targets is not separately colour coded.

Figure 2.

(a) Variation of the efficiency, η, with α for L = 512. The different symbols used are for N = 1024, γ = 2.5 (triangle-up); N = 512, γ = 2.5 (circle); N = 256, γ = 2.5 (triangle-down); N = 512, γ = 2.0 (triangle-right); and N = 512, random regeneration of targets (triangle-left). The inset of (a) corresponds to N = 512 and γ = 3.5. (b) Scaling collapse of η for different values of N with γ = 2.5 and L = 512. The collapse results with β₁ = 0.15 and β₂ = 0.70. The inset of (b) shows the dependence of α_m on N in a log–log plot. The dashed straight line having slope ζ = 0.65 shows the power-law nature of the variation. (Online version in colour.)

Figure 3.

Plots of the fraction of walkers executing targeted walks at any instant f_t (a) and the flocking order parameter Φ (b) against α for L = 512. The different parameter sets denoted by the different symbols in the main areas and the insets are the same as those in figure 2a. (Online version in colour.)

Figure 4.

Snapshots of two different patterns exhibited by the model corresponding to regrowth rules for targets and for the parameters N = 512, L = 512 and α = 0. A stable moving band, observed when targets regenerate randomly across the lattice (a) and wedge formation for γ = 3.5 (b) (only the section of the lattice where foragers and targets are concentrated is shown). In (a), the foragers are marked with blue and the targets are marked with green. The path of a typical forager is marked with red. The arrow points to the direction of motion of the band. The multiple occupation of sites by foragers or targets is not colour coded. The inset shows a section of the trajectory where the RWs appear in red and TWs in black. In (b), the foragers are marked with blue (multiple occupation ignored), and the path of a typical forager is shown with a colour scheme similar to the inset of (a). A colouring scheme for the multiple occupation of sites by targets is used (provided in legend).

Figure 5.

The plots of probability density function, P(l), of the length of targeted walks. (a–d) The different sets of values of N, L, γ and α, as indicated in the legends. The dashed line in (a) is a guide to the eye and indicates the power-law nature in the region for the curve with N = 512, L = 512, γ = 2.5 and α = 0. Regression fit with gives the value of μ = 2.80. In (c), the values of μ corresponding to different values of N (=L) resulting from regression fits are provided in the legend. (Online version in colour.)

Figure 6.

(a) The plot of average number of neighbours n as a function of m, where . The bars indicate the standard deviations obtained from averaging over time snapshots and initial conditions. The straight line corresponds to a power-law fit having exponent equal to 2.23. (b) The plots of probability density function, P(l), of the length of targeted walks when α = 0, corresponding to different values of m as indicated in the legend. (Online version in colour.)

Figure 7.

The plots of probability density function, P(τ), of waiting times between forager–target encounters. (a–d) Different sets of values of N, L, γ and α, as indicated in the legends. The dashed line in (a) is a guide to the eye and indicates the power-law nature in the region for the curves with N = 512, L = 512, γ = 2.5 and with the values of α = 0, 5.0 × 10⁻⁴, 5.0 × 10⁻³. Regression fits with give the value of δ = 1.67 for all three values of α. In (c), the values of δ corresponding to different values of N (=L) resulting from regression fits are provided in the legend. (Online version in colour.)