Optimal foraging: Lévy pattern or process?

Abstract

Many different species have been suggested to forage according to a Lévy walk in which the distribution of step lengths is heavy-tailed. Theoretical research has shown that a Lévy exponent of approximately 2 can provide a higher foraging efficiency than other exponents. In this paper, a composite search model is presented for non-destructive foraging behaviour based on Brownian (i.e. non-heavy-tailed) motion. The model consists of an intensive search phase, followed by an extensive phase, if no food is found in the intensive phase. Quantities commonly observed in the field, such as the distance travelled before finding food and the net displacement in a fixed time interval, are examined and compared with the results of a Lévy walk model. It is shown that it may be very difficult, in practice, to distinguish between the Brownian and the Lévy models on the basis of observed data. A mathematical expression for the optimal time to switch from intensive to extensive search mode is derived, and it is shown that the composite search model provides higher foraging efficiency than the Lévy model.

Figure 1

Two sample foraging paths, both starting at x=x₀. On the path represented by the dashed line, the forager finds the food item at x=0 during the intensive search phase; on the path represented by the solid line, the forager fails to find food during the intensive phase and would continue to perform an extensive search.

Figure 2

Distribution of distance travelled L before finding food (S(l)=P(L>l)), calculated from SDE simulations and analytically from equation (2.4). (a) Early changeover from intensive to extensive search mode (τ=50 s). (b) Late changeover (τ=1000 s). Other parameter values: v_I=4 ms⁻¹, x₀=10 m and d=10 000 m. Solid line, SDE simulations; dashed line, analytic.

Figure 3

Distribution of observed step lengths Y (S(y)=P(Y>y)), calculated from SDE simulations and analytically from equation (2.8). (a) Early changeover from intensive to extensive search mode (τ=50 s). (b) Late changeover (τ=1000 s). Other parameter values are v_I=4 m s⁻¹, v_E=10 m s⁻¹, σ_E=2.5 m s⁻¹, x₀=10 m, d=10 000 m and t_s=3 s. Solid line, SDE simulations; dashed line, analytic.

Figure 4

(a) Mean efficiency (reciprocal of distance travelled to find food) against τ, calculated from SDE simulations and analytically from equation (2.9): v_I=1, d=1000 (solid line, SDE simulations x₀; dashed line, analytic x₀; dot-dashed line, SDE simulations x₀; dotted line, analytic x₀). (b) Optimal switching time ${\hat{\tau }}^{*}=v_I{\tau }^{*}/d$ against ϵ (solid line, numerical; dashed line, analytical). The curves are calculated by numerical minimization of E(L) according to equation (2.9) and using the approximation (2.10). If $ϵ<0.19$, there is a global minimum in E(L) at a value of τ closely approximated by (2.10). If ϵ>0.19, this local minimum ceases to be a global minimum—the optimal value of τ is zero.

Figure 5

Distribution of step lengths Y ($S(y)=P(Y>y)$), under the SDE model (τ=1000) and the Lévy random walk model with μ=1.9 and x_min=0.01. For both curves, v_I=v_E=1, σ_E=0.1, x₀=10, d=1000 and t_s=1. Solid line, SDE simulations; dashed line, analytic.

Figure 6

Mean maximum efficiency (reciprocal of distance travelled to find food) against x₀, for the composite strategy and the Lévy strategy. The analytical results for the SDE model were calculated using equations (2.9) and (2.10). The analytical results for the Lévy model were calculated according to equation (2.1) of Viswanathan et al. (2000) (with r_v=x_min), maximized over all values of μ. Parameter values: x_min=0.01, d=1000. Solid line, SDE simulations; dashed line, SDE analytic; dot-dashed line, Lévy RW simulations; dotted line, Viswanathan analytic.

Figure 7

Graphs of $F(\hat{\tau })$ (solid lines) and $G(\hat{\tau })$ (dashed lines) against $\hat{\tau }$ for (a) ϵ=0.3 and (b) ϵ=0.2. In case of (a), $(\text{d}E(L))/(\text{d}\hat{\tau })>0$ for all $\hat{\tau }$. In case of (b), there is a range of $\hat{\tau }$ (between the two roots of $F(\hat{\tau })=G(\hat{\tau })$) for which $(\text{d}E(L))/(\text{d}\hat{\tau })<0$.