Optimal Lévy-flight foraging in a finite landscape

Abstract

We present a simple model to study Lévy-flight foraging with a power-law step-size distribution [P(l) ∞ l-μ] in a finite landscape with countable targets. We find that different optimal foraging strategies characterized by a wide range of power-law exponent μopt, from ballistic motion (μopt → 1) to Lévy flight (1 < μopt < 3) to Brownian motion (μopt ≥ 3), may arise in adaptation to the interplay between the termination of foraging, which is regulated by the number of foraging steps, and the environmental context of the landscape, namely the landscape size and number of targets. We further demonstrate that stochastic returning can be another significant factor that affects the foraging efficiency and optimality of foraging strategy. Our study provides a new perspective on Lévy-flight foraging, opens new avenues for investigating the interaction between foraging dynamics and the environment and offers a realistic framework for analysing animal movement patterns from empirical data.

Keywords: Lévy flight; optimal foraging; random search.

Figure 1.

A schematic diagram of the model. This diagram shows a foraging process with N = 4 steps. The red dots represent targets. The cylinder formed by black dashed boundaries indicates the detection area during an exploration step. In step one, the forager leaves a target and detects no new target during this step. Note that the forager ignores the departure target. In step two, the forager decides to undertake exploration and detects a target during this step. Therefore, the original probabilistic step (the green dash-dotted line) is truncated to a shorter actual step l₂ (the green solid line). Step three is similar to step one. In step four, the forager decides to return and it flies straight back to the departure target in step one. In a return step, the forager is not attempting to detect targets.

Figure 2.

(a) The number of discovered targets S_n versus step number n. The curves in different colours from top to bottom correspond to various μ ∈ [1.1, 2.5] with an interval of 0.2. Here, we use the landscape size L = 200 and the number of targets K = 200. The dots represent simulation results averaged for 100 realizations, and the solid lines represent the corresponding mean-field solution given by equation (3.3). (b) The mean number of steps between the discovery of the Sth target and the (S + 1)th target versus the number of undiscovered targets K − S. The dots represent simulation results, and the lines represent the corresponding linear regression fit. (c) The mean number of steps between two consecutive detections n_d as a function of μ for different values of L and K. Here, K is adjusted to obtain the corresponding λ given L. The solid lines represent the nonlinear fitting . (d) The constant coefficient γ as a function of μ for different values of L and K. The solid lines represent the cubic polynomial fitting.

Figure 3.

(a) The rescaled search efficiency η/η_max versus the power-law exponent μ for different values of the total number of steps N from numerical simulation with the intensity of stochastic returning β = 0, the landscape size L = 1000, the number of targets K = 5000 and the termination condition Θ_n = δ(n − N). The results are averaged over 100 realizations. The rescaling factor η_max = η(μ = μ_opt, N = 5000) is the overall maximum search efficiency. (b) η/η_max versus μ from mean-field calculation. The black dots indicate the peaks of the curves. The inset of panel (b) shows μ_opt versus N. The curves in panels (a,b) with different colours from top to bottom correspond to various N ∈ [5 × 10³, 5 × 10⁴] with an interval of 5000. Note that the discrepancy between the simulation and the mean-field approach here is mostly due to the slight deviation of equation (3.4), as shown in figure 7b. This can be improved by better calibrating the form of equation (3.4), e.g. using curve fitting as we obtain the form of continuous function for n_d(μ) and γ(μ).

Figure 4.

Panels (a–c) show the rescaled search efficiency η/η_max versus the power-law exponent μ for different values of the landscape size L and the number of targets K from numerical simulation with the intensity of stochastic returning β = 0 and the termination condition Θ_n = δ(n − 20 000). The results are averaged over 100 realizations. The curves in panel (a) from bottom to top correspond to (L, K) = (100, 50), (200, 200), (400, 800), (800, 3200) such that the mean free path λ = 100 remains constant. The curves in panel (b) from top to bottom correspond to L = 200, 400, 600, 800 with K = 800 kept constant. The curves in panel (c) from top to bottom correspond to K = 800, 1600, 2400, 3200 with L = 400 kept constant. Panels (d–f) show η/η_max evaluated by the mean-field solution, corresponding to the results from numerical simulation in panels (a–c), respectively. The black dots indicate the peak of the curves.

Figure 5.

(a) The search efficiency η versus the power-law exponent μ for different values of the threshold moving distance from numerical simulation with the intensity of stochastic returning β = 0, the landscape size L = 1000, the number of targets K = 5000 and the termination condition where is the accumulated moving distance. The results are averaged over 100 realizations. (b) η evaluated by the mean-field solution by setting the total number of steps N by in equation (3.3).

Figure 6.

(a) η/η_max versus μ for various intensities of stochastic returning β from numerical simulation with the landscape size L = 1000, the number of targets K = 5000 and the termination condition Θ_n = δ(n − 50 000). The results are averaged over 100 realizations. The dots represent simulation results, and the dashed lines are a guide to the eye. (b) The total number of discovered targets S_N versus μ. The curves in panels (a,b) with different colours from top to bottom corresponding to various β with increasing values as indicated in the legend of panel (b).

Figure 7.

(a) The mean step-size versus the power-law exponent μ. (b) The mean step-size in exploration versus μ. The red solid line represents equation (3.4). (c) The mean step-size in return versus μ. (d) The ratio of the number of return steps N^ret to the total number of steps N versus μ. In panels (a–d), the dots represent simulations results, and the dashed lines are a guide to the eye. Different colours correspond to different values of β, as indicated in the legend of panel (a). The results are obtained from numerical simulation with the landscape size L = 1000, the number of targets K = 5000 and the termination condition Θ_n = δ(n − 50 000) and are averaged over 100 realizations.

Figure 8.

(a) The total moving distance versus the power-law exponent μ. (b) The total moving distance in exploration versus μ. (c) The total moving distance in return versus μ. (d) The ratio of to the total number of steps versus μ. In panels (a–d), the dots represent simulations results and the dashed lines are a guide to the eye. Different colours correspond to different values of β, as indicated in the legend of panel (a). The results are obtained from numerical simulation with the landscape size L = 1000, the number of targets K = 5000 and the termination condition Θ_n = δ(n − 50 000) and are averaged over 100 realizations.