Optimal Lévy-flight foraging in a finite landscape

Abstract: We present a simple model to study L\'{e}vy-flight foraging in a finite landscape with countable targets. In our approach, foraging is a step-based exploratory random search process with a power-law step-size distribution $P(l) \propto l^{-\mu}$. We find that, when the termination is regulated by a finite number of steps $N$, the optimum value of $\mu$ that maximises the foraging efficiency can vary substantially in the interval $\mu \in (1,3)$, depending on the landscape features (landscape size and number of targets). We further demonstrate that subjective returning can be another significant factor that affects the foraging efficiency in such context. Our results suggest that L\'{e}vy-flight foraging may arise through an interaction between the environmental context and the termination of exploitation, and particularly that the number of steps can play an important role in this scenario which is overlooked by most previous work. Our study not only provides a new perspective on L\'{e}vy-flight foraging, but also opens new avenues for investigating the interaction between foraging dynamics and environment as well as offers a realistic framework for analysing animal movement patterns from empirical data.