Lévy flights, optimal foraging strategies, and foragers with a finite lifespan

Serena Dipierro¹ ; Giovanni Giacomin¹ ; Enrico Valdinoci¹

¹ Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA6009 Crawley, Australia

Mathematical modelling of natural phenomena, Tome 19 (2024), article no. 17

Résumé

In some recent work, we have introduced some efficiency functionals to account for optimal dispersal strategies of predators in search of food. The optimization parameter in this framework is given by the Lévy exponent of the dispersal of the predators. In this paper, we apply our model to the case of foragers with finite lifetime (i.e., foragers which need to eat a certain amount of food in a given time, otherwise they die). Specifically, we consider the case in which the initial distribution of the forager coincides with a stationary distribution of the targets and we determine the optimal Lévy exponent for the associated efficiency functional. Namely, we show that if the Fourier transform of the prey distribution is supported in a sufficiently small ball, then the optimizer is given by a Gaussian dispersal, and if instead the Fourier transform of the prey distribution is supported in the complement of a suitable ball, then the ballistic diffusion provides an optimizer (precise conditions for the uniqueness of these optimizers are also given).

Détail

DOI : 10.1051/mmnp/2024015

Bibliographie

[1] G.M. Viswanathan, M.G.E. da Luz, E.P. Raposo and H.E. Stanley, The Physics of Foraging. Cambridge University Press, Cambridge (2011) MR2814265.

[2] E. Kagan and I. Ben-Gal, Search and Foraging. CRC Press, Boca Raton, FL (2015) MR3379124.

[3] M. Van Dartel, E. Postma, J. Van Den Herik, G. De Croon Macroscopic analysis of robot foraging Behaviour Connect. Sci. 2004 169 181

[4] D. Brockmann Following the money Phys. World 2010 31 34

[5] A. Baronchelli, F. Radicchi Lévy flights in human behavior and cognition Chaos Solitons Fractals 2013 101 105

[6] E. Gelenbe and O.H. Abdelrahman, Search in Random Media with Lévy Flights (2014).

[7] A. Brabazon and S. McGarraghy, Foraging-inspired Optimisation Algorithms. Natural Computing Series. Springer, Cham (2018) MR3838316.

[8] G. Estrada-Rodriguez, H. Gimperlein Interacting particles with Lévy strategies: limits of transport equations for swarm robotic systems SIAM J. Appl. Math. 2020 47 498

[9] S. Duncan, G. Estrada-Rodriguez, J. Stocek, M. Dragone, P.A. Vargas, H. Gimperlein Efficient quantitative assessment of robot swarms: coverage and targeting Lévy strategies Bioinspir. Biomim. 2022 1 16

[10] G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G. Da Luz, E.P. Raposo, H.E. Stanley Optimizing the success of random searches Nature 1999 911 914

[11] G.H. Pyke Understanding movements of organisms: it’s time to abandon the Lévy foraging hypothesis Meth. Ecol. Evol. 2015 1 16

[12] S. Dipierro, G. Giacomin, E. Valdinoci The Lévy flight foraging hypothesis: comparison between stationary distributions and anomalous Diffusion J. Phys. A 2023 35

[13] S. Dipierro, G. Giacomin, E. Valdinoci Analysis of the Lévy flight foraging hypothesis in ℝn and unreliability of the most rewarding Strategies SIAM J. Appl. Math. 2023 1935 1968

[14] S. Bury Energy expenses on prey processing are comparable, but paid at a higher metabolic scope and for a longer time in ambush vs active predators: a multispecies study on snakes Oecologia 2021 61 70

[15] S. Dipierro, G. Giacomin, E. Valdinoci Efficiency functionals for the Lévy flight foraging hypothesis J. Math. Biol. 2022 50

[16] O. Bénichou, S. Redner Depletion-controlled starvation of a diffusing forager Phys. Rev. Lett. 2014 1 5

[17] P.A.M. Dirac, Bakerian lecture – the physical interpretation of quantum mechanics. Proc. R. Soc. Lond. A (1942) 1801–1840.

[18] R.P. Feynman, Negative Probability, Quantum Implications: Essays in Honour of David Bohm (1987) 235–248.

[19] S. Dipierro, G. Giacomin and E. Valdinoci, The Lévy flight foraging hypothesis in bounded regions’ subordinate Brownian motions and high-risk/high-gain strategies. Memoirs of the European Mathematical Society, Vol. 10. European Mathematical Society (EMS), Berlin (2024) MR4711702.

[20] P.E. Lestrel (Ed.), Fourier Descriptors and Their Applications in Biology. Cambridge University Press, Cambridge (1997).

[21] M. Kot, Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001) MR2006645.

[22] J.D. Murray, Mathematical Biology I, 3rd ed. Interdisciplinary Applied Mathematics, Vol. 17. Springer-Verlag, New York (2002) MR1908418.

[23] J. Sueur, Sound Analysis and Synthesis with R, Use R!. Springer Cham (2018).

[24] E.L. Charnov Optimal foraging, the marginal value theorem Theor. Popul. Biol. 1976 129 136