The evolutionary maintenance of Lévy flight foraging

Lévy flight is a type of random walk that characterizes the behaviour of many natural phenomena studied across a multiplicity of academic disciplines; within biology specifically, the behaviour of fish, birds, insects, mollusks, bacteria, plants, slime molds, t-cells, and human populations. The Lévy flight foraging hypothesis states that because Lévy flights can maximize an organism’s search efficiency, natural selection should result in Lévy-like behaviour. Empirical and theoretical research has provided ample evidence of Lévy walks in both extinct and extant species, and its efficiency across models with a diversity of resource distributions. However, no model has addressed the maintenance of Lévy flight foraging through evolutionary processes, and existing models lack ecological breadth. We use numerical simulations, including lineage-based models of evolution with a distribution of move lengths as a variable and heritable trait, to test the Lévy flight foraging hypothesis. We include biological and ecological contexts such as population size, searching costs, lifespan, resource distribution, speed, and consider both energy accumulated at the end of a lifespan and averaged over a lifespan. We demonstrate that selection often results in Lévy-like behaviour, although conditional; smaller populations, longer searches, and low searching costs increase the fitness of Lévy-like behaviour relative to Brownian behaviour. Interestingly, our results also evidence a bet-hedging strategy; Lévy-like behaviour reduces fitness variance, thus maximizing geometric mean fitness over multiple generations.

" Is the terminology of intrinsic vs. extrinsic hypotheses new? This should be clearly stated. I have only heard of these previously referred to as the adaptationist vs. the emergentist viewpoints, and so forth." Response: This terminology has been used at least once before, for example, in ref. 30 (Sims, 2019). In the interest of scientific clarity, the introduction now states "(the intrinsic or adaptionist hypothesis)" and "(extrinsic or emergentist hypothesis)" (lines 17 & 19, and lines 438 & 446, respectively).

Reviewer 2
"It is said in the abstract that "Lévy flight is a type of random walk that models the behaviour of many phenomena across a multiplicity of academic disciplines". I disagree with this sentence since LF are a natural phenomenon and not models. I suggest instead 'Lévy flight is a type of random walk present in the behaviour of many natural phenomena across a multiplicity of academic disciplines.'" Response: Agreed, with the slight tweak "that characterizes the behaviour of many natural phenomena studied across a multiplicity of academic disciplines".
"Even though the possibility of the emergent nature of Lévy flight is mentioned, its treatment is somewhat sloppy. I suggest in order to give more punch to the paper and to show that the authors handle well the literature, that the following papers on deterministic walks be read and discussed either at the introduction or the conclusions: Response: Both papers do not seem to be testing the Lévy flight foraging hypothesis per se (there are no evolutionary components), but I do understand their importance in demonstrating Lévy-like behaviour (or self-avoidance) as being not strictly intrinsic. I discuss this in the following response, too. I've included a brief discussion of these papers in the conclusion (lines 441-447).
"Line 425: "Organisms with Brownian behaviour have a higher probability of re-visiting the same resource and their behaviour may be optimal for shorter lifespans, but with sufficient time they would diffuse into empty space reducing the probability of re-visits. A Lévy-like exponent, however, would have a higher probability of leaving that empty space and eventually encountering resources." There is a paper demonstrating this fact but that has been ignored:[Ref3] Dannemann, Teodoro, Denis Boyer, and Octavio Miramontes. "Lévy flight movements prevent extinctions and maximize population abundances in fragile LotkaVolterra systems." Proceedings of the National Academy of Sciences 115.15 (2018): 3794-3799." Response: This paper was very interesting to read and clearly has complementary results. It is now included in the revised manuscript (1. 433-436).
"First sentence of the conclusion is misleading and there is indeed lack of general evidence through the paper to claim this as true: "These results provide evidence that the intrinsic (please correct the typo) hypothesis is a sufficient explanation for Lévy-like behaviour: Lévy flight is the result of selection for behavioural adaptations, rather than an emergent phenomenon due to the encounters within an environments distribution of resources." [Ref2] gives enough theoretical evidence that the distribution of resources modulates the searching strategy resulting in emergent movement patterns. Perhaps the authors would like to explain under which particular and specific circumstances their claim is true and tone it down consequently." Response: I've encountered [Ref2] before, and I really like that paper. True, it does provide evidence for emergent Lévy behaviour, although it is not strictly a test of the Lévy flight foraging hypothesis (because in [Ref2], "the forager knows the location and size of all targets in the system." whereas in Lévy flight, a forager has no information about the distribution of resources), and the conclusion has been toned down accordingly (linens 438-447). Typo corrected -thanks.

Additional Changes
The following equation: Should instead be: This was a typo, and does not affect the other equations.