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Diverse Stochasticity Leads a Colony of Ants to Optimal Foraging

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Diverse Stochasticity Leads a Colony of Ants to Optimal Foraging 1805.05598 Diverse Stochasticity Leads a Colony of Ants to Optimal Foraging Masashi Shiraishi1;3, Rito Takeuchi4, Hiroyuki Nakagawa2, Shin I Nishimura5, Akinori Awazu1, and Hiraku Nishimori1∗ 1Department of Mathematical and Life Sciences, Hiroshima University, Kagamiyama, Higashi-hiroshima 739-8526, Japan 2Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan 3CREST, JST, Tokyo, Japan 4Yahoo Japan Corporation, Tokyo, Japan and 5JamGuard Corporation, 1-5-6 Kudam-Minami, Chiyoda, Tokyo, Japan (Dated: January 7, 2019) Abstract A mathematical model of garden ants (Lasius japonicus) is introduced herein to investigate the relationship between the distribution of the degree of stochasticity in following pheromone trails and the group foraging efficiency. Numerical simulations of the model indicate that depending on the systematic change of the feeding environment, the optimal distribution of stochasticity shifts from a mixture of almost deterministic and mildly stochastic ants to a contrasted mixture of almost deterministic ants and highly stochastic ants. In addition, the interaction between the stochasticity and the pheromone path regulates the dynamics of the foraging efficiency optimization. Stochas- ticity could strengthen the collective efficiency when the variance in the sensitivity to pheromone for ants is introduced in the model. PACS numbers: Valid PACS 82.39.Rt, 07.05.Tp, 89.75.Kd Keywords: Stochastic foraging, Optimization, Collective motion, Ants arXiv:1805.05598v2 [nlin.AO] 4 Jan 2019 ∗ [email protected] 1 I. INTRODUCTION Ants efficiently shuttle between their nest and food sources over long distances by deposit- ing and following pheromone trails[2]. As a result of the collective usage of pheromones, ants are able to construct a trail that might be accomplished by accurate pheromone usage. Fol- lowing the trail ensures that ants can accurately reach food sources. However, a certain frac- tion of ants in each colony does not accurately follow pheromone trails[3, 4]. The concept of stochastic foraging or strategy of error is introduced and validated through experiments and mathematical models to interpret the ecological significance of this stochasticity[3, 5, 7, 10{ 15, 17]. Deneubourg et al. performed numerical experiments to understand the meaning of the very low percentage of Tetramorium impurum reaching a food source following a pheromone trail[3]. Using a simple phenomenological model for the recruiting dynamics of ants, they estimated the optimized degree of stochasticity for each ant to be recruited to an already-found food source in a multi-food-source environment. They found that a less accurate recruitment led the colony to a more efficient foraging as the number of location of food sources increased. These studies recognized the stochasticity of ants in following pheromone trails as a factor that increased their chance to find new food sources and avoid excessive persistence to previously established trails. Nowadays, the concept of \strategy of error" is also known as \explore{exploit" trade-off problem in diverse research fields. Studies share the similar structures and solutions, even though research subjects are different from each other[1]. However, many basic problems on stochasticity-induced efficient foraging remain un- solved. One among them is the main issue being focused on herein: what type of population distribution of stochasticity enables optimized foraging in a colony of ants? Previous models of stochastic foraging have implicitly or explicitly introduced the uniform population distri- bution of stochasticity because of their simple forms[3]. However, experiments with existing ants have shown widely dispersed population distributions of stochasticity. As an extreme example, only 18% of Tetramortium inpurum in a colony can reach the food sources by fol- lowing pheromone trails, and the same quantity decreases down to 9% in the case without a leader[3]. As another example, typically more than 40% individuals in a colony of garden ants species Lasius niger, which usually make use of both visual and chemical cues in their foraging trip, fail to follow pheromone trails if visual cues are made imperfectly available[4]. We believe that the population distribution of stochasticity in following pheromone trails is not always restricted to a Gaussian-like unimodal form. We evaluate this presumption by performing a model-based study on what type of distribution of stochasticity leads a colony of ants to efficient foraging. We focus on the effects of the intensity of stochasticity of the stochastic ants and their population rate in the colony under different environment condi- 2 tions. Our study aims to mathematically understand how the stochasticity in a complex collective system based on real biological systems works to optimize the collective behaviors and analyze how the specific stochasticity in the ant colony model works compared to the stochasticity in previous studies[3, 5, 7, 10{15, 17]. This paper is organized as follows: Section 2 introduces our agent-based model; Section 3 explains the evaluation procedure of the foraging behaviors with the model; Section 4 shows the numerical results and discusses some combinations of stochasticity intensity and their population rate in the colony; and Section 5 concludes and summarizes the results. II. MODEL We employ herein a computational model categorized in multi-agent models [15, 16, 18]. The unique task for each agent (i.e., ant) in the present model is to find food sources and bring food to the nest. The peculiar feature of this model is that in addition to using trail pheromone, ants make use of visual cues to roughly grasp the homing direction, where this non-local walking rule of ants is introduced to make a qualitative correspondence to the behavior of the existing species of ants, namely the Lasius japonicus. The foraging field is a hexagonal two-dimensional lattice measuring 150 × 150 with a periodic boundary condition, on which one nest site and two sites of food sources are designated. The dynamical states for each ant, which are labeled as k 2 f1; 2;:::Ntotalg, are represented by a positional vector, ~xk, consisting of a lattice index for the hexagonal two-dimensional lattice and a facing direction vector, ~ηk, consisting of one of the unit vectors of the hexagonal two-dimensional lattice. Each ant at each time step is located at one site, but does not exclusively occupy it, and faces toward one of its nearest six sites. A randomly selected single ant among a total of Ntotal ants moves to the facing site, its right site, or its left site (the detailed process to select a site is explained below). The same processes are repeated for Ntotal times and constituted a unit Monte Carlo (MC) step. This unit MC step is accompanied by a single step of a difference equation used to update the amount of food and another difference equation used for the evolution of the pheromone field, which will be explained in Sections 2F and C, respectively. The distance R between the nest and the food source sites and the relative angle θ between two food source sites as seen from the nest are given as the feeding environmental parameters (Fig. 1). Increasing the food supplies per unit time is also an important feeding environment parameter, which characterizes the dynamical aspects of this model. The dynamics of food supply and the evaluation methods of the system are explained in the sub-sections that follow. 3 Food � � �⃗# ℓ % �⃗# �⃗# Nest �⃗# (a) (b) FIG. 1: (a) The distance R between the nest and the food source points and the relative angle θ between two food sources as seen from the nest are considered as the feeding environmental parameters. Food resource sites are the closest hexagonal sites (blue meshed ` f r hexagonal sites) to the food source points (blue dots). (b) Movable sites f~xk; ~xk; ~xkg of ants. A. Chemotactic Dynamics In this model, each ant moves to the facing site or its adjacent sites, which are stochasti- cally decided upon depending on the trail pheromone on the lattice field. ~xk is the position of the kth ant, and ~ηk is its facing direction vector. f ` r Therefore, the site to which the ant faces is represented by ~xk = ~xk + ~ηk, and ~xk and ~xk are the left and the right sites of the facing site, respectively. The response probability, pj, j denoting the move to one of the three movable sites, ~xk (j 2 f`; f; rg), is presented as: g φ ~xj p (φ(~x` ); φ(~xf ); φ(~xr )) = k ; (1a) j k k k P3 i i2f`;f;rg g (φ (~xk)) h g(y) = (αky + z) ; (1b) j j where φ ~xk is the trail pheromone density at ~xk; a parameter h = 10 corresponds to the Hill coefficient, z = 0:05; and g(y) is a function defining the individual's susceptibility to the pheromone.Unfortunately, this response function is not directly experimentally determined; thus, we adopted a function with a strong non-linear effect to make the response function of ` f r the pheromone a sigmoid-like function of φ(~xk), φ(~xk), and φ(~xk). Note that αk is the key parameter controlling the stochasticity of the kth ant; hence, the sensitivity of the kth ant to the trail pheromone increases as αk increases. In this sense, we call αk as the \sensitivity parameter." Figure 2 shows the effect of αk on the probability to choose a higher density cell. In the case of − log10 αk = 4, the probability remains at 0.5 even though ∆φ increases 4 (Fig. 2(5)). Aside from introducing the sensitivity parameter in this manner, the forms of Eqs. (1a) and (1b) follow those of the previous studies[5, 6]. We define \normal ants," whose sensitivity parameter is fixed as αk = αn = 1, as a criterial parameter.
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