Hamming distance and mobility behavior in generalized rock-paper-scissors models

This work reports on two related investigations of stochastic simulations which are widely used to study biodiversity and other related issues. We first deal with the behavior of the Hamming distance under the increase of the number of species and the size of the lattice, and then investigate how the mobility of the species contributes to jeopardize biodiversity. The investigations are based on the standard rules of reproduction, mobility and predation or competition, which are described by specific rules, guided by generalization of the rock-paper-scissors game, valid in the case of three species. The results on the Hamming distance indicate that it engenders universal behavior, independently of the number of species and the size of the square lattice. The results on the mobility confirm the prediction that it may destroy diversity, if it is increased to higher and higher values. Introduction. – Despite the many ways to study complex systems, stochastic simulations represent an important tool that has been largely used to investigate collective behavior in nature. The procedure is based on a set of simple rules that are assessed randomly and, in particular, has been employed to model and understand biodiversity in nature; see, e.g., Refs. [1–11] and references therein. In this work we deal with generalized rock-paper-scissors models to describe stochastic network simulations of the May and Leonard type [3,6]. We consider a square lattice of size N × N and follow three recent investigations, the first [8] describing how local dispersal may promote biodiversity in a real-life game, in which the predictions based on the rock-paper-scissors rules are empirically tested using a non-transitive model community containing three populations of Escherichia coli, the second [10] suggesting that population mobility may be central feature to describe real ecosystems, and the third [12] that shows how to use the Hamming distance [13] to unveil the presence (a)Corresponding author; [email protected] of chaos in complex biological systems. These studies develop simulations that engender cyclic competition, modeled as in the rock-paper-scissors game, which is controlled by the simple rules where paper wraps rock, rock crushes scissors and scissors cut paper [4], and the generalizations that are illustrated in Fig. 1. An interesting behavior of the stochastic simulations is the formation of spiral patterns, which indicate a subjacent law that was recently investigated in [12]. The study unveiled the presence of chaos as an important component to guide evolution of natural life, taking the Hamming distance [13] as a way to measure the difference between the time evolution corresponding to two slightly distinct initial states, via an algorithm developed in [12]. In the current work we further explore the Hamming distance concept, to confirm that it acquires the universal qualitative behavior identified in Ref. [12]. To do this, we consider three distinct lattice sizes and several distinct systems, labeled by n, the first with three distinct species a, b, and c, with n = 3, the second with four distinct species, a, b, c, and d, with n = 4, the third with five distinct species, a, b, c, d, and e, with n = 5, and so on until the case with n = 10, p-1 ar X iv :1 71 1. 02 75 4v 1 [ qbi o. PE ] 7 N ov 2 01 7