Interfaces with internal structures in generalized rock-paper-scissors models

Abstract

In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships, in the context of cyclic predator-prey models with an even number of species N ≥ 8. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behaviour associated to the symmetric or asymmetric evolution of the interface profiles depending on whether N/2 is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive internal structure of the interfaces. Although this work focus on cyclic predator prey-models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.

Coarsening with nontrivial in-domain dynamics: Correlations and interface fluctuations

“…3, a similar picture seems to hold for the (6,3) model when identifying as one of the states the pattern emerging from the interactions between the different members forming one of the two teams. In some recent studies of related many-species predator-prey models [42,43,49,50,54] the relevant length scale was found to increase as a square root of time, similar to what is found in the coarsening regime of the Ising model. These studies, however, focused (with one exception on which we comment below) on cases without the formation of large-scale dynamic structures inside the domains.…”

“…The collisions of spirals at the domain boundaries strongly slow down the elimination of empty sites that are originally formed through interactions between the different teams. We also remark that the postulated simple relationship [42,43,49,50,54] between the exponent x C of the correlation length and the exponent x E of the density of empty site, x E = −1/x C , does not hold for the (6,3) model with non-trivial internal dynamics.…”

“…These studies, however, focused (with one exception on which we comment below) on cases without the formation of large-scale dynamic structures inside the domains. Whereas the studies in [42,43,49,50,54] only measured the density of empty sites and derived from this quantity the typical length under the assumption that they are inversely proportional, we will in the following investigate a wide range of quantities that have been extensively tested in the past for the Ising and related models. It follows from our results that in the presence of non-trivial internal dynamics the values of the exponents governing coarsening, aging, and interface fluctuations differ from those expected from a system with curvature driven dynamics.…”

“…Much work has been devoted to cyclic cases as for example the three-species cyclic game [21] or the corresponding game with four species where each species is preying on one other species while being at the same time the prey of another species [22][23][24][25][26][27][28][29]. Whereas some earlier papers have considered spatial and stochastic effects in systems with a larger number of species [30][31][32][33][34][35][36][37][38][39][40][41], it is only in the last few years that systematic theoretical studies of more complicated food networks with five or more species 2 have become available [19,[42][43][44][45][46][47][48][49][50][51][52][53][54]. One of the intriguing results of these studies has been the discovery of a rich variety of space-time patterns, including spirals where each wavefront is formed by a single species, fuzzy spirals due to the mixing of different species inside the waves, coarsening domains where every domain is formed by an alliance of mutually neutral species as well as coarsening processes where inside every domain spirals are formed, thus yielding non-trivial dynamics inside the coarsening domains [19,48,51,55].…”

From pairwise to group interactions in games of cyclic dominance

“…In general, if the mobility in the population is sufficiently small [7], the spatial rock-paper-scissors game leads to the stable coexistence of all three competing strategies, whereby the coexistence is maintained by the spontaneous formation of complex spatial patterns. Most intriguing recent examples of this phenomenon include the observation of labyrinthine clustering [57] and interfaces with internal structure [58].…”