When does cyclic dominance lead to stable spiral waves?

Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional threespecies population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system’s phase diagram is inferred from the underlying complex Ginzburg–Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion. Introduction. – In nature, organisms live in areas much larger than the distances they typically travel and thus they interact with a finite number of individuals in their neighbourhood. Space and mobility are therefore crucial ingredients in understanding how populations evolve and how ecosystems self-organise. Even in the presence of sources of randomness and inhomogeneities, spatial degrees of freedom and movement can lead to the formation of characteristic spatio-temporal patterns [1], whose origin in ecosystems has been a subject of intense research for decades [1–3]. In his pioneering work, Turing showed that pattern-forming instabilities can be caused by diffusion [4]. While Turing patterns have been found in ecology and biology [2], the requirements of Turing’s theory (e.g. separation of scales in diffusivities) appear to be too restrictive to explain pattern formation in many ecosystems, see e.g. Ref [5]. Another important problem concerns the mechanisms promoting the maintenance of biodiversity [6]. In this context, cyclic dominance has been recently proposed as an intriguing motif facilitating the coexistence of diverse species in ecosystems. Examples of cyclic competition between three species can be found in coral reef (a)Electronic address: [email protected] (b)Electronic address: [email protected] (c)Electronic address: [email protected] invertebrates, Uta stansburiana lizards, and communities of E.coli [3, 7–9]. In the experiments of Ref. [7], the cyclic competition of three bacterial strains on twodimensional plates was shown to yield patterns sustaining species coexistence. Such competition is metaphorically described by rock-paper-scissors (RPS) games, where “rock crushes scissors, scissors cut paper, and paper wraps rock” [10]. While non-spatial RPS-like models often evolve towards extinction of all but one species in finite time [11], their spatial counterparts are generally characterised by the long-term coexistence of species and by the formation of complex spatio-temporal patterns [12–16]. Recently, various two-dimensional versions of the model introduced by May and Leonard [17] have received much attention [13–16]. When mobility is implemented by pairexchange among neighbours, species coexistence is longlived and populations form non-Turing spiralling patterns below a certain mobility threshold, whereas biodiversity is lost when that threshold is exceeded [13]. In this Letter, we characterise the intricate patterns emerging from the dynamics of a generic model of a cyclically competing three-species population, and study how these patterns affect the maintenance of biodiversity in two dimensions. The basic evolutionary processes considered here are the most general form of cyclic dominance between three species obtained by combining and unifyp-1 ar X iv :1 21 0. 83 76 v2 [ qbi o. PE ] 1 2 A pr 2 01 3