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Register a new userCharacterization of spiraling patterns in spatial rock-paper-scissors games
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The spatio-temporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatio-temporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising rock-paper-scissors interactions via dominance removal and replacement, reproduction, mutations, pair-exchange and hopping of individuals. By combining analytical and numerical methods, we obtain the models phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away far from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the models Hopf bifurcation. Our results allows us to clarify when spatial rock-paper-scissors competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.
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We consider a two-dimensional model of three species in rock-paper-scissors competition and study the self-organisation of the population into fascinating spiraling patterns. Within our individual-based metapopulation formulation, the population composition changes due to cyclic dominance (dominance-removal and dominance-replacement), mutations, and pair-exchange of neighboring individuals. Here, we study the influence of mobility on the emerging patterns and investigate when the pair-exchange rate is responsible for spiral waves to become elusive in stochastic lattice simulations. In particular, we show that the spiral waves predicted by the systems deterministic partial equations are found in lattice simulations only within a finite range of the mobility rate. We also report that in the absence of mutations and dominance-replacement, the resulting spiraling patterns are subject to convective instability and far-field breakup at low mobility rate. Possible applications of these resolution and far-field breakup phenomena are discussed.
We investigate the impact of parity on the abundance of weak species in the context of the simplest generalization of the rock-paper-scissors model to an arbitrary number of species -- we consider models with a total number of species ($N_S$) between 3 and 12, having one or more (weak) species characterized by a reduced predation probability (by a factor of ${mathcal P}_w$ with respect to the other species). We show, using lattice based spatial stochastic simulations with random initial conditions, large enough for coexistence to prevail, that parity effects are significant. We find that the performance of weak species is dependent on whether the total number of species is even or odd, especially for $N_S le 8$, with odd numbers of species being on average more favourable to weak species than even ones. We further show that, despite the significant dispersion observed among individual models, a weak species has on average a higher abundance than a strong one if ${mathcal P}_w$ is sufficiently smaller than unity -- the notable exception being the four species case.
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.
In this letter, we investigate the population dynamics in a May-Leonard formulation of the rock-paper-scissors game in which one or two species, which we shall refer to as weak, have a reduced predation or reproduction probability. We show that in a nonspatial model the stationary solution where all three species coexist is always unstable, while in a spatial stochastic model coexistence is possible for a wide parameter space. We find, that a reduced predation probability results in a significantly higher abundance of weak species, in models with either one or two weak species, as long as the simulation lattices are sufficiently large for coexistence to prevail. On the other hand, we show that a reduced reproduction probability has a smaller impact on the abundance of weak species, generally leading to a slight decrease of its population size -- the increase of the population size of one of the weak species being more than compensated by the reduction of the other, in the two species case. We further show that the species abundances in models where both predation and reproduction probabilities are simultaneously reduced may be accurately estimated from the results obtained considering only a reduction of either the predation or the reproduction probability.
We review recent results obtained from simple individual-based models of biological competition in which birth and death rates of an organism depend on the presence of other competing organisms close to it. In addition the individuals perform random walks of different types (Gaussian diffusion and L{e}vy flights). We focus on how competition and random motions affect each other, from which spatial instabilities and extinctions arise. Under suitable conditions, competitive interactions lead to clustering of individuals and periodic pattern formation. Random motion has a homogenizing effect and then delays this clustering instability. When individuals from species differing in their random walk characteristics are allowed to compete together, the ones with a tendency to form narrower clusters get a competitive advantage over the others. Mean-field deterministic equations are analyzed and compared with the outcome of the individual-based simulations.
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