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In a diverse population, where many species are present, competitors can fight for surviving at individual and collective levels. In particular, species, which would beat each other individually, may form a specific alliance that ensures them stable coexistence against the invasion of an external species. Our principal goal is to identify those general features of a formation which determine its vitality. Therefore, we here study a traditional Lotka-Volterra model of eight-species where two four-species cycles can fight for space. Beside these formations, there are other solutions which may emerge when invasion rates are varied. The complete range of parameters is explored and we find that in most of the cases those alliances prevail which are formed by equally strong members. Interestingly, there are regions where the symmetry is broken and the system is dominated by a solution formed by seven species. Our work also highlights that serious finite-size effects may emerge which prevent observing the valid solution in a small system., Comment: 10 double-column pages, 11 figures

Multistability is common feature resulting in nonlinear dynamical systems, and its characteristic can be generally depicted by investigating basin structures of initial conditions for give parameter settings. In this paper, we explore the formation of extinction states according to the change of strength of competition levels in the toy model for three species. Through the linear stability analysis, we find that the extinction state can be stable which is persistent. For specific conditions between intensities of two different competitions, we also found that the extinction state can be either bistable or tristable. In each case, the final state of the system can be characterized sensitively depending on initial conditions. To validate our results, we investigate basin structures of parameters for interspecific competition associated to a strength of intraspecific competition. In addition, we found that coexistence becomes robust as intraspecific competition is intensified relatively to the interspecific competition level. We hope our results can be a chance to suggest the emergence of the multistability according to complex competition structures on systems of many populations.

Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities.

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.