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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.

One of the common assumptions in previous spatial dynamics of cyclic competition is that, regardless of competing structure and strength among species, the spatial size of a network is considered as large as possible to avoid finite size effect for species biodiversity. In real ecosystems, however, species richness, which can be defined by spatial size and competition strength, can sensitively affect species coexistence as a competition among individuals becomes complicated. In this paper, we investigate the structural stability of coexistence of mobile species in three cyclic competition games due to network complexity in which imposes a size of a square lattice and competition strength among species. By exploiting the coexistence probability, our computations quantitatively reveal that the network complexity due to changes in the competition rate and lattice size can strongly affect the structural stability of coexistence in each model. In particular, intense intraspecific competition can yield the robust coexistence at small-sized lattices regardless of mobility, and strengthening interspecific competition simultaneously induces changes in critical mobility that hampers coexistence and in spatial size for stable coexistence. Qualitatively, we find that such structural stability of coexistence relates to the degree of stability of fixed points in deterministic systems. Our finding can be useful to gain insights into species coexistence on spatially extended systems with respect to network complexity.

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.