Robust coexistence with alternative competition strategy in the spatial cyclic game of five species

Emerging solutions from the battle of defensive alliances

Competing strategies in an evolutionary game model, or species in a biosystem, can easily form a larger unit which protects them from the invasion of an external actor. Such a defensive alliance may have two, three, four or even more members. But how effective can be such formation against an alternative group composed by other competitors? To address this question we study a minimal model where a two-member and a four-member alliances fight in a symmetric and balanced way. By presenting representative phase diagrams, we systematically explore the whole parameter range which characterizes the inner dynamics of the alliances and the intensity of their interactions. The group formed by a pair, who can exchange their neighboring positions, prevail in the majority of the parameter region. The rival quartet can only win if their inner cyclic invasion rate is significant while the mixing rate of the pair is extremely low. At specific parameter values, when neither of the alliances is strong enough, new four-member solutions emerge where a rock-paper-scissors-like trio is extended by the other member of the pair. These new solutions coexist hence all six competitors can survive. The evolutionary process is accompanied by serious finite-size effects which can be mitigated by appropriately chosen prepared initial states.

Correlation between the formation of new competing group and spatial scale for biodiversity in the evolutionary dynamics of cyclic competition

Securing space for species breeding is important in the evolution and maintenance of life in ecological sciences, and an increase in the number of competing species may cause frequent competition and conflict among the population in securing such spaces in a given area. In particular, for cyclically competing species, which can be described by the metaphor of rock-paper-scissors game, most of the previous works in microscopic frameworks have been studied with the initially given three species without any formation of additional competing species, and the phase transition of biodiversity via mobility from coexistence to extinction has never been changed by a change of spatial scale. In this regard, we investigate the relationship between spatial scales and species coexistence in the spatial cyclic game by considering the emergence of a new competing group by mutation. For different spatial scales, our computations reveal that coexistence can be more sensitive to spatial scales and may require larger spaces for frequencies of interactions. By exploiting the calculation of the coexistence probability from Monte-Carlo simulations, we obtain that certain interaction ranges for coexistence can be affected by both spatial scales and mobility, and spatial patterns for coexistence can appear in different ways. Since the issue of spatial scale is important for species survival as competing populations increase, we expect our results to have broad applications in the fields of social and ecological sciences.

Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link

The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.

Environment driven oscillation in an off-lattice May-Leonard model

Cyclic dominance of competing species is an intensively used working hypothesis to explain biodiversity in certain living systems, where the evolutionary selection principle would dictate a single victor otherwise. Technically the May–Leonard models offer a mathematical framework to describe the mentioned non-transitive interaction of competing species when individual movement is also considered in a spatial system. Emerging rotating spirals composed by the competing species are frequently observed character of the resulting patterns. But how do these spiraling patterns change when we vary the external environment which affects the general vitality of individuals? Motivated by this question we suggest an off-lattice version of the tradition May–Leonard model which allows us to change the actual state of the environment gradually. This can be done by introducing a local carrying capacity parameter which value can be varied gently in an off-lattice environment. Our results support a previous analysis obtained in a more intricate metapopulation model and we show that the well-known rotating spirals become evident in a benign environment when the general density of the population is high. The accompanying time-dependent oscillation of competing species can also be detected where the amplitude and the frequency show a scaling law of the parameter that characterizes the state of the environment. These observations highlight that the assumed non-transitive interaction alone is insufficient condition to maintain biodiversity safely, but the actual state of the environment, which characterizes the general living conditions, also plays a decisive role on the evolution of related systems.