Game Dynamics Of Spatial Rock-Paper-Scissors With Periodic Environmental Stress

Earth’s ecosystem is made up of many competing species which are constantly striving to coexist since the species have to face limited resources.The mechanism of the coexistence of multiple competing species has been the subject of long-term research in ecology,and Voltera proposed the predator-prey model in 1925.In 1975,May and Leonard proposed the symmetric May-Leonard model based on the model of perfect competition among the three populations.The form of competition among these three species is gradually being referred to as the rock-paper-scissors（RPS）model.There have been many studies on the coexistence factors in this model over the past century.The periodic changes in the environment,such as sunlight,CO₂concentration,precipitation,seasonal climate,and even human activity,affect the growth and other behaviours of organisms,which in turn affects the mortality or reproduction rate of species.The environment can greatly affect the coexistence of multiple species by modulating the mortality or/and birth rate of individuals,but we know little about the mechanism,so we investigate the periodic environmental stress factors in the population model of the rock-paper-scissors game in this paper Impact.We use the May-Leonard model to study the spatial evolutionary dynamics of inter-species cyclic competition,where the periodic changes in the external envi-ronment can affect the death rate of one species.Through extensive Monte Carlo simulations,we found that the evolutionary dynamics of inter-species cyclic compe-tition have specific responses to external environmental periodic changes of different frequencies.Specifically,we calculated the signal-to-noise ratio of species density fluctuations under periodic environmental stresses with different frequencies and found that the best response occurs when the frequency of environmental change is close to the natural frequency of species density,which indicates the phenomenon of periodic death rate and stochastic resonance caused by inherent fluctuations.When this resonance occurs,there is a significant increase in the probability of species ex-tinction,and the oscillation amplitude of their population density is also significantly enhanced during the coexistence of three species.This leads to the destruction of species diversity when stochastic resonance occurs.In order to study this stochastic resonance phenomenon through analytical analysis,we hope to propose an analyti-cal approximation method to describe the fluctuation of species density during the evolution process.By analysing the distribution of individual species in space during the evolution process and comparing the proportion of each biological behaviour in the simulation and analysis,we use the non-Markov process to describe the preda-tion behaviour that occurs on the boundary of the population cluster.At the same time,the Markov process is used to describe the reproduction and death behaviours in the whole space.We use these processes to describe the behaviours of the pop-ulation in the simulation.Furthermore,the mean field approximation equation of the implied spatial structure is established.The results in this paper indicate that the intensity and frequency of peri-odic environmental changes may play a crucial role in determining the stability of multiple species coexistence.The phenomenon that periodic environmental stress enhances the oscillation amplitude of population density also corresponds to the fluc-tuation of microbial communities and environmental factors in soil microorganisms.From this phenomenon,we can assume that the periodic fluctuations of popula-tion density in the May-Leonard model is an intrinsic property of the model.The mean field approximation equation of the implied spatial structure by introducing non-Markovian models provides a clue for explaining various new phenomena in the simulations through analytical solution methods.