Stochastic Evolution Analysis Of Asymmetric Game Models

Game theory is a classical mathematical theory and method to discuss the problem of population struggle or competition.It is an important part of modern mathematics and operational research.With the development of theory,evolutionary game theory is formed by combining classical game theory with evolutionary thought in biological theory.Because this method can better describe the population evolution dynamics,evolutionary game theory has become one of the most important theories in modern evolutionary biology,and has gradually been applied to social,economic and other fields.However,in previous studies,evolutionary game analysis is mostly based on the assumption that the environmental conditions are constant and the participants interact symmetrically.These two assumptions cannot well simulate the real environment.First of all,in real life,the environment is always uncertain,and random environmental disturbance will significantly affect the behavior of the game system,and introducing randomness into the research will probably better explain the real system.Secondly,the asymmetric phenomenon in life is universal,but the research on the evolution of asymmetric game models has not received enough attention,largely because its mixed strategy Nash equilibrium point is not evolutionarily stable,and an important reason for this situation is that the game model ignores the influence of randomness.This thesis studies the evolution dynamics of asymmetric random game model from theoretical analysis and numerical simulation.In theory,based on stochastic differential equation,the stability characteristics of stochastic asymmetric game model in continuous time are discussed.In this part,the asymmetric differential equation is obtained according to the equivalent transformation relationship between the game model replication dynamic equation and the population competition model to describe the system motion trajectory;Brown motion,a random disturbance term,is introduced to represent the influence of environmental randomness;Finally,the appropriate Lyapunov function is constructed to analyze the dynamic characteristics of the system.This thesis assumes that the environmental random disturbance is reflected in the species growth rate of the dynamic system.Through mathematical derivation,the dynamic characteristics of the global positive solution of the asymmetric game model with random disturbance,such as the existence and uniqueness,stochastic persistence,and stochastic ultimate boundedness,are discussed,and the stochastic stability conditions at the boundary equilibrium point of the system related to random parameters are obtained.In the part of numerical simulation,in order to explore the impact of random environmental noise on asymmetric game models,this thesis takes four classic game models: gender war game,asymmetric hawk and pigeon game,smart pig game,and market entry obstruction game as examples,respectively constructs deterministic system and stochastic differential system,and uses mathematical software Matlab to carry out numerical simulation and draw the system motion trajectory in combination with the results of stability theory analysis above.Through comparative analysis,it is found that after adding random disturbance to the differential system corresponding to the deterministic game model,the unstable equilibrium point of the deterministic system will reach random evolution stability under the influence of the random environment fluctuation,such as gender war game;It may also make the system appear newThe stable point of the game,such as asymmetric eagle and pigeon game and intelligent pig game;It may also make the stable point of the original deterministic system become unstable,such as market entry blocking game.The main reason for the above diversified evolution results is the existence and intensity change of random interference.This thesis studies the evolution dynamics of asymmetric random game model from the aspects of mathematical theory derivation and numerical simulation,explores the influence of random environment disturbance on the stability of asymmetric game model,and draws the conclusion that different random disturbance intensities will make the system reach different stable states.The research results enrich the theoretical research of game models from the perspective of randomness and asymmetry,and have some implications for the application research in related fields.