Stochastic analysis and optimal control of a donation game system with non-uniform interaction rates and Gram–Schmidt orthogonalization procedure

• Published in: Computational & applied mathematics Vol. 42; no. 5
• Main Authors: Yuan, Hairui, Meng, Xinzhu, Alzahrani, Abdullah Khames, Zhang, Tonghua
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2023
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; 60H10; 91A22; Stochastic replicator equation; 34A34; Nonlinear fitness function; Non-uniform interaction rates; Saddle-node bifurcation; Evolutionary game theory
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-023-02369-9

This paper investigates an evolutionary donation game with non-uniform interaction rates in well-mixed populations. Further, we consider that the costs and benefits of the game are subject to stochastic disturbances and explore the stochastic replicator dynamics. Firstly, the system has two interior equilibrium points, one of which is stable. In other words, cooperators and defectors coexist when the interaction rates satisfy certain conditions. And the number of cooperators may exceed the number of defectors, which changes the final steady state of the traditional donation game system. Secondly, according to It o ^ ′ s formula and Gram-Schmidt orthogonalization procedure, we obtain the stochastic replicator equation of the game. Since the different interaction rates between players lead to the emergence of the interior equilibria of the system, we give the conditions for the stochastic stability of equilibria. The relationship between interaction rate and disturbance value is shown, and we explore the optimal path from the area of stochastic stability to the area of stochastic instability. In short, the donation game system has two stable states, and we can control the cooperators in the population by the non-uniform interaction rates. Finally, we conduct numerical simulations and find that it is consistent with the theoretical results described previously.