Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, Budapest, 1525, Hungary.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China.
Sci Rep. 2023 May 25;13(1):8472. doi: 10.1038/s41598-023-35746-9.
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.
在进化博弈模型中的竞争策略，或者生物系统中的物种，可以很容易地形成一个更大的单元，保护它们免受外部因素的入侵。这样的防御联盟可能有两个、三个、四个甚至更多的成员。但是，这样的形成对于由其他竞争者组成的替代群体能有多大的效果呢？为了回答这个问题，我们研究了一个最小模型，其中两个成员和四个成员的联盟以对称和平衡的方式进行战斗。通过呈现代表性的相图，我们系统地探索了联盟内部动态和它们相互作用强度的整个参数范围。由一对可以交换它们相邻位置的成员组成的群体，在大多数参数区域中占据优势。如果对手四重奏的内部循环入侵率显著，而对的混合率极低，则只有四重奏才能获胜。在特定的参数值下，当没有一个联盟足够强大时，会出现新的四重奏解决方案，其中一个石头剪刀布式的三人组由对的另一个成员扩展。这些新的解决方案共存，因此所有六个竞争者都可以生存。进化过程伴随着严重的有限尺寸效应，可以通过适当选择的准备初始状态来减轻。
