AI 中文总结
本文针对表型识别不完善的情况，扩展合作的多维表型空间模型，推导选择促进合作丰度的广义阈值，发现区分度、表型空间维度及突变类型对合作演化有不同影响，揭示了不完善识别下合作演化的规律。
AI 中文摘要
表型相似性是合作演化的经典机制，多数现有模型采用个体仅与相同表型者合作的二元规则，该假设在生物学上存在局限性，因为识别与区分往往是渐进式而非全有或全无的。本文扩展了合作的多维表型空间模型，允许帮助概率随表型距离降低。在弱选择、策略与表型均存在突变的大种群中，我们推导了选择促进合作丰度的广义阈值，并采用新的拉普拉斯型变换表示；结果显示该阈值随区分度提升严格降低，说明在这类识别规则中，精确表型匹配是最有利的极限情况。我们还发现阈值随表型空间维度升高严格降低，意味着即使识别不完善，更高维的表型空间仍会促进合作。渐近分析进一步表明，低表型突变会强烈抑制合作，而足够高的表型突变会使阈值趋近于最小值；不过高策略突变会使合作更难维持。
英文摘要
Phenotypic similarity is a classical mechanism for the evolution of cooperation. Most existing models assume a binary rule in which individuals cooperate only with others of the same phenotype. This assumption is biologically restrictive, since recognition and discrimination are often gradual rather than all-or-nothing. In this paper, we extend the multidimensional phenotype-space model of cooperation by allowing the probability of helping to decline with phenotypic distance. In a large population under weak selection with mutation in both strategy and phenotype, we derive a generalized threshold for selection to favor the abundance of cooperation and express it using a new Laplace-type transform. We show that this threshold decreases strictly as discrimination becomes sharper, meaning that exact phenotype matching is the most favorable limit within this family of recognition rules. We also show that the threshold decreases strictly with phenotype-space dimension, meaning that higher-dimensional phenotype spaces promote cooperation even when recognition is imperfect. Our asymptotic analysis further shows that low phenotype mutations strongly inhibit cooperation, whereas sufficiently high phenotype mutations drive the threshold toward its minimal value. However, high strategy mutation makes cooperation harder to maintain.
