非退化复制器系统的进化数学模型

中文摘要 AI 辅助

我们提出并分析了非退化（永久）复制器系统的进化适应数学模型，其中适应度景观矩阵在慢时间尺度——进化时间上演变，而物种动态在快时间尺度上展开。在由蒂霍诺夫定理证明的双时间尺度分离下，适应问题简化为在适应度景观矩阵的凸容许集上最大化稳态平均适应度。我们推导了适应度变化公式并建立了适应度最大值的充要条件，表明优化在每一步都简化为线性规划问题。该算法应用于四个典型复制器系统：超循环、双超循环、蚁丘系统和RNA分子网络。在所有情况下，进化过程遵循普遍的三相模式：适应度增长且无平衡转移的初始阶段，在此期间纯粹利他复制让位于混合利他 - 自私行为；优势物种出现的第二阶段；以及类似于准物种模型中误差灾难阈值的稳定阶段。一个关键结果是所有进化系统都获得了对寄生物种的抗性。我们进一步证明，没有非退化约束时，该过程会导致物种相继灭绝，通过降维可证明适应度增加的谱下界。

英文摘要

We propose and analyse a mathematical model of evolutionary adaptation for non-degenerate (permanent) replicator systems, in which the fitness landscape matrix evolves on a slow timescale -- the evolutionary time -- while the species dynamics unfold on a fast timescale. Under a two-timescale separation justified by Tikhonov's theorem, the adaptation problem reduces to maximising the mean fitness at steady state over a convex admissible set of fitness landscape matrices. We derive a fitness variation formula and establish necessary and sufficient conditions for a fitness maximum, showing that the optimisation reduces at each step to a linear programming problem. The algorithm is applied to four canonical replicator systems: the hypercycle, the bi-hypercycle, the anthill system, and the RNA molecule network. In all cases the evolutionary process follows a universal three-phase pattern: an initial phase of fitness growth without equilibrium shift, during which purely altruistic replication gives way to mixed altruistic-selfish behaviour; a second phase of dominant species emergence; and a stabilisation phase analogous to the error catastrophe threshold in quasispecies models. A key consequence is that all evolved systems acquire resistance to parasitic species. We further prove that without non-degeneracy constraints the process leads to sequential species annihilation, with a provable spectral lower bound on fitness increase by dimension reduction.

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