AI 中文总结
本研究针对同域成种Derrida–Higgs模型中的临界基因组长度预测问题，修正无约束矩方程并推导其显式闭式表达式，通过渐近分析得到不同主导机制下的标度关系，经模拟验证，揭示了瞬态谱系方差的核心作用并提供实用解析预测。
AI 中文摘要
在同域成种的Derrida–Higgs模型中，具有有限二元基因组的有性繁殖种群仅当两个个体的遗传重叠超过阈值$q_{\text{min}}$时才能交配。根据基因组长度$L$的不同，种群要么保持遗传连通性，要么分裂为生殖隔离的物种。因此一个核心问题是预测分裂开始时的临界基因组长度$L_c$。在有限$L$下，涨落会展宽重叠分布，并使种群中遗传距离较远的区域保持连通。此前提出的瞬态方差准则捕捉到了这一效应，但其计算需要对耦合矩方程进行数值迭代。本文首先通过移除一个此前隐含的假设修正了无约束矩方程，随后推导了$L_c$的显式闭式表达式。所得公式表明，当平均重叠达到$q_{\text{min}}$时，临界基因组长度由无约束平衡的确定性分离与重叠分布的瞬态谱系方差之间的竞争决定。该表达式支持系统的渐近分析：当确定性贡献占主导时，$L_c$本质上与种群规模$M$无关，且标度为$\u03bc^{-2}$；当瞬态谱系方差占主导时，$L_c$随$M^{3/2}$或$\u221aM/\u03bc$增长，具体取决于$M$与突变率$\u03bc$的共同变化方式。模拟结果支持所有预测的行为。本研究明确了瞬态谱系方差是将有限基因组涨落与生殖分裂起始联系起来的核心机制，并为临界基因组长度提供了实用的解析预测。
英文摘要
In the Derrida--Higgs model of sympatric speciation, a sexually reproducing population with finite binary genomes can mate only when the genetic overlap between two individuals exceeds a threshold $q_{\min}$. Depending on the genome length $L$, the population either remains genetically connected or fragments into reproductively isolated species. A central problem is therefore to predict the critical genome length $L_c$ at which fragmentation begins. At finite $L$, fluctuations broaden the overlap distribution and allow genetically distant regions of the population to remain connected. A previously proposed transient variance criterion captures this effect, but its evaluation requires numerical iteration of coupled moment equations. Here we first correct the unrestricted moment equations by removing a previously implicit assumption and then derive an explicit closed form expression for $L_c$. The resulting formula shows that the critical genome length is determined, at the time the mean overlap reaches $q_{\min}$, by the competition between deterministic separation from the unrestricted equilibrium and the transient genealogical variance of the overlap distribution. This expression permits a systematic asymptotic analysis. When the deterministic contribution dominates, $L_c$ becomes essentially independent of the population size $M$ and scales as $μ^{-2}$. When the transient genealogical variance dominates, $L_c$ grows as $M^{3/2}$ or as $\sqrt{M}/μ$, depending on how $M$ and the mutation rate $μ$ jointly vary. Simulations support all predicted behaviors. Our results identify transient genealogical variance as the principal mechanism linking finite genome fluctuations to the onset of reproductive fragmentation and provide a practical analytical prediction for the critical genome length.
Comments56 pages, 9 figures, + appendices
