How biodiversity is maintained is one of the most important problems in biology. It, seemingly, challenges the Darwinian evolutionary theory which assumes that only the fittest species survive. Most previous researches try to solve the puzzle based on deterministic dynamics. However, it is not clear how demographic stochasity plays a role in biodiversity, even though randomness cannot be ignored in real biological systems. In contrast with deterministic methods, we propose that biodiversity is measured by the time that the species coexist. Modeled by the cyclic Rock-Paper-Scissors game, we concentrate on the Fermi process, where randomness is captured by the selection intensity. We find that there is a non-monotonic relationship between the selection intensity and the time that the species coexist. Intrinsic differences between stochastic dynamics and deterministic dynamics are shown to be present, even if the selection is in its strong limit which mirrors the deterministic dynamics. Furthermore, we prove that these results are robust for general stochastic imitation processes beyond the Fermi process. Our work highlights the importance of transient dynamics on biodiversity, which is absent in the deterministic counterpart, and opens an avenue to calculate biodiversity under stochastic evolutionary process.

如何维持生物多样性是生物学中最重要的问题之一。它似乎挑战了达尔文进化论，该理论认为只有最适合的物种才能生存。大多数先前的研究试图解决基于确定性动力学的难题。然而，尚不清楚人口随机性如何在生物多样性中发挥作用，尽管随机性在真实的生物系统中不容忽视。与确定性方法相反，我们建议通过物种共存的时间来衡量生物多样性。通过循环岩石剪刀布游戏建模，我们专注于费米过程，其中随机性由选择强度捕获。我们发现选择强度与物种共存时间之间存在非单调关系。随机动力学和确定性动力学之间的内在差异被证明是存在的，即使选择处于反映确定性动力学的强极限。此外，我们证明了这些结果对于费米过程之外的一般随机模仿过程是稳健的。我们的工作强调了瞬态动力学对生物多样性的重要性，这在确定性对应物中是不存在的，并开辟了一条在随机进化过程下计算生物多样性的途径。