In the May–Leonard model of three cyclically competing species, we analyze the statistics of rare events in which all three species go extinct due to strong but rare fluctuations. These fluctuations are from the tails of the probability distribution of species concentrations. They render a coexistence of three populations unstable even if the coexistence is stable in the deterministic limit. We determine the mean time to extinction (MTE) by using a WKB-ansatz in the master equation that represents the stochastic description of this model. This way, the calculation is reduced to a problem of classical mechanics and amounts to solving a Hamilton–Jacobi equation with zero-energy Hamiltonian. We solve the corresponding Hamilton’s equations of motion in six-dimensional phase space numerically by using the Iterative Action Minimization Method. This allows to project on the optimal path to extinction, starting from a parameter choice where the three-species coexistence-fixed point undergoes a Hopf bifurcation and becomes stable. Specifically for our system of three species, extinction events can be triggered along various paths to extinction, differing in their intermediate steps. We compare our analytical predictions with results from Gillespie simulations for two-species extinctions, complemented by an analytical calculation of the MTE in which the remaining third species goes extinct. From Gillespie simulations we also analyze how the distributions of times to extinction (TE) change upon varying the bifurcation parameter. Our results shed some light on the sensitivity of the TE to system parameters. Even within the same model and the same dynamical regime, which allows a stable coexistence of species in the deterministic limit, the MTE depends on the distance from the bifurcation point in a way that contains the system size dependence in the exponent. It is challenging and worthwhile to quantify how rare the rare events of extinction are.

在三个周期性竞争物种的 May-Leonard 模型中，我们分析了三个物种由于强烈但罕见的波动而灭绝的罕见事件的统计数据。这些波动来自物种浓度概率分布的尾部。它们使三个种群的共存不稳定，即使共存在确定性极限内是稳定的。我们通过在代表该模型随机描述的主方程中使用 WKB-ansatz 来确定平均灭绝时间 (MTE)。这样，计算就简化为经典力学问题，相当于用零能哈密顿量求解哈密顿-雅可比方程。我们使用迭代动作最小化方法在六维相空间中数值求解相应的哈密顿运动方程。这允许从三个物种共存固定点经历 Hopf 分叉并变得稳定的参数选择开始，预测灭绝的最佳路径。特别是对于我们的三个物种系统，灭绝事件可以沿着不同的路径触发，它们的中间步骤不同。我们将我们的分析预测与 Gillespie 模拟两种物种灭绝的结果进行比较，并辅以 MTE 的分析计算，其中剩余的第三种物种灭绝。根据 Gillespie 模拟，我们还分析了在改变分岔参数时消光时间 (TE) 的分布如何变化。我们的结果揭示了 TE 对系统参数的敏感性。即使在相同的模型和相同的动态范围内，这允许物种在确定性极限内稳定共存，MTE 取决于距分叉点的距离，其方式包含指数中的系统大小依赖性。量化罕见的灭绝事件的罕见程度是具有挑战性的，也是值得的。