1. In ecology, one of the most fundamental questions relates to the persistence of populations, or conversely to the probability of their extinction. Deriving extinction thresholds and characterizing other critical phenomena in spatial and stochastic models is highly challenging, with few mathematically rigorous results being available for discrete‐space models such as the contact process. For continuous‐space models of interacting agents, to our knowledge no analytical results are available concerning critical phenomena, even if continuous‐space models can arguably be considered to be more natural descriptions of many ecological systems than lattice‐based models.
2. Here we present both mathematical and simulation‐based methods for deriving extinction thresholds and other critical phenomena in a broad class of agent‐based models called spatiotemporal point processes. The mathematical methods are based on a perturbation expansion around the so‐called mean‐field model, which is obtained at the limit of large‐scale interactions. The simulation methods are based on examining how the mean time to extinction scales with the domain size used in the simulation. By utilizing a constrained Gaussian process fitted to the simulated data, the critical parameter value can be identified by asking when the scaling between logarithms of the time to extinction and the domain size switches from sublinear to superlinear.
3. As a case study, we derive the extinction threshold for the spatial and stochastic logistic model. The mathematical technique yields rigorous approximation of the extinction threshold at the limit of long‐ranged interactions. The asymptotic validity of the approximation is illustrated by comparing it to simulation experiments. In particular, we show that species persistence is facilitated by either short or long spatial scale of the competition kernel, whereas an intermediate scale makes the species vulnerable to extinction.
4. Both the mathematical and simulation methods developed here are of very general nature, and thus we expect them to be valuable for predicting many kinds of critical phenomena in continuous‐space stochastic models of interacting agents, and thus to be of broad interest for research in theoretical ecology and evolutionary biology.

中文翻译：

---

用于推导相互作用因子的空间模型和随机模型中的灭绝阈值的数学和模拟方法

1. 在生态学中，最基本的问题之一与人口的持久性有关，或者与人口灭绝的可能性有关。在空间模型和随机模型中得出灭绝阈值并表征其他关键现象非常具有挑战性，对于离散空间模型（如接触过程），几乎没有严格的数学结果。就交互作用剂的连续空间模型而言，据我们所知，即使可以认为连续空间模型比基于格的模型更能自然地描述许多生态系统，但没有关于临界现象的分析结果。
2. 在这里，我们介绍了基于数学和模拟的方法，用于在称为时空点过程的广泛基于代理的模型中推导灭绝阈值和其他关键现象。数学方法基于围绕均值场模型的扰动展开，这是在大规模相互作用的极限下获得的。模拟方法基于检查平均灭绝时间与模拟中使用的域大小之间的关系。通过利用适合模拟数据的约束高斯过程，可以通过询问何时到灭绝时间的对数和域大小从次线性变为超线性来确定关键参数值。
3. 作为案例研究，我们推导了空间和随机逻辑模型的灭绝阈值。这种数学技术可以在远程相互作用的极限处严格逼近灭绝阈值。通过将其与仿真实验进行比较来说明近似的渐近有效性。特别是，我们表明，竞争内核的短期或长期空间尺度促进了物种的持久性，而中间尺度使物种容易灭绝。
4. 这里开发的数学方法和模拟方法都具有非常普遍的性质，因此我们希望它们对于预测相互作用因子的连续空间随机模型中的多种临界现象具有重要的价值，因此对于理论研究具有广泛的兴趣。生态学和进化生物学。