Agent-based models are used to study complex phenomena in many fields of science. While simulating agent-based models is often straightforward, predicting their behaviour mathematically has remained a key challenge. Recently developed mathematical methods allow the prediction of the emerging spatial patterns for a general class of agent-based models, whereas the prediction of spatio-temporal pattern has been thus far achieved only for special cases. We present a general and mathematically rigorous methodology that allows deriving the spatio-temporal correlation structure for a general class of individual-based models. To do so, we define an auxiliary model, in which each agent type of the primary model expands to three types, called the original, the past and the new agents. In this way, the auxiliary model keeps track of both the initial and current state of the primary model, and hence the spatio-temporal correlations of the primary model can be derived from the spatial correlations of the auxiliary model. We illustrate the agreement between analytical predictions and agent-based simulations using two example models from theoretical ecology. In particular, we show that the methodology is able to correctly predict the dynamical behaviour of a host–parasite model that shows spatially localized oscillations.

中文翻译：

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一种预测基于代理模型的时空相关性的通用数学方法

基于代理的模型用于研究许多科学领域中的复杂现象。虽然模拟基于代理的模型通常很简单，但从数学上预测它们的行为仍然是一个关键挑战。最近开发的数学方法允许为一般类别的基于代理的模型预测新出现的空间模式，而迄今为止，时空模式的预测仅适用于特殊情况。我们提出了一种通用且数学上严格的方法，该方法允许为一般类别的基于个体的模型推导出时空相关结构。为此，我们定义了一个辅助模型，其中主要模型的每个代理类型扩展为三种类型，称为原始代理、过去代理和新代理。这样，辅助模型跟踪主模型的初始状态和当前状态，因此主模型的时空相关性可以从辅助模型的空间相关性中推导出来。我们使用来自理论生态学的两个示例模型来说明分析预测和基于代理的模拟之间的一致性。特别是，我们表明该方法能够正确预测显示空间局部振荡的宿主 - 寄生虫模型的动态行为。