Abstract The triangle conceptual model is a construct that is foundational across several fields of the natural sciences including the study of diseases, invasive species, and fire. The invasion triangle incorporates the complex ecological and evolutionary interactions between the qualities of the abiotic environment, the invader, and the biotic interactions that describes or predicts the impacts of the invasive species. Although the triangle concept is widely used among fields, to date there has not been an analytical implementation of the model. Current modelling in invasion biology often only considers the effects of one or two factors on the outcomes of species introductions. A mathematical implementation of the triangle model will allow a more comprehensive consideration of the various ecological factors. Here, we provide the first mathematical theorem for an interpretation of the invasion triangle that allows for the consideration of time. This new analytical development of the triangle is flexible, and can be used to model the spatial and temporal population dynamics observed in invasions. We also describe the conditions under which invasion is maintained when factors change with opposing effects. In this interpretation, the lower limits for invasion are explicitly defined and each component can move independently. The complexity of the interactions between factors contributing to invasions is integrated into the single model, such as those suggested by major invasion hypotheses. We briefly describe how the theorem can be applied to account for various phenomena in range dynamics using rapid range expansion and the time lag in invasions as examples. Future work can explicitly define the interdependence among components to suit more specific questions.

中文翻译：

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入侵三角形定理及其在入侵生物学中的适用性

摘要 三角形概念模型是一种跨自然科学多个领域的基础结构，包括疾病、入侵物种和火灾的研究。入侵三角形包含了非生物环境质量、入侵者和描述或预测入侵物种影响的生物相互作用之间复杂的生态和进化相互作用。尽管三角形概念在各个领域中被广泛使用，但迄今为止还没有对该模型进行分析实现。当前入侵生物学的建模通常只考虑一两个因素对物种引入结果的影响。三角形模型的数学实现将允许更全面地考虑各种生态因素。这里，我们提供了第一个数学定理来解释入侵三角形，它允许考虑时间。这种三角形的新分析发展是灵活的，可用于模拟入侵中观察到的空间和时间种群动态。我们还描述了当因素随着相反的影响而变化时维持入侵的条件。在这种解释中，入侵的下限被明确定义，每个组件可以独立移动。导致入侵的因素之间相互作用的复杂性被整合到单一模型中，例如主要入侵假设所建议的模型。我们以快速范围扩展和入侵中的时间滞后为例，简要描述如何应用该定理来解释范围动态中的各种现象。