A theorem for the invasion triangle and its applicability for invasion biology

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.