Normal-form analysis of the cusp-transcritical interaction: applications in population dynamics

Bistability, the presence of alternative stable states, is an important feature of population models as it indicates that long-term predictions are dependent on the current population density. Two distinct kinds of bistability re-occur in population modelling studies, Allee Bistability and Positive Bistability. In this article, we show that a novel codimension-3 bifurcation, the cusp-transcritical interaction, can act as an organising centre for ordinary differential equations that exhibit both Allee Bistability and Positive Bistability. We first show how a normal form for cusp-transcritical interactions emerges from the unfolding of a particular one-dimensional degeneracy. We then illustrate the ecological relevance of the cusp-transcritical interaction. Finally, we provide a comprehensive example of normal-form analysis of an existing population model that demonstrates the occurrence of the codimension-3 bifurcation. We note that Allee Bistability and Positive Bistability may manifest unexpectedly in complex, ecological models, and therefore, this bifurcation-focused approach can provide valuable insight into the behaviour of newly developed ecosystem models.