Three-dimensional bifurcation analysis of a predator-prey model with uncertain formulation

An issue in mathematical modeling is that multiple models describe the same data based on different but all acceptable theoretical assumptions. To take this uncertainty in model construction into account, the classical parameter sensitivity analysis needs to be extended to consider alternative choices in model formulation. This is particularly important in population biology, where a single function often summarizes mechanisms from the physiological to the population level. Predictions based on population models were recently shown to be sometimes highly sensitive to the function that models predation. This so-called structural sensitivity extends the classical structural stability. It arises from differences between the bifurcation diagrams based on alternative model formulations. Here, we ask through an example of a predator-prey system that exhibits complex bifurcations (involving limit cycles and homoclinic orbits) how the qualitative changes in bifurcations occur during a small continuous switch between two acceptable model formulations. In this model, fixed parameter values are estimated from empirical data, free parameters allow one to explore a wide range of environmental conditions and species, and a remaining parameter makes a switch between two predation formulations. Thus, the change in model formulation becomes a parameter sensitivity problem that is studied by bifurcation analysis. A three-dimensional analysis revealed that most changes that occur with the change in formulation are related to a codimension-three degenerated Bogdanov-Takens bifurcation. Its canonical unfolding is derived, and its analysis highlights the differences in the phase portraits predicted with the two model formulations. Differences that are not related to this bifurcation are also discussed. The discussion ends with some perspectives toward the quantification of prediction uncertainty due to uncertainty in model structure in ecology, and possibly in other fields involving the modeling of complex systems.