Bautin bifurcations in a forest-grassland ecosystem with human-environment interactions

Ecosystems may be characterized by a complex dynamical behaviour where external disturbances and/or internal perturbations may trigger sudden/irreversible changes, called catastrophic shifts. Simple mathematical models in the form of ordinary and/or partial differential equations have been proposed to approximate in a qualitatively manner the observed complex phenomena, where catastrophic shifts are determined by bifurcation points. In this work, we show that in ecosystems, gradual/smooth changes may be transformed in sudden/catastrophic shifts as a consequence of codimension-2 bifurcations. We stress the importance of using the full arsenal of numerical bifurcation theory to systematically identify and characterize criticalities in ecological models in the 2D parameter space. For our demonstrations, we revisit the analysis of a simple model of a forest-grassland mosaic ecosystem constructing the 2D bifurcation diagram with respect to the impact of human influence and that of natural causes. Our numerical analysis reveals that this simple model is able to approximate both abrupt (catastrophic) and smooth transitions as the system undergoes Bautin bifurcations.