  We perform both analytical and numerical bifurcation analysis of a
forest-grassland ecosystem model coupled with human interaction. The model
consists of two nonlinear ordinary differential equations incorporating the
human perception of forest/grassland value. The system displays multiple steady
states corresponding to different forest densities as well as regimes
characterized by both stable and unstable limit cycles. We derive analytically
the conditions with respect to the model parameters that give rise to various
types of codimension-one criticalities such as transcritical, saddle-node, and
Andronov-Hopf bifurcations and codimension-two criticalities such as cusp and
Bogdanov-Takens bifurcations. We also perform a numerical continuation of the
branches of limit cycles. By doing so, we reveal turning points of limit cycles
marking the appearance/disappearance of sustained oscillations. These
far-from-equilibrium criticalities that cannot be detected analytically give
rise to the abrupt loss of the sustained oscillations, thus leading to another
mechanism of catastrophic shifts

Numerical Analysis;Numerical Analysis;Dynamical Systems;Pattern Formation and Solitons