Analytical and numerical bifurcation analysis of a forest ecosystem model with human interaction

Abstract

We perform both analytical and numerical bifurcation analysis of an alternating forest and grassland ecosystem model coupled with human interaction. The model consists of two nonlinear ordinary differential equations incorporating the human perception of the value of the forest. 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 at which homoclinic orbits occur. 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. Such critical points that cannot be detected analytically give rise to the abrupt loss of the sustained oscillations, thus leading to another mechanism of catastrophic shifts.

Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

“…For example, in several studies it has been shown, that Turing instabilities may experience secondary bifurcations leading to far-from-equilibrium oscillating solutions [22,23], spatio-temporal chaos [24,25] and symmetry-breaking bifurcations [26]. In such regimes, nonlinearities play a key role not only in stabilizing a pattern, but also in producing unsuspected bifurcations lined with catastrophic transitions [27,23,28]. Thus, to systematically investigate such phenomena systematically, the exploitation of the full arsenal of numerical bifurcation theory is of out-most importance [29,30,27,28]).…”

“…In such regimes, nonlinearities play a key role not only in stabilizing a pattern, but also in producing unsuspected bifurcations lined with catastrophic transitions [27,23,28]. Thus, to systematically investigate such phenomena systematically, the exploitation of the full arsenal of numerical bifurcation theory is of out-most importance [29,30,27,28]).…”