Fig. 4. Phase representations of the socio-ecological system for strategies in the viable and collapse regions of the strategy spectrum.

The plots correspond to particular landscape planes of the three-dimensional phase space. The chosen planes are the ones containing the system’s viable equilibrium, hence determined by setting the population at its viable equilibrium value. The dotted grey lines are projections of the null planes for the population, the natural land and the agricultural land on the chosen landscape plane. The black solid lines are projections of simulated trajectories. The vector field depicted with blue arrows indicates the landscape’s direction of change for each landscape composition given a population at equilibrium. On the top: before (a) and after (b) the first transition to collapse. A subcritical Hopf bifurcation causes the stability loss of the viable equilibrium explaining the transition to collapse. On the bottom: emergence of a stable limit cycle (c) from a stable focus node (d) after a supercritical Hopf bifurcation. Parameter values: r = 1.0, d = 1.0, k = 4.5, e = 1.0, k ₀ = 0.5, q = 1.0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)