Fig. 4: Numerical investigation of the transient dynamics of a consumer-renewable resource system that contains a stable (spiral) equilibrium.

a Graph showing the isoclines of the system and the basin of attraction of the stable equilibrium point (green shaded region; numerically derived with c = 0.0088 year⁻¹, h = 0.0035 individual⁻¹ year^-1, and other parameters as in Table 1). The relatively small size of this basin of attraction suggests that any given resource availability strongly constrains the population range that will develop towards the stable equilibrium, and vice versa. b Transient dynamics show that collapsing systems may undergo a relatively long phase of apparent stability, before a relatively rapid collapse occurs. In contrast, initial conditions that are within the basin of attraction will develop towards a stable equilibrium point. c Example showing how an immigration event (adding 10% to the population at t = 200 years after initialization) can move the system into the sustainable equilibrium’s basin of attraction (green shaded region), and save the system from collapsing on the longer term. However, when the same perturbation is applied at a different moment in time (t = 2000 years) it accelerates the decline of consumers and resources. In panels c and d, the gray dotted line shows the default simulation from panel b, with initial population size P(t = 0) = 30,000 individuals. d Example showing how a resource culling event (removing 10% of the resource at t = 200 years after initialization) can move the system into the sustainable equilibrium’s basin of attraction (green shaded region) save the system from collapsing on the longer term. However, when the same perturbation is applied at a different moment in time (t = 2000 years) it accelerates the decline of consumers and resources.