Figure 2

This Figure shows the state evolution in four examples of nonlinear ecological systems with mutualistic interactions⁴⁷ of dimension \(n=3\). Panels (a and b) highlight the differences between the dynamics of stable and of unstable equilibria. In panel (a) the three state evolutions are at equilibrium until time \(t=10\), after which they are slightly perturbed. Because this equilibrium configuration is stable, the state returns to the equilibrium configuration after a small transient. In panel (b), instead, a small perturbation of the state generates a large deviation from the equilibrium, which drives the system towards a different configuration where one of the three species is absent. Panels (c and d) highlight the difference between a robust and a fragile system. Panel (c) shows the evolution of a robust system: the change of parameters modifies the equilibrium but not its stability properties, allowing the state to converge to a new equilibrium. Panel (d) shows the evolution of a fragile system, where the parameters variation modifies the equilibrium point and its stability properties, rendering the trajectories unstable and leading to the extinction of one specie.