Analytical methods for ecosystem resilience: A hydrological investigation

In recent years a number of papers have quantitatively explored multiple steady states and resilience within a wide range of hydrological systems. Many have identified multiple steady states by conducting simulations from different initial state variables and a few have used the more advanced technique of equilibrium or limit cycle continuation analysis to quantify how the number of steady states may change with a single model parameter. However, like resilience investigations into other natural systems, these studies often omit explanation of these fundamental resilience science techniques; rely on complex numerical methods rather than analytical methods; and overlook use of more advanced techniques from nonlinear systems mathematics. In the interests of wider adoption of advanced resilience techniques within hydrology, and advancing resilience science more broadly, this paper details fundamental methods for quantitative resilience investigations. Using a simple model of a spatially lumped unconfined aquifer, one and two parameter continuation analysis was undertaken algebraically. The shape of each steady state attractor basin was then quantified using Lyapunov stability curves derived at a range of precipitation rates, but was found to be inconsistent with the resilience behavior demonstrated by stochastic simulations. Most notably, and contrary to standard resilience concepts, the switching between steady states from wet or dry periods (and vice versa) did not occur by crossing of the threshold between the steady states. It occurred by exceedance of the two steady-state domain, producing a counterclockwise hysteresis loop. Additionally, temporary steady states were identified that could not have been detected using equilibrium continuation with a constant forcing rate. By combining these findings with the Lyapunov stability curves, new measures of resilience were developed for endogenous disturbances to the model and for the recovery from disturbances exogenous to the model.