Regime shifts in bistable water-stressed ecosystems due to amplification of stochastic rainfall patterns

We develop a framework that casts the point water-vegetation dynamics under stochastic rainfall forcing as a continuous-time random walk (CTRW), which yields an evolution equation for the joint probability density function (PDF) of soil-moisture and biomass. We find regime shifts in the steady-state PDF as a consequence of changes in the rainfall structure, which flips the relative strengths of the system attractors, even for the same mean precipitation. Through an effective potential, we quantify the impact of rainfall variability on ecosystem resilience and conclude that amplified rainfall regimes reduce the resilience of water-stressed ecosystems, even if the mean annual precipitation remains constant.

Figure 1

Behavior of the deterministic system [Eqs. (1a) and (1b)]. The dynamical system is deterministic when the infiltration rate is constant, and the Gaussian noise terms are neglected. For parameter values that are typical of arid and semiarid environments, the system exhibits two alternative stable states. (a) Sample trajectories for $E=0.004,I=0.004$. Furthermore, we set $\beta =0.004,\gamma =0.02,\delta =0.004,\mu =0.008,k=0.2,b_0=0.4$, and $K_s=0.002$. The system is bistable, and we may arrive at either of the attractors depending on the initial condition. The region of initial values in phase space that converges to a given attractor is called the basin of attraction of that stable point. (b) Bifurcation diagrams using the recharge rate, $I$, as the bifurcation parameter. The two curves represent the biomass at equilibrium that corresponds to two evaporation rates, $E=0.004$ and $E=0.016$. For small infiltration rates, the system has two alternative stable states: a vegetated and a nonvegetated state (red dots).

Figure 2

Stochastic ecosystem trajectory for a power-law interstorm waiting time with exponent $a=0.6$ and characteristic time scale ${\tau }_c=10$ and average recharge depth $\alpha =0.04$ such that $\alpha /{\tau }_c=0.004$ is equal to the recharge rate for the deterministic model illustrated in Fig. 1. The remaining model parameters are the same as described in the caption of Fig. 1 with $E=0.004$. The trajectory starts in the vegetated basin and ends asymptotically in the nonvegetated basin.

Figure 3

Stochastic ecosystem trajectories for (a) frequent rain events with low amplitude, and (b) rare storm events with large amplitude. The model parameters are the same as in Fig. 1 with $E=0.004$ and a mean daily preciptation of $I=0.004$. The mean daily precipitation is the same for both cases, but the mean lag time between storms and mean precipitation depths are different: $\alpha =0.016,\lambda =1/4$ (a), and $\alpha =0.04,\lambda =1/10$ (b).

Figure 4

(a), (c) Sample realizations of the model Eqs. (5a) and (5b), coupling rainfall, soil moisture, and vegetation biomass with stochastic rainfall and Gaussian perturbations. The model parameters are the same as described in the caption of Fig. 1, with $E=0.004$ and the mean daily precipitation $I=0.004$. In these two scenarios, the mean annual precipitation is the same, but the mean lag time between storms and mean precipitation depths are different: $\alpha =0.012,\lambda =1/3$ (cases a and b), and $\alpha =0.032,\lambda =1/8$ (cases c and d). (b), (d) Steady-state joint probability density functions of soil moisture and biomass, calculated with our stochastic model, Eq. (27). In the case of frequent, small storm events, the PDF has a clear maximum around the vegetated stable state (b). In contrast, for infrequent, larger precipitation events, the probability of the nonvegetated system states increases (d). The cumulative probability to be in the vegetated basin is $0.92$ for scenario 1 (b) and $0.6$ in scenario 2 (d).

Figure 5

Empirical joint probability density functions from Monte Carlo simulation of the model Eqs. (5a) and (5b) for the same scenarios as described in the caption of Fig. 4. (b), (d) Steady-state joint probability density functions of soil moisture and biomass as in Fig. 4. (a), (c) Scatter plot of system states during the time evolution of soil moisture and biomass. Right panels, normalized histograms of system states.

Figure 6

Long-time biomass time series for the systems described in the caption of Fig. 4. The shift from the vegetated to the nonvegetated state leads to larger system fluctuations and intermittent behavior. The model parameters are the same as in Fig. 4. In these two scenarios, the mean annual precipitation is the same, but the mean lag time between storms and mean precipitation depths are different: $\alpha =0.012,\lambda =1/3$ (case a), and $\alpha =0.032,\lambda =1/8$ (case b).

Figure 7

Quantifying resilience using effective potential analysis. The effective potentials reflect the effective system dynamics for mean daily precipitation $\alpha \times \lambda =0.004$ and (left) frequency of storms $\lambda =1/2$ and (right) $\lambda =1/10$. Superposed are sample trajectories determined from Eq. (5) for the same model parameters as described in the caption of Fig. 4.

Figure 8

Depths of the vegetated well for $\alpha \times \lambda =0.004$ as a function of the mean interstorm waiting time $1/\lambda$. The model parameters are the same as described in the caption of Fig. 4. The depth of the well can be interpreted as the attracting strength of the vegetated state and is a measure of its resilience against random perturbations. In the case of shallow wells, which are characteristic of the larger events separated by larger intervening periods [large $\lambda$, Fig. 7] the system can easily fall into the basin of attraction of the nonvegetated state. These systems are also characterized by higher variability and intermittency [Fig. 6].