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Stochastic Modeling of Spatially Extended Ecosystems and Ecological and Climate Data Analysis

Final Report Summary - SMSEE (Stochastic Modeling of Spatially Extended Ecosystems and Ecological and Climate Data Analysis)

This project’s research involved two different fields: climate dynamics and ecological modeling. Within the climate dynamics research, we focused on two different aspects, specifically, geophysical fluid dynamics and the improvement of climate predictions using sequential learning algorithms. Our ecological modeling focused on the dynamics of water-limited vegetation.
We studied the statistics of wind-driven open ocean currents. Using the Ekman layer model for the integrated currents, we investigated, analytically and numerically, the relationship between the wind-stress distribution and its temporal correlations and the statistics of the open ocean currents. We found that long-range temporally correlated winds result in currents whose statistics is proportional to the wind-stress statistics. On the other hand, short-range correlated winds lead to Gaussian distributions of the current components, regardless of the stationary distribution of the winds and, therefore, to a Rayleigh distribution of the current amplitude, if the wind stress is isotropic. We found that the second moment of the current speed exhibits a maximum as a function of the correlation time of the wind stress for a non-zero Coriolis parameter. The results were validated using an oceanic general circulation model. This study was later extended to show that the seminal model for the effect of winds on surface ocean currents, which was proposed by Ekman more than a century ago to demonstrate the non-trivial effect of the earth’s rotation on surface ocean currents driven by constant wind, is ill-defined when more realistic stochastic wind is considered. We showed that the component of the stochastic wind that resonates with the Coriolis frequency leads to the divergence (singularity) of the surface and depth-integrated currents. The addition of a linear friction term to the model suppresses this unphysical singularity. We presented explicit solutions for the surface and depth-integrated currents for wind stress with exponentially decaying and oscillating temporal correlations and showed that the wind’s temporal correlations and the friction drastically affect, and can even diminish, the resonance. Winds and currents from the Gulf of Elat were compared with the model’s predictions.
Simulated climate dynamics, initialized with observed conditions, is expected to be synchronized, for several years, with the actual dynamics. However, the predictions of climate models are not sufficiently accurate. Moreover, there is a large variance between simulations initialized at different times and between different models. One way to improve climate predictions and to reduce the associated uncertainties is to use an ensemble of climate model predictions, weighted according to their past performances. We showed that skillful predictions, for a decadal time scale, of the 2m temperature can be achieved by applying a sequential learning algorithm to an ensemble of decadal climate model simulations. The predictions generated by the learning algorithm were shown to be better than those of each of the models in the ensemble, the better performing simple average and a reference climatology. In addition, the uncertainties associated with the predictions were shown to be reduced relative to those derived from an equally weighted ensemble of bias-corrected predictions. The results demonstrate that learning algorithms can help to better assess future climate dynamics.
Ecosystem regime shifts are regarded as abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or by global disturbances. Spatially extended ecosystems, however, can also respond to local disturbances by the formation of small domains of the alternative state. Such a response can lead to gradual regime shifts involving front propagation and the coalescence of alternative state domains. When one of the states is spatially patterned, a multitude of intermediate stable states appears, giving rise to step-like gradual shifts with extended pauses at these states. To study the characteristics of regime shifts in spatially extended systems, we first used a minimal model describing pattern formation and the bistability of uniform and patterned states. We proposed indicators to probe gradual regime shifts, and proposed that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. Our results are particularly relevant to desertification in drylands where transitions to bare soil take place from spotted vegetation, and the degradation process appears to involve step-like events of local
vegetation mortality caused by repeated droughts.
Drylands are pattern-forming systems showing self-organized vegetation patchiness, a multiplicity of stable states and fronts separating domains of alternative stable states. We used the context of dryland vegetation to study a general problem of complex pattern-forming systems: multiple pattern-forming instabilities that are driven by distinct mechanisms but share the same spectral properties. We found that the co-occurrence of two Turing instabilities when the driving mechanisms counteract each other in some region of the parameter space results in the growth of a single mode rather than two interacting modes. The interplay between the two mechanisms compensates for the simpler dynamics of a single mode by inducing a wider variety of patterns, which implies higher biodiversity in dryland ecosystems.
Pattern dynamics, induced by droughts or disturbances, can result in desertification shifts from patterned vegetation to bare soil. Pattern formation theory suggests various scenarios for such dynamics: an abrupt global shift involving a fast collapse to bare soil, a gradual global shift involving the expansion and coalescence of bare soil domains and an incipient shift to a hybrid state consisting of stationary bare soil domains in an otherwise periodic pattern. Using models of dryland vegetation, we addressed the question of which of these scenarios can be realized. We found that the models can be split into two groups: models that exhibit a multiplicity of periodic patterned and bare soil states, and models that exhibit, in addition, a multiplicity of hybrid states. Furthermore, in all models, we could not identify parameter regimes in which bare soil domains expand into vegetated domains. The significance of these findings is that, while models belonging to the first group can only exhibit abrupt shifts, models belonging to the second group can also exhibit gradual and incipient shifts.
Using empirical data and mathematical modeling, we investigated the dynamics of the Namibian fairy circle ecosystem as a case study of regime shifts in a pattern-forming ecosystem. Our results provide new support, based on the dynamics of the ecosystem, for the view of fairy circles as a self-organization phenomenon driven by water–vegetation interactions. The study further suggested that fairy circle birth and death processes correspond to spatially confined transitions between alternative stable states. Cascades of such transitions, possible in various pattern-forming systems, result in gradual, rather than abrupt, regime shifts.
Many mathematical models have been proposed to explain the emergence of vegetation patterns in arid and semiarid environments, but only a few of them take into account the heterogeneity in the system properties. We presented a rigorous study of the effects of heterogeneous soil-water diffusivity on vegetation patterns, using two mathematical models. The two models differ in the pattern-forming feedback that they capture; one model captures the infiltration contrast between vegetated and bare soil domains, whereas the other model captures the increased growth rate of denser vegetation due to an enhanced ability to extract water from the soil. In both models, the most significant effect of the heterogeneity on the soil-water diffusivity is the increased durability of patterned vegetation to a reduced precipitation rate. An additional effect is that the heterogeneity makes the desertification process, namely, the transition from a spotted vegetation pattern to a bare soil state, more gradual than in the homogeneous system. Our findings suggest that the heterogeneity cannot be neglected in the study of critical transitions in heterogeneous ecosystems and, particularly, in the study of the desertification process due to climate changes or anthropogenic disturbances.
In summary, our work advanced the understanding of regime shifts in spatially extended ecosystems, pointed out the crucial role of temporal correlations in wind-driven ocean current statistics and provided a practical tool for the improvement of decadal climate predictions.