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Abstract. Robustness and resilience are concepts in systems thinking that have grown in importance and popularity. For many complex social-ecological systems, however, robustness and resilience are difficult to quantify and the connections and trade-offs between them difficult to study. Most studies have either focused on qualitative approaches to discuss their connections or considered only one of them under particular classes of disturbances. In this study, we present an analytical framework to address the linkage between robustness and resilience more systematically. Our analysis is based on a stylized dynamical model that operationalizes a widely used conceptual framework for social-ecological systems. The model enables us to rigorously delineate the boundaries of conditions under which the coupled system can be sustained in a long run, define robustness and resilience related to these boundaries, and consequently investigate their connections. The results reveal the trade-offs between robustness and resilience. They also show how the nature of such trade-offs varies with the choice of certain policies (e.g., taxation and investment in public infrastructure), internal stresses, and uncertainty in social-ecological settings.

Conflicts over water resources can be highly dynamic and complex due to the various factors which can affect such systems, including economic, engineering, social, hydrologic, environmental and even political, as well as the inherent uncertainty involved in many of these factors. Furthermore, the conflicting behavior, preferences and goals of stakeholders can often make such conflicts even more challenging. While many game models, both cooperative and non-cooperative, have been suggested to deal with problems over utilizing and sharing water resources, most of these are based on a static viewpoint of demand points during optimization procedures. Moreover, such models are usually developed for a single reservoir system, and so are not really suitable for application to an integrated decision support system involving more than one reservoir. This paper outlines a coupled simulation-optimization modeling method based on a combination of system dynamics (SD) and game theory (GT). The method harnesses SD to capture the dynamic behavior of the water system, utilizing feedback loops between the system components in the course of the simulation. In addition, it uses GT concepts, including pure-strategy and mixed-strategy games as well as the Nash Bargaining Solution (NBS) method, to find the optimum allocation decisions over available water in the system. To test the capability of the proposed method to resolve multi-reservoir and multi-objective conflicts, two different deterministic simulation-optimization models with increasing levels of complexity were developed for the Langat River basin in Malaysia. The later is a strategic water catchment that has a range of different stakeholders and managerial bodies, which are however willing to cooperate in order to avoid unmet demand. In our first model, all water users play a dynamic pure-strategy game. The second model then adds in dynamic behaviors to reservoirs to factor in inflow uncertainty and adjust the strategies for the reservoirs using the mixed-strategy game and Markov chain methods. The two models were then evaluated against three performance indices: Reliability, Resilience and Vulnerability (R-R-V). The results showed that, while both models were well capable of dealing with conflict resolution over water resources in the Langat River basin, the second model achieved a substantially improved performance through its ability to deal with dynamicity, complexity and uncertainty in the river system.

So far many optimization models based on Nash Bargaining Theory associated with reservoir operation have been developed. Most of them have aimed to provide practical and efficient solutions for water allocation in order to alleviate conflicts among water users. These models can be discussed from two viewpoints: (i) having a discrete nature; and (ii) working on an annual basis. Although discrete dynamic game models provide appropriate reservoir operator policies, their discretization of variables increases the run time and causes dimensionality problems. In this study, two monthly based non-discrete optimization models based on the Nash Bargaining Solution are developed for a reservoir system. In the first model, based on constrained state formulation, the first and second moments (mean and variance) of the state variable (water level in the reservoir) is calculated. Using moment equations as the constraint, the long-term utility of the reservoir manager and water users are optimized. The second model is a dynamic approach structured based on continuous state Markov decision models. The corresponding solution based on the collocation method is structured for a reservoir system. In this model, the reward function is defined based on the Nash Bargaining Solution. Indeed, it is used to yield equilibrium in every proper sub-game, thereby satisfying the Markov perfect equilibrium. Both approaches are applicable for water allocation in arid and semi-arid regions. A case study was carried out at the Zayandeh-Rud river basin located in central Iran to identify the effectiveness of the presented methods. The results are compared with the results of an annual form of dynamic game, a classical stochastic dynamic programming model (e.g. Bayesian Stochastic Dynamic Programming model, BSDP), and a discrete stochastic dynamic game model (PSDNG). By comparing the results of alternative methods, it is shown that both models are capable of tackling conflict issues in water allocation in situations of water scarcity properly. Also, comparing the annual dynamic game models, the presented models result in superior results in practice. Furthermore, unlike discrete dynamic game models, the presented models can significantly reduce the runtime thereby avoiding dimensionality problems.

In this study, a continuous model of stochastic dynamic game for water allocation from a reservoir system was developed. The continuous random variable of inflow in the state transition function was replaced with a discrete approximant rather than using the mean of the random variable as is done in a continuous model of deterministic dynamic game. As a result, a new solution method was used to solve the stochastic model of game based on collocation method. The collocation method was introduced as an alternative to linear-quadratic (LQ) approximation methods to resolve a dynamic model of game. The collocation method is not limited to the first and second degree approximations, compared to LQ approximation, i.e. Ricatti equations. Furthermore, in spite of LQ related problems, consideration of the stochastic nature of game on the action variables in the collocation method would be possible. The proposed solution method was applied to the real case of reservoir operation, which typically requires considering the effect of uncertainty on decision variables. The results of the solution of the stochastic model of game are compared with the results of a deterministic solution of game, a classical stochastic dynamic programming model (e.g. Bayesian Stochastic Dynamic Programming model, BSDP), and a discrete stochastic dynamic game model (PSDNG). By comparing the results of alternative methods, it is shown that the proposed solution method of stochastic dynamic game is quite capable of providing appropriate reservoir operating policies.

Resilience-based approaches have been attracting attention in governing social-ecological systems facing rapid social and environmental changes. In this article, we investigate the governance policies that focus on resilience. Our analysis is built on a stylized dynamical model that mathematically operationalizes a widely used conceptual framework, which links social components, natural resources, and infrastructure in social-ecological systems. Specifically, we numerically solve the Hamilton-Jacobi-Bellman (HJB) equation to determine policies-in the form of investment in public infrastructure-that maximize a quantitative metric of the system's resilience. For comparison purposes, we also derive policies that maximize the system's performance and discuss the differences between and implications of the two policies. The results showed that a policy that maximizes performance results in sub-optimal resilience and vice versa. Moreover, our sensitivity analysis suggests that managing resilience requires that one be more responsive to changes in external forcing.