Sustainable reservoir management is not only a hydrological problem. Although water inflow, storage, evaporation, leakage, and planned withdrawals are fundamental components of reservoir dynamics, the long-term performance of a water storage system can also depend strongly on the quality of governance, monitoring, enforcement, and management. In many real-world reservoir systems, physical water losses occur simultaneously with institutional and management-related losses. Understanding the interaction between these processes is therefore essential for assessing reservoir sustainability and resilience.

In our study, “Modeling and Stability of Reservoir Systems under Physical Losses and Governance,” we develop a nonlinear dynamical framework that explicitly couples reservoir water volume with management quality. The main objective is to investigate how hydrological conditions and governance performance jointly determine the long-term behavior and stability of a reservoir system. The study was published in the International Journal of Energy and Water Resources in 2026.

The proposed framework considers two principal state variables. The first is the reservoir water volume, denoted by (V(t)), while the second is the management or governance quality, represented by (s(t)). The management variable is normalized between zero and one, where a value close to one represents strong and effective control, while a value close to zero represents weak or severely degraded management.

The reservoir-volume equation incorporates the major mechanisms that influence the water balance. These include natural or transferred inflow, designed outflow, leakage, evaporation, and unauthorized or illegal water withdrawals. Importantly, the model does not treat these losses as completely independent processes. Leakage depends on reservoir volume and structural condition, evaporation depends on the available water surface, and illegal withdrawals are explicitly connected to the governance level. Consequently, deterioration in management quality can increase unauthorized extraction and indirectly accelerate reservoir depletion.

The second equation describes the evolution of management quality. Management quality naturally deteriorates over time in the absence of intervention, while investments, monitoring, training, technological improvements, and enforcement can improve the management state. This formulation allows governance to be treated as a dynamic component of the reservoir system rather than as a fixed external parameter.

A major contribution of the study is its mathematical analysis of the resulting nonlinear system. Before studying stability, we establish fundamental properties such as positivity, boundedness, and global existence of the model solutions. Under the stated assumptions, reservoir volume remains nonnegative and bounded, while the management variable remains within an appropriate bounded region. These properties are important because they guarantee that the mathematical trajectories remain physically meaningful and that the model does not produce unrealistic finite-time blow-up behavior.

We then derive the equilibrium state of the coupled reservoir-management system and investigate its local stability using Jacobian analysis and eigenvalue conditions. For the considered case, the equilibrium is locally asymptotically stable when the relevant hydrological and management parameters satisfy the required positivity conditions. A particularly important quantity is the net inflow margin (R), which measures whether available inflow is sufficient to compensate for designed withdrawals and unavoidable losses.

The condition (R>0) has a direct physical interpretation. When (R>0), a positive steady-state reservoir volume can exist. If (R=0), the system reaches a degenerate equilibrium corresponding to a dry reservoir. More critically, when (R<0), the total losses exceed the available inflow even under the modeled management conditions, meaning that no positive equilibrium can exist and reservoir volume tends toward depletion. Thus, the mathematical condition for equilibrium existence also provides a useful criterion for assessing the hydrological feasibility of sustainable reservoir operation.

Beyond local stability, the study investigates global asymptotic stability. Using Lyapunov’s direct method together with LaSalle’s invariance principle, we establish conditions under which the equilibrium is globally asymptotically stable. In other words, under the specified assumptions, the system converges to its equilibrium state from all admissible initial conditions rather than only from states sufficiently close to equilibrium. This result is particularly relevant for reservoir systems because real reservoirs can experience substantial deviations from their nominal operating states.

To complement the analytical results, numerical simulations are performed using a classical fourth-order Runge–Kutta method with a fixed time step, with the simulations implemented in MATLAB. The computational analysis examines five representative scenarios designed to demonstrate how changes in inflow, management effort, and illegal withdrawals affect the reservoir dynamics.

The baseline scenario represents normal operating conditions, with an inflow of 2.0, designed outflow of 0.7, and management effort of 0.8. Under these conditions, the equilibrium management quality is approximately 0.64 and the corresponding equilibrium reservoir volume is approximately 4.47 in the normalized model.

The second scenario examines a substantial reduction in inflow, decreasing the inflow from 2.0 to 0.6. The results show that this reduction makes the net inflow margin negative, eliminating the possibility of a positive equilibrium. This scenario demonstrates how sufficiently severe hydrological stress can drive the system toward instability and reservoir depletion.

The third scenario investigates deterioration in management by reducing management effort from 0.8 to 0.2. Under these conditions, the equilibrium management quality falls to approximately 0.16, while the equilibrium reservoir volume decreases to approximately 2.18. This result demonstrates that governance deterioration can have a substantial indirect impact on physical water availability.

The fourth scenario increases illegal withdrawal sensitivity from 0.1 to 0.4. The resulting equilibrium volume decreases to approximately 3.88. Although the reservoir remains within a stable regime in this scenario, the increased unauthorized extraction significantly reduces the long-term storage level.

Finally, the fifth scenario considers improved management by increasing management effort from 0.8 to 1.0. In this case, management quality rises to approximately 0.8 and the equilibrium reservoir volume increases to approximately 6.08. Compared with the baseline condition, the improvement in governance therefore produces a substantial increase in sustainable storage capacity.

These simulations highlight one of the central findings of the study: governance is not merely an administrative factor; it can directly influence the dynamical stability and long-term water availability of a reservoir system. The analysis indicates that inflow reliability, management effort, and control of illegal extraction are particularly influential parameters. Small changes in these factors can potentially move the system from a sustainable regime with (R>0) to a depletion regime with (R<0).

The proposed framework therefore provides a mathematical perspective for understanding reservoir sustainability under coupled environmental and institutional pressures. Rather than analyzing hydrological losses independently from governance, the model integrates these mechanisms into a unified nonlinear dynamical system. This makes it possible to investigate not only whether a reservoir reaches equilibrium, but also whether that equilibrium is stable and how management interventions can alter the long-term state of the system.

The broader significance of the framework extends beyond reservoirs. Similar mathematical structures can arise whenever physical processes interact with institutional or management quality. Potential applications include irrigation networks, energy-grid management, infrastructure maintenance, and other resource-management systems in which physical degradation and governance performance evolve simultaneously.

Overall, the study demonstrates that sustainable reservoir operation requires attention to both physical water losses and governance effectiveness. Mathematical stability analysis provides a rigorous way to identify critical conditions for sustainable operation, while scenario simulations illustrate how changes in inflow, management effort, and unauthorized withdrawals can reshape the long-term behavior of the system. The results emphasize that strengthening monitoring, enforcement, investment, and management capacity can play a crucial role in increasing the resilience of water-storage systems under environmental and socio-economic pressures.

This work combines nonlinear dynamical systems, stability theory, Lyapunov analysis, LaSalle’s invariance principle, and numerical simulation to provide a mathematically grounded framework for reservoir sustainability. By connecting hydrological dynamics with governance quality, it offers a useful perspective for researchers, engineers, policymakers, and water-resource managers interested in understanding and improving the resilience of complex water systems.