Nonlinear Mathematical Model for Sustainable Water Resource Management under Drought Conditions in Somalia

Somalia faces one of the most severe and protracted water security crises in sub-Saharan Africa, driven by consecutive seasons of deficient rainfall, transboundary river flow reductions, and rapid population growth. The country’s only perennial rivers, the Juba and Shabelle, sustain the majority of agricultural and domestic water demand across southern and central Somalia, yet quantitative mathematical frameworks capable of capturing the nonlinear coupling between water storage dynamics and drought stress accumulation remain absent from the Somali water governance literature. This study develops and calibrates a nonlinear coupled ordinary differential equation system, the Water–Drought Interaction Model (WDIM), to characterise the temporal evolution of water storage and drought stress intensity under variable recharge conditions. The WDIM incorporates a power-law storage decay term (where water loss increases nonlinearly with depletion), a bilinear water–drought coupling interaction (representing feedback between stress and accelerated depletion), and a threshold-driven drought stress accumulation mechanism (stress only accumulates when storage falls below a critical level). Formal mathematical analysis, including existence and uniqueness of solutions, equilibrium characterisation, and global asymptotic stability verification through a quadratic Lyapunov function approach (an energy-like function that decreases along system trajectories), is rigorously established. An optimal control formulation derived via the Pontryagin Maximum Principle enables the identification of time-optimal water allocation strategies. Model calibration against observed streamflow data and the Standardised Precipitation Evapotranspiration Index record (1981–2023) yields a Nash–Sutcliffe Efficiency of 0.82 and a coefficient of determination of 0.94, outperforming comparable linear models. Optimal control interventions reduce cumulative water deficits by 43.7 percent, and bifurcation analysis identifies a critical recharge threshold below which the system collapses into persistent extreme drought. These findings provide a robust decision-support framework for evidence-based water policy in data-scarce, conflict-affected dryland contexts.