A Nonlinear Ordinary Differential Equation Model for a Society Strongly Dependent on Non-Renewable Resources

In this work, we investigate an ODE system designed to describe the interplay between human society and the environment, with a strong focus on the role of non-renewable resources. Specifically, our model captures how the depletion and replenishment of non-renewable resources (along with renewable ones) and dependence on wealth drive population and resource dynamics in a society. We prove that the solutions of the system remain in the non-negative cone and are bounded, implying that indefinite unbounded growth in wealth or population cannot occur within this model. Next, we compute and classify all equilibrium points, exploring which equilibria can be stable and physically relevant. In particular, we show that, depending on parameter regimes, the system may admit a stable equilibrium with positive levels of population, renewable resources, non-renewable resources, and wealth, suggesting a possible sustainable long-term outcome for a heavily resource-dependent society.