Interactions Between Wealth and Natural Resources: A Nonlinear ODE Model

In this paper we study an ODE model for the interaction between nature and society where the system dynamics is driven largely by the social wealth. The relevant variables are renewable resources, non-renewable ones, and wealth, while population depends on wealth and does not act on the other variables. We first obtain that the relevant trajectories are positive and cannot grow arbitrarily large. Then we compute all the equilibria and study their stability. We also give numerical simulations of some trajectory.