On the structure of the solutions of a resource optimal model

The purpose of this paper is to illustrate that stable limit cycles represent a possible, generic equilibrium strategy in a renewable resource model. Following Wirl (2004), Bella (2010) Kogan, Venturi, and Shnaiderman (2017), Bella, Mattana, and Venturi (2017), we consider an optical system with an increasing pollution and a quick depletion of the reserves. We put our model in a reduced form: a system of three first order non –linear differential equations with one pre-determined variable (a combination of the state variables) and two no pre-determined variables (related to the control variables). By using instruments of the global analysis: bifurcation theory, we are able to show as a stable cycle and complex dynamics can occur in a set for a change in some parameters. A numerical simulation is given to support our theoretical results.