Stability analysis and optimal control of the logistic equation

We consider the unexploited/exploited logistic equation and study the stability of equilibrium points through Lyapunov functions. Then, we apply first and second order optimality conditions for the optimal control of the total biomass yield. Finally, we note that the time-scale logistic equation present in the literature lacks biological significance and we propose a new version of a dynamic logistic equation, valid on an arbitrary time scale, for which any trajectory of the system, beginning with a positive initial condition, remains nonnegative.
Comment: Accepted 30-Aug-2024; 20 pages, 5 figures