  Optimal control of harvested population at the edge of extinction in an
unprotected area, is considered. The underlying population dynamics is governed
by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and
space-dependent coefficients while the control consists of transporting
individuals from a natural reserve. The nonlinear optimal control problem is
approximated by means of a Galerkin scheme. Convergence result about the
optimal controlled solutions and error estimates between the corresponding
optimal controls, are derived. For certain parameter regimes, nearly optimal
solutions are calculated from a simple logistic ordinary differential equation
(ODE) with a harvesting term, obtained as a Galerkin approximation of the
original partial differential equation (PDE) model. A critical allowable
fraction $\underline{\alpha}$ of the reserve's population is inferred from the
reduced logistic ODE with a harvesting term. This estimate obtained from the
reduced model allows us to distinguish sharply between survival and extinction
for the full PDE itself, and thus to declare whether a control strategy leads
to success or failure for the corresponding rescue operation while ensuring
survival in the reserve's population. In dynamical terms, this result
illustrates that although continuous dependence on the forcing may hold on
finite-time intervals, a high sensitivity in the system's response may occur in
the asymptotic time. We believe that this work, by its generality, establishes
bridges interesting to explore between optimal control problems of ODEs with a
harvesting term and their PDE counterpart.

Optimization and Control;Dynamical Systems