  We consider fully discrete finite element approximations for a semilinear
optimal control system of partial differential equations in two cases: for
distributed and Robin boundary control. The ecological predator-prey optimal
control model is approximated by conforming finite element methods mimicking
the spatial part, while a discontinuous Galerkin method is used for the time
discretization. We investigate the sensitivity of the solution distance from
the target function, in cases with smooth and rough initial data. We employ
low, and higher-order polynomials in time and space whenever proper regularity
is present. The approximation schemes considered are with and without control
constraints, driving efficiently the system to desired states realized using
non-linear gradient methods.
