Optimization with an Exact Discrete Adjoint for a Diffusive Lotka–Volterra Harvesting Problem

We study a diffusive predator--prey system controlled by a bounded fishing effort. The objective contains tracking, effort, and harvest terms. A continuous comparison argument gives explicit state bounds and a weighted population identity. Two fully discrete schemes are analyzed. The reference IMEX scheme is fast but has a conditional positivity property. A triangular transfer-implicit scheme is positive for every time step and satisfies an exact discrete population balance. Exact discrete adjoints are derived for both schemes. For the IMEX discretization, tangent equations give a matrix-free reduced Hessian. Under strict complementarity, the local linearization of the relaxed forward--backward sweep on the free control set is IF-θ HF/λ, namely a Richardson iteration matrix. This yields a sharp local stability interval and the best constant relaxation. In the reference test, matrix-free Lanczos estimates the free-Hessian spectrum as $[0.04737,10.41638]$. Hence θ<0.009600 is locally stable and θₒₚₜ=0.009557. A sweep with θ=0.0093 reaches a projected residual below 10⁻⁶ in $1192$ iterations. Anderson acceleration reduces this count to $234$, spectral projected gradient needs $187$ iterations, and L-BFGS-B needs $81$. Both adjoints pass second-order Taylor tests. Fixed-control refinement gives order one in time and order two in space. Five starts produce stationary controls whose objectives agree to twelve digits.