You can model harvesting a population in two ways. Additive: remove individuals at a fixed rate per unit effort (the harvest is proportional to effort alone). Multiplicative: remove a fraction of the standing stock (the harvest is proportional to effort times population size). Both are reasonable descriptions. Both appear in the literature. The choice looks cosmetic — a modeling convention, not a structural commitment.
The paper shows the choice is structural. Under additive harvesting, the optimal control problem yields local Pontryagin conditions — the control at time t depends on the state at time t and the costate at time t. The problem is solvable by marching forward and backward in time, matching conditions at the boundaries. Standard.
Under multiplicative harvesting, the optimal control becomes nonlocal. The harvest rate at time t depends on the total standing stock — an integral over all ages — which couples every age class to every other age class simultaneously. The costate equation acquires an integral term. The control at any given moment depends on the entire population structure, not just the local state.
One bit of modeling choice — additive versus multiplicative — transforms the mathematical character of the entire optimal control problem. Local becomes nonlocal. Separable becomes coupled. Tractable becomes hard. The complexity wasn't in the population dynamics or the objective function. It was in the interaction between the control and the state, specified by a single structural choice made before any analysis begins.
The through-claim: modeling assumptions are not neutral frames placed over reality. They are causal commitments that determine which mathematical tools apply. Two models that appear to describe the same system in slightly different notation can have fundamentally different solution structures. The choice of how effort enters the dynamics is as consequential as the dynamics themselves.