Biologic characteristics of schooling fish species explain why the rates of harvesting in pelagic fisheries are not proportional to the existent stock size and may exhibit no variation between the periods of fish abundance and scarcity. Therefore, the stock-dependent nonlinearities in catchability must be reflected in the design of flexible fishing policies, which target the sustainable exploitation of this important natural resource. In this study, such nonlinearities are expressed through eventual variability of the "catch-to-stock" parameter that measures the sensitivity of an additional catch yield to marginal changes in the fish-stock level. Using the optimal control modeling framework, we establish that each value of the "catch-to-stock" parameter generates a unique steady-state size of the fish stock and the latter engenders an optimal fishing policy that can be sustained as long as the "catch-to-stock" parameter remains unchanged. We also prove the continuous dependence of the steady-state stock and underlying fishing policy upon the mentioned "catch-to-stock" parameter and then focus on the analysis of the equilibrium responses to changes in this parameter induced by external perturbations.