Differential movement and movement bias models for marine protected areas

Marine protected areas (MPAs) are promoted as a tool to protect overfished stocks and increase fishery yields. Previous models suggested that adult mobility modified effects of MPAs by reducing densities of fish inside reserves, but increasing yields (i.e., increasing densities outside of MPAs). Empirical studies contradicted this prediction: as mobility increased, the relative density of fishes inside MPAs (relative to outside) increased or stayed constant. We hypothesized that this disparity between theoretical and empirical results was the result of differential movement of fish inside versus outside the MPA. We, therefore, developed a model with unequal and discontinuous diffusion, and analyzed its steady state and stability. We determined the abundance in the fishing grounds, the yield, the total abundance and the log ratio at steady-state and examined their response to adult mobility (while keeping the relative inequity in the diffusion constant). Abundance in the fishing grounds and yield increased, while total abundance and log-ratio decreased, as mobility increased. These results were all qualitatively consistent with the previous models assuming uniform diffusivity. Thus, the mismatch between empirical and theoretical results must result from other processes or other forms of differential movement. Therefore, we modified our original model by assuming that species located on the boundary of the MPA will preferentially move towards the MPA. This localized movement bias model gives rise to steady state profiles that can differ radically from the profiles in the unbiased model, especially when the bias is large. Moreover, for sufficiently large bias values, the monotonicity of the four measures with increased mobility is reversed, when compared with our original model. Thus, the movement bias model reconciles empirical data and theoretical results.