  Overfishing can lead to the reduction or elimination of fish populations and
the degradation or even destruction of their habitats. This can be prevented by
introducing Marine Protected Areas (MPA's), regions in the ocean or along
coastlines where fishing is controlled. MPA's can also lead to larger fish
densities outside the protected area through spill-over, which in turn may
increase the fishing yield. A natural question in this context, is where
exactly to establish an MPA, in order to maximize these benefits. This problem
is addressed along a one-dimensional stretch of coast-line, by first proposing
a model for the fish dynamics. Fish are assumed to move diffusively, and are
subject to recruitment, natural death and harvesting through fishing. The
problem is then cast as an optimal control problem for the steady state
equation corresponding to the PDE which models the fish dynamics. The
functional being maximized is a weighted sum of the average fish density and
the average fishing yield. It is shown that optimal controls exist, and that
the form of an optimal control -and hence the location of the MPA- is
determined by two key model parameters, namely the size of the coast, and the
weight of the average fish density appearing in the functional. If these
parameters are large enough -and precisely how large, can be calculated
exactly- the results indicate when and where an MPA should be established. The
results indicate that an MPA always takes the form of a Marine Reserve, where
fishing is prohibited. The main mathematical tool used to prove the results is
Pontryagin's maximum principle.
