Marine reserves and optimal harvesting

Author: M. Neubert, 2003.

Elegantly reasoned and written, this paper looks at a one-dimensional fishery whose total fish-density is given by a PDE with Dirichlet boundary conditions, diffusion and growth terms, and a harvest term. Under this model, the paper looks at equilibrium harvest strategies that maximize total revenue, i.e.

$ y(u,h) = \int h(x) u(x) dx $

Using Pontryagin's principle, finding the optimal control $h$ is reduced to solving a two-point boundary value problem, which the authors do without comment (though my own attempts to do the same resulted in resounding failure (on the level of convergence, not comment.)) The results show that there are more and more "reserves" (i.e. regions in which $h$ is 0) as the domain expands. Indeed, as it passes some critical threshhold, there is a region of "singular control" which is entered and exited by "chattering control", i.e. the control goes to 0 an infinite number of times in a finite interval.

Questons/Items of Interest:

1. The numerical problem of how to solve for this control on a computer. Apparently, a lot is known. Is there any freeware that can solve this class of control problems?

2. There is no formula for how the number of zones increases as a function of domain size. It should be something like $C/(L - x)$, I suppose, where C is a scaling constant, $L$ is the critical domain size, and $x$ is the size of some particular domain. Is such a formula even a remote hope? Perhaps easier: is it even the case that the number of reserves goes to infinity as x --> L, or is there a jump from finite to infite?

3. Is there a stochastic variant to this formulation?