Optimal Dynamic Harvest of a Mobile Renewable Resource

Authors: H. Joshi, G. Herrera, S. Lenhart, and M. Neubert, 2008.

The practical examples in this paper are very similar to those in Neubert (2003), with two exceptions: one, they add an advection term, and two, they consider the case where equilibrium has not been reached (i.e. the system changes as a function of time.) The equation that drives the numerical examples is:

$ u_t = ru \left( 1 - \frac{u}{K}\right) - hu + a_{xx} + bu_x $

The authors solve this using forward-backward iteration. For every set of parameters they show, they get a reserve (sometimes multiple reserves), though the shape and position of the reserves varies with the parameters. There are no chattering controls and no singular controls.

The bulk of the paper is devoted to proving the existence of an optimal control. Apparently, one can't just apply Pontryagin naively in the case of PDEs.

To Do:

1. as usual, try to reproduce the graphics. Are these forward-backward solvers standard issue? Can I write a generic interface? How stable are the numerics?

2. I'd like to understand why Pontryagin can't be applied in this case, and to really understand the proofs. The proofs are not particularly well done, alas, so a better bet might be to revisit the paper from which they are drawn, namely Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer.)

Triple benefits from spatial resource management

Authors: M. Neubert and G. Herrera, 2008.

Considers the same model as Neubert (2003), i.e.

$ u_t(t,x) = ru(t,x)(1-u(t,x)/k) + d\frac{\partial^2 u(t,x)}{\partial x^2} - qE(t,x)u(t,x) $

but instead of approaching the problem from the perspective of a single owner, i.e. maximizing yield

$\int_0^L h(x)u(x)dx$,

this paper maximizes rents, which happen to be modeled as

$ \int_0^L [pqu(x) - (w_0 + w_1E(x))]E(x) dx $.

(Note the odd cost structure. Idea: quadratic captures the fact that high effort density means more fishermen are tripping over one another's lines.)

Solutions are numeric and graphical, but show several things:

1. Effort concentrated near boundaries, where population levels are low.

2. At rent-maximizing equilibrium, stock levels and effort levels are often both higher than at open access equilibrium.

3. Conditions that support this phenomenon basically boil down to having heterogeneity and misaligned incentives (i.e. harvestors want one thing, regulators another.)


Questions/TODO

1. Are these numerics any stabler than the one for Neubert (2003)? I'll want to try to reproduce the graphics.

2. Can some of the corresponding patch or age-structured models be easily worked out? Worth taking a look.