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.)

Annual migrations, diving behavior, and thermal biology of Atlantic bluefin tuna, Thunnus thynnus, on their Gulf of Mexico breeding grounds

Authors: S. Teo, et. mucho al, 2006.

This paper is an empirical analysis of data collected via pop-up and archival tags implanted in breeding tuna. The tags record internal and external temperature of the tuna, as well as their depth, every 120 seconds. (Not all tags do this, but this is the gist of the data.) The authors compile this data for a sample of 28 mature tuna, and use the data to draw striking conclusions about tuna breeding behavior, and how it relates to their thermal biology. Tuna are normally cold water fish, and as a consequence normally have a relatively low thermal exchange coefficient $K$ (i.e. they retain heat well.) Since their breeding ground in the Gulf of Mexico are quite warm, however, and they need to spend a lot of time near the surface, the thermal exchange coefficient rises as they begin to breed, and the regulate their temperature by diving. This partially exchanges the high mortality rate of bluefin tuna caught in the GOM on pelagic long-lines: unable to dive, they become thermally stressed, which when added to the stress of capture does them in.