The political economy of marine reserves: Understanding the role of opportunity costs

Authors: M. Smith, J. Lynham, J. Sanchirico, and J. Wilson, 2009.

This paper takes as its point of departure an elementary model for the costs and benefits of working in a fishery. The meat of the work consists of a qualitative analysis of how changes in certain model parameters yield changes in incentives for fishermen. The basic "willingness to pay" model (WTP) is

$\displaystyle{ E(WTP) = \ln \left[ e^{\alpha} + \sum_{j=1}^J \exp \left( p_t q_j X_{jt} - c - \phi z_j\right) \right] - \ln \left[ e^{\alpha} + \sum_{j=1}^{J-1} \exp \left( p_t q_j X_{jt} - c - \phi z_j\right) \right] .}$

By adjusting these parameters, the authors generate Table 1 and Figure 1 in the paper. Figure 2 considers the "long term", i.e. analyzes how these preferences change over time. The data are produced by a meta-population growth model of the form

$\displaystyle{ X_{jt+1} - X_jt = f(X_{jt}) X_{jt} - d(X_{1t}, X_{2t}, X_{3t}) - \sum_i^N q_j E_{ijt} X_{jt} }$

.
Matlab code is here.

Comments/Questions:

1. This is a toy model, and it doesn't involve data. I find it interesting that this was of sufficiently wide-spread interest that it found its way into PNAS.

2. In the long-term model, are there only three patches (i.e. "choices" for the fisherman), or are there subpatches within the meta-patches?

The Effects of Population Heterogeneity on Disease Invasion

Authors: Dushoff and Levin, 1994.

Vaguely intolerable paper. All text with the exception of a few misbegotten line graphs, and although it has no ecology that I can tell, it also has no theorems, at least not defined as such. Locally it reads intelligibly, but is globally impenetrable.

To Do:

1. Re-read and update this review.

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Authors: van den Driessche and Watmough, 2005.


This paper defines the basic reproductive number as the spectral radius of a certain matrix,

$R_0 = \rho (FV^{-1})$.

The matrices $F$ and $V$ emerge in the computation of the Jacobian. To compute the Jacobian, the authors decompose the state equations into two parts, one corresponding to new infections and the other corresponding to 'everything else':

$\displaystyle{x^{\prime}_{i} = \hat{G}_i(x) - \hat{V}_i(x)}$

If the state has the form $x = (x_1, \cdots, x_m, x_{m+1}, \cdots, x_n)$, where only $x_1$ through $x_m$ admit infection, then the Jacobian breaks into blocks:

$\displaystyle{ J \hat{G} = \left( \begin{array}{cc} F + V & 0 \\ J_3 & J_4 \end{array} \right)}$

Although this decomposition is not unique, in many cases it is apparently intuitive, and yields satisfying results in the sense that the corresponding $R_0$ is actually a functional threshhold parameter (i.e. you get one kind of behavior for $R_0 >1$, another for $R_0 < 1$.)

To Do:

1. Sections 1-4 are transparent, but section 5 contains a bunch of stuff having to do with "center manifolds". This is apparently standard fare in dynamical systems--have a look at the relevant text, see if you can make head or tail of it.

2. Starting with the examples, see if you can calculate the corresponding number $R_0$ without looking at the derivations in the paper.

3. Can you get your hands on some data and try estimating this number? If so, it would be interesting to compare the estimates to those obtained from other methodologies.

The effects of averaging on the basic reproduction ratio

Authors: F. Adler (1992).

Beautiful paper written by Adler for his thesis at Cornell under the direction of Simon Levin. Ultimately it is a bit of relatively simple matrix analysis. The paper has one theorem, which basically says that, under appropriate modeling assumptions, the largest eigenvalue of one matrix is bigger than that of another.

To Do:

1. Understand the proof on purely mathematical terms (i.e. as a statement about two matrices that live in some relation to one another.)

2. Push the model in some way: allow not just group combinations, but other sorts of distortions (though see next paper.) Joe's ideas: allow differences in susceptibilities, or heterogeneities in the $\gamma$.

3. This paper answers a question about invasability. Is the concept well defined? Can the same techniques be used to answer other questions, perhaps not at a DFE?

4. When do you get equality?

5. Is it possible to get the wrong qualitative answer under aggregation? (i.e. suppose under aggregation $R_0 < 1$, while in fact $R_0 >1$.)

Optimal spatial management of renewable resources: matching policy scope to ecosystem scale

Authors: J. Sanchirico and J. Wilen, 2004.


This paper applies the formal apparatus of optimal control theory to a particular open access fisheries model. The optimization is to maximize rents, i.e. the problem is being solved from an economic viewpoint. The objective functional has the form:

$ \max_{\tau_i, \pi_i} \int_0^{\infty} e^{-\delta t} \sum_i R_i(E_i,x_i) dt$

where $R_i$ represents the rent from spatial patch $i$ with fish population $x_i$ and effort level $E_i$. The regulatory mechanisms are taxes, in this case: $\tau_i$ and $\pi_i$ represent landing and shipping taxes, respectively. The constraint equations have the form

$ x_i^{\prime} = f_i(x_i)x_i + ND_i(x_1, x_2, \cdots , x_n) - h_i(E_i,x_i)$

and

$ E_i^{\prime} = s_i R(E_i,x_i, \tau_i, \pi_i) + \sum s_{ij}[R_i(E_i, x_i, \tau_i, \pi_i) - R_j(E_j, x_j, \tau_j, \pi_j)]$,

where the rent function is

$ R(x_i, E_i, \tau_i, \pi_i) = [(p_i - \tau_i) h_i(x_i, E_i) - (c_i + \pi_i) E_i]$,

with $p_i$ the ex-vessel price and $c_i$ the per-unit effort cost. The problem is linear in the control variables, and thus leads to bang-bang control.

From here I get lost, unfortunately. There is a lot of text, a few sketches, a long discussion, but I miss the main point.

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.

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?