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.)
Friday, October 29, 2010
Wednesday, October 6, 2010
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.
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.
Tuesday, September 28, 2010
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
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
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?
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?
Friday, September 24, 2010
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.
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':
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:
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.
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.
Thursday, September 23, 2010
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 $R_0 >1$.)
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.
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.
Subscribe to:
Posts (Atom)