external/input_devel/model-overview.md

In PrincetonUniversity/agroEcoTradeoff: Landuse Tradeoff analysis model

title: "Model overview" author: "Lyndon Estes" date: "06 January 2015" output: html_document: highlight: tango theme: spacelab toc: yes bibliography: trade.bib

The model (trademod) has 5 major modules:

1. Master module, where flexible parameters are set and runs controlled. The master module calls the following modules/functions in order:
2. Targets, which calculates the amount of each crop (in tonnes) that needs to be produced.
3. This may also readjust the cropland fractional share as a function of the relative demand increase for each crop.

4. Constraints, which applies the various land use constraints on top of the main constraint, which is the land's productivity for a given crop. The formulation of the land use constraints follows Koh and Ghazoul [-@koh_spatially_2010]:

   $$ \begin{aligned} C_{ij} = \beta_0Y_{ij} \beta_1c1_{ij} \beta_2c2_{ij} \beta_3c3_{ij} \end{aligned} $$

   Where $C_{ij}$ are the total constraints on production for crop i in pixel j, $Y$ is crop i's yield potential, $c1-3$ are constraints for carbon, biodiversity, and transport costs, respectively, and $\beta_0$ is a positive coefficent (from 0 to some arbitary multiple, but the maximum should be line with realistic potential yield gains) that modifies the yield potential of crop i in some way. Here we will use grids representing climate change impacts and irrigation-based increases, respectively, to provide these values, so $\beta_0$ will be the product of two coefficients. $\beta_{1-3}$ range between 0-1, where 0 means the constraint is not applied, and 1 that it is fully applied, with values in between representing partial weights. Theoretically, this value could be greater than 1, which would simply place a greater weight on that constraint. This means that:

   $$ \begin{aligned} P_{ij} = C_{ij} \end{aligned} $$

   That is, the probability $P$ that pixel j is converted to crop i is equivalent to the product of the constraints $C$. Let's demonstrate whether this is the correct way of going about things or not, that is, multiplicatively as opposed to additively. First, we'll set up a dummy matrix representing three constraints (c1-3) on three pixels (p1-3).

   r options(width = 90) set.seed(3) pmat <- sapply(1:3, function(x) sample(1:100, 3) / 100) pmat

   ```

   [,1] [,2] [,3]

   [1,] 0.17 0.33 0.13

   [2,] 0.80 0.60 0.30

   [3,] 0.38 0.99 0.57

   ``` Here's are the rankings when we assume that all pixels have a $\beta$ of 1 and we multiply, and also when we add constraints and divide by 3. Here we'll just see whether each pixel's probability rank is the same under the different schemes.

   r unname(rank(apply(pmat, 1, prod)) == rank(rowSums(pmat) / 3))

   ```

   [1] TRUE TRUE TRUE

   ``` So they produce the same ranks. But let's try this a bit more robustly, with more pixels and more random permutations of probabilities.

   r set.seed(3) rank_diffs <- sapply(1:1000, function(x) { pmat <- t(sapply(1:10, function(x) sample(1:100, 10) / 100)) d <- rank(apply(pmat, 1, prod), ties.method = "first") - rank(rowSums(pmat) / 10, ties.method = "first") d[d != 0] <- 1 sum(d) }) hist(rank_diffs)

   

   So these are not at all the same. In any case, let's go with multiplying, since we are creating a probability, and probabilities should be multiplied.

5. Convert, which converts the area of cropland necessary to hit $T_i$, which is the target production in tonnes for crop i:

   $$ \begin{aligned} A_i = \sum_{j=max(P_i)}^{t} fa_i \end{aligned} $$

   Where $A_i$ is the total area converted, as determined by summing the area of each pixel $a$ converted for crop i, multiplied by the fraction $f$ available in that pixel for converting to that crop (predetermined and possibly modified by the Targets module). Conversion begins in the pixel where $P_i$ is greatest, and ends in pixel $t$, which is the last pixel where:

   $$ \begin{aligned} T_i \leq \sum_{j=max(P_i)}^{t} fa_iY_i \end{aligned} $$

6. Impacts, which, in addition to the total area converted will assess some impacts for each scenario, which must still be developed, such as the total $CO_2$ emissions, some sort of biodiversity impact metric, some water use metric, etc. Ideas for biodiversity

7. Species loss metric (e.g. matrix weighted species area curve; [@koh_spatially_2010]$)

8. Fragmentation metric
9. Mean richness metric per converted pixel

References

PrincetonUniversity/agroEcoTradeoff documentation built on May 8, 2019, 3:12 p.m.