• Only take into account species (10% coverage)

• Add Agriculture as cost (min cost)

• Start with protected areas, eliminate cities (HFP)

• Instead of protected not protected we can model for:
  • Management
  • Landuse
  • categories

• Ampl Software
  • Optimization speciallized software
• TroublemakeR package
  • Build problems for AMPL

• Total richness
• Rarity weighted richness
• Phylogenetic richness
• Add contrains for rare species

\[H'=-\sum _{i=1}^{R}p_{i}\ln p_{i}\]

• Most of us have done this in a quadrat

• Think of every cell as a square
• Calculate composition
• Calculate whole diversity of denmark

• Package SDMWorlflow
• ~30.000 Species

• For each species we can model landuse change efect

set Cells;   # vertex set of the spacial graph
set Species; # Names of Species in the problem
set Landuses; # Name of possible Landuses

param SpeciesSuitabilityLanduse {Species, Landuses,Cells} ; #suitability for each species in each cell in each Landuses
param TransitionCost {Cells, Landuses}; # Cost of transforming a cell to this landuse
param b; #budget

var LanduseDecision {c in Cells,l in Landuses} binary; # decision on which landuse to use for cell Cell

minimize InvShanonDiv{s in Species}: 
(sum{c in Cells, l in Landuses} LanduseDecision[c,l]*SpeciesSuitabilityLanduse[s,l,c]/
 sum{t in Species, d in Cells, m in Landuses} LanduseDecision[d,m]*SpeciesSuitabilityLanduse[t,m,d])*
 log(sum{c in Cells, l in Landuses} LanduseDecision[c,l]*SpeciesSuitabilityLanduse[s,l,c]/
 sum{t in Species, d in Cells, m in Landuses} LanduseDecision[d,m]*SpeciesSuitabilityLanduse[t,m,d]);

subj to PropotionalUse{c in Cells}:
  sum{l in Landuses} LanduseDecision[c,l] = 1;

subj to Budget:
  sum{c in Cells, l in Landuses} LanduseDecision[c,l]*TransitionCost[c,l] <= b;

Given a set of spatial cells \(c\), a set of species \(s\), and a set of land uses \(l\), we have:

• A parameter for species suitability land use \(SSL_{s,c,l}\), representing the suitability of a given land use \(l\) in cell \(c\) for species \(s\).

• A parameter for transition cost \(TransitionCost_{c,l}\), representing the cost of transforming a cell \(c\) to a land use \(l\).

• A budget constraint \(b\).

where \(p_s\) is defined as:

\[p_s = \frac{\sum_{c \in C} \sum_{l \in L} SSL_{s,c,l} \cdot LandUseDecision_{c,l}}{\sum_{t \in S} \sum_{d \in C} \sum_{m \in L} SSL_{t,d,m} \cdot LandUseDecision_{d,m}}\]

We want to determine which land use should be assigned to each cell in order to minimize the inverse Shannon diversity index. Formally, we have:

\[\min \sum_{s \in S} p_s \log(p_s)\]

The decision variable \(LandUseDecision_{c,l}\) is a binary variable that indicates whether cell \(c\) is assigned to land use \(l\):

\[LandUseDecision_{c,l} \in {0,1}\]

Subject to the following constraints:

PropotionalUse: each cell must use exactly one landuse

Budget: the total cost of land use change must be within the budget limit