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https://github.com/gottacatchenall/ms_corridor_optimization


https://github.com/gottacatchenall/ms_corridor_optimization

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README 

        

# Introduction



Human activity has rapidly reshaped the face of Earth's surface, leaving

fragments of patchy habitat. Although there is no shortage of debate as to the

effects of fragmentation _per se_ on biodiversity and ecosystem function [@cite

], it is generally accepted that the combination of habitat and ensuing

subdivision produce negative outcomes for ecosystem function and services

[@resasco review].



In order to mitigate the consequences of landscape change on ecosystems,

developing landscape _corridors_ has seen much attention in the last several

decades. Bit more evidence for corridors here. But still, the spatter of

fragments in a landscape, where should ecologists choose to use their limit

resources to build a corridor?



Here we propose to answer that question by proposing an algorithm to estimate

the landscape modification that results in optimizing a specific ecosystem

process (in this paper maximizing the time until extinction of a metapopulation,

although the algorithm and associated software can be generally applied to any

process-based model with a quantifiable target state).



Although algorithms have been proposed for this [@peterman etc], they are

focused on finding the where the paths of least existance for a given species is

given data on that species dispersal.



# An algorithm for optimizing corridor placement



Start with some definitions and notation.



Define the set of possible landscape modifications, $\mathfrak{M}$,

in optimization language called the _search-space_.

Introduce uncountability argument of this space.



Because we cannot test every possible modification in $\mathfrak{M}$, we

use simulated annealing, a method for estimating the global optimum of functions

with NP search-spaces.



## Proposing landscape modifications



This is really important. We propose (no pun intended) several algorithms for

generating landscape modifications. Some of the details here might have to go

in a supplement/appendix.



### Graph-based



Consider only modifications that consist of connecting nodes. TODO: only choose

topologies from the minimum spanning tree of the nodes. 



#### The two stage approach



Stage-one: accept a new topology of connected nodes with probability in proportion

to chain temperature (see next section).



Stage-two: modify way that the connection for a given topological structure is

chosen. Because we are working in a 2D raster, all distances between points are

Manhattan distances, and any link between points is composed of $x$ horizontal

steps and $y$ vertical steps. There are thus $2^{\min(x,y)}$ ways to connect two

nodes that far apart.



### Not graph-based



The reason to avoid this is because the search-space grows much faster with lattice

size and budget. That being said, we can use some simply heuristics to weight

proposals using "common-sense".   



## Simulated annealing to explore the space of landscape modifications



The transition probability function, $q$, which gives the probability of moving from one

modification $i \in \mathbb{M}$ to a new proposed state $j \in \mathbb{M}$, as a

function of a chains temperature.



Here we define $q(i,j)$ using a logistic function,



$$q(i,j, \alpha) = \frac{1}{1 + e^{\alpha (s(j) - s(i))}}$$



$s(i)$ is the function that gives the score of a proposed modification. Here,

the mean time to extinction.



Simulated annealing can be written described as the following.



A markov-chain, denoted $\pi_\alpha$



***Figure 1: concept fig***



## Process-based optimization (using occupancy dynamics)



Here we use occupancy dynamics as the process, although we emphasize that this

method works for arbitrary process models and is instead limited only by the

computational demands of a given process model.



Compute new resistance surface which gives pairwise potential values for each

pair of points.



### Occupancy model



This pairwise potential value becomes normalized dispersal potential in

spatially-explicit metapopulation dynamics model [@hanski2000, @ovask2003]. Done

using `MetacommunityDynamics.jl`.



There might be attempts at analytic stuff but maybe not here?



***Figure 2: MTE versus epoch fig***: shows the chains move toward higher

*extinction times over time, i.e. it works.



# Simulation of data for testing the algorithm



In this section we describe the generation of simulated land cover and sites

for testing the algorithm.



## Simulation of landscapes



### Generation of landcover maps



DiamondSquare with high autocorrelatoin (0.7). Binned into $N_{cat}$ land cover

categories.



### Generation of points



Random Poisson process rounded to be integer coordinates. Padding around the edges

because real data doesn't have points on the edges, 10% on each side.



### Resistance values assigned to each land cover type



Each simulated land cover has $N_{cat}$ categories. The values of resistance



***Some type of performance fig vs. raster size and budget figure***



# Actual data St. Lawrence lowlands



# Discussion