Least-cost links between patches

So far the gap between two forests was a straight distance, as if 800 m of wheat, 800 m of pasture and 800 m of town were the same obstacle. They are not, for almost any forest species. In this part you give every land cover class a resistance, the cost of crossing one of its cells, and measure the distance between patches as the cost of the cheapest route between them. You build the resistance from CORINE's classes, make the forest graph with least-cost distances, and compare its components and the importance of its patches with those of the straight graph. At the end you will know when least-cost links change the answer, how to choose the distance that goes with them, and one trap they hide.

The idea#

Resistance#

A resistance surface is a raster in which every cell holds the difficulty, cost or risk of moving through it for a species: low in the habitat, higher in land it avoids or where it dies more often, very high for barriers such as motorways or lakes. Resistances are relative: what matters is that arable land is, say, twenty times harder to cross than forest, not the absolute numbers. Zeller et al. (2012) reviewed how they are made:

• Expert opinion, as here: a value for each land cover class, from the literature and from people who know the species. Quick, transparent and the most common, but subjective.
• Habitat suitability: the inverse of a species distribution model's suitability (often transformed, since a cell of medium suitability is usually much easier to cross than its value suggests; Keeley et al., 2016).
• Movement or genetic data: resistances fitted so that the costs explain observed movements (GPS tracks, step selection) or genetic distances between populations (Spear et al., 2010).

The circuit connectivity series has a whole part on building and testing resistance surfaces.

The least-cost distance#

The least-cost distance between two places is the smallest sum of resistances along any route between them on the grid, each step weighted by its length (a diagonal step is √2 times longer). Adriaensen et al. (2003) made it popular in landscape ecology as a "functional" distance: two woods 1 km apart across pasture can be closer, for a forest bird, than two woods 600 m apart across a town.

The same model of movement underlies the grid distances series, which maps accumulated costs and draws the paths themselves; here we only need their costs, between every pair of patches.

When to use it#

• When the matrix is not uniform: farmland with towns, motorways, rivers, a mix of pasture and crops. In a landscape of one kind of matrix (a forest in a sea of wheat), straight distances say almost the same thing.
• When barriers matter: a lake or a city between two forests makes them far apart however close they are.
• When the result will be compared with movement or genetic data, which follow the land, not straight lines.

What it is not: a least-cost distance assumes the disperser knows the best route, which a juvenile in unknown country does not. Circuit theory spreads movement over all routes instead (circuit connectivity), and Terraflux's habitat graphs can also use commute-time distances, from a random walk over the same resistance (distance = "commute-time").

What you need#

• The land cover of the series, clc_100m.tif.
• A resistance per class. The values below are for a generalist forest mammal that uses woody vegetation readily, crosses grassland and mixed farming with some risk, open arable land with more, and avoids towns and water. They are an example: for your species, take them from the literature and test alternatives.

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Build the model#

1. The resistance#

Add raster.reclassify after the raster.read of the land cover. Its rules are one line per rule, the classes before the arrow and the new value after; * takes every value not yet given. In the Node pane the rules are a table (A rule per class lists the map's classes); in the model file they are a text with \n between lines:

23 24 25 -> 1
29 -> 2
21 26 27 32 35 36 -> 5
18 20 -> 10
12 -> 20
10 11 -> 30
7 8 9 -> 50
1 2 3 4 5 6 40 41 -> 100
* -> 20

2. The graph with least-cost distances#

Connect the land cover to the map of conn.graph_from_raster, as before, and the resistance to its resistance input. Set distance = "least-cost". The other settings of least-cost distances:

• cost_directions: 8 (default) or 4 neighbours for the steps.
• cost_function: makurhini (default), the cost of a step is the resistance of the cell entered (Makurhini's rule); mean, the mean of the two cells' resistances (gdistance's usual rule).
• unreachable: pairs that no route joins (a patch on an island) get their straight distance (euclidean, the default) or no link (drop).

The least-cost distances run between the centres of the patches (as Makurhini computes them), so they include the cheap crossing of each patch's own forest.

format = "terraflux/1"

[[nodes]]
id = "land"
op = "raster.read"
params = { path = "clc_100m.tif" }
note = "CORINE Land Cover, cells of 100 m."

[[nodes]]
id = "resistance"
op = "raster.reclassify"
inputs = { map = "land.raster" }
params = { rules = "23 24 25 -> 1\n29 -> 2\n21 26 27 32 35 36 -> 5\n18 20 -> 10\n12 -> 20\n10 11 -> 30\n7 8 9 -> 50\n1 2 3 4 5 6 40 41 -> 100\n* -> 20" }
note = "The cost of crossing each class, 1 (forest) to 100 (towns, water)."

[[nodes]]
id = "straight"
op = "conn.graph_from_raster"
inputs = { map = "land.raster" }
params = { habitat = [23, 24, 25], area_unit = "ha", distance = "edge", landscape_area = 250000 }
note = "Gaps between edges, in metres."

[[nodes]]
id = "least_cost"
op = "conn.graph_from_raster"
inputs = { map = "land.raster", resistance = "resistance.map" }
params = { habitat = [23, 24, 25], area_unit = "ha", distance = "least-cost", landscape_area = 250000 }
note = "Cheapest routes between the patches' centres, in metres of forest."

[[nodes]]
id = "imp_straight"
op = "conn.node_importance"
inputs = { graph = "straight.graph" }
params = { distance = 1000, probability = 0.5 }
note = "Half cross 1 km."

[[nodes]]
id = "imp_cost"
op = "conn.node_importance"
inputs = { graph = "least_cost.graph" }
params = { distance = 20000, probability = 0.5 }
note = "Half cross a cost of 20,000: 1 km of arable land."

[[nodes]]
id = "dpc_cost"
op = "conn.to_map"
inputs = { graph = "least_cost.graph", table = "imp_cost.table" }
params = { column = "dPC" }

3. Matching the two distances#

To compare the two graphs, the species must be the same in both. A species that crosses 1 km of open land with probability one half (part 3's straight graph at 1 km) crosses 1 km of arable land, the commonest matrix here, at a cost of 20,000. So the least-cost graph is read at distance = 20000, and at 10,000, 40,000 and 100,000 for the straight 500 m, 2 km and 5 km. The ratio is a modelling choice: say what it is.

conn.distances with the same settings gives the pairs table itself, to see how the costs compare with the gaps: here the least cost of a pair is a median 6.6 times its straight gap in metres (from 4 to 12 for most pairs). Crossing pasture is cheaper than 20 per metre and towns dearer, and part of each cost is the forest of the two patches.

Read the results#

The network as a whole#

The equivalent connected area is almost the same with both distances: the matching above works, and in this landscape, where arable land dominates the matrix, the straight gap is a fair proxy on average. The components differ more (46 instead of 56 at 1 km: some pairs are cheaper through pasture than their gap suggests, others dearer around villages).

The striking difference is the step fraction: from 17–43% with straight gaps to under 1% with least costs. This is not a property of the landscape but of the distances. A least-cost route between two patches may run through a third patch on the way, at the low cost of forest; so the cost from A to C is never more than the cost from A to B plus the cost from B to C, and the best path in the graph is almost always the direct link. The stepping-stone role has moved into the distances.

The importance of each patch#

Drag the handle to compare the importance of each patch with straight gaps (left) and with least-cost links (right), on the same scale:

Straight gaps, 1 km

Least cost, 20,000

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The rankings agree (Spearman's rank correlation 0.95): nine of the ten most important patches are the same in both, in nearly the same order. The patches that fall are the small stepping stones of part 3, 9, 77, 81, 94 and 132 (24 to 55 ha): their dPCconnector was 1.0 to 1.6% with straight gaps and is about 0 with least costs.

Which is right?#

Both, for different questions, and here lies the trap. With least-cost distances, the routes between patches were computed once, with every patch in place: the cost from A to C is low because the route runs through forest B. When conn.node_importance removes B, it removes B as a node but does not recompute the costs, so the route from A to C still benefits from B's forest. The connector role of B is lost from dPC. This is the same in Conefor and Makurhini, which also compute the distances first.

So:

• to rank patches by their area and flux, either distance works, and least costs are more realistic;
• to find stepping stones, use straight (edge) distances, or compute the costs without the patch you are testing (remove it from the resistance by giving its cells the matrix's resistance, and build the graph again).

Settings that change the answer#

• The resistance values. Least-cost results are more sensitive to the ratios between resistances than to their absolute values (Etherington, 2016): doubling every value doubles every cost and changes nothing once the dispersal distance is doubled too, but making towns 1,000 instead of 100 moves routes. Test two or three plausible sets.
• The cost distance of the kernel. It sets everything, and there is no single right conversion. Bracket it.
• cost_function. makurhini (the cell entered) and mean (the mean of the two cells) give slightly different costs at the edges of barriers.
• Centroids. Least costs run between patch centres; for a long, curved patch the centre can fall outside it, and the cost then includes matrix that a disperser leaving the patch's edge would not cross.

Common pitfalls#

• A dispersal distance in metres on a graph in cost units. With distance = 1000 on the least-cost graph, almost nothing is linked (1 km of forest). Always convert.
• Zero or negative resistances. Resistances must be positive: give the habitat 1, not 0. Cells with no value cannot be crossed.
• Barriers at the border. A route that would go around a town just outside the map cannot; analyse a margin around the study area.
• Reading dPCconnector from a least-cost graph as the stepping-stone value of a patch (see above).

Going further#

• Part 5: the MSPA network: structural corridors from the shape of the forest itself.
• Grid distances: the accumulated cost, the paths and the corridors between two forests.
• Resistance surfaces: how to make and test them.
• The operators: conn.graph_from_raster, conn.distances, raster.reclassify, conn.graph_from_samc (links as the probability that a random walker leaving one patch settles in another).

References#

• Adriaensen, F., Chardon, J. P., De Blust, G., Swinnen, E., Villalba, S., Gulinck, H., & Matthysen, E. (2003). The application of 'least-cost' modelling as a functional landscape model. Landscape and Urban Planning, 64(4), 233–247. https://doi.org/10.1016/S0169-2046(02)00242-6
• Etherington, T. R. (2016). Least-cost modelling and landscape ecology: concepts, applications, and opportunities. Current Landscape Ecology Reports, 1(1), 40–53. https://doi.org/10.1007/s40823-016-0006-9
• Galpern, P., Manseau, M., & Fall, A. (2011). Patch-based graphs of landscape connectivity: a guide to construction, analysis and application for conservation. Biological Conservation, 144(1), 44–55. https://doi.org/10.1016/j.biocon.2010.09.002
• Keeley, A. T. H., Beier, P., & Gagnon, J. W. (2016). Estimating landscape resistance from habitat suitability: effects of data source and nonlinearities. Landscape Ecology, 31(9), 2151–2162. https://doi.org/10.1007/s10980-016-0387-5
• Spear, S. F., Balkenhol, N., Fortin, M.-J., McRae, B. H., & Scribner, K. (2010). Use of resistance surfaces for landscape genetic studies: considerations for parameterization and analysis. Molecular Ecology, 19(17), 3576–3591. https://doi.org/10.1111/j.1365-294X.2010.04657.x
• Zeller, K. A., McGarigal, K., & Whiteley, A. R. (2012). Estimating landscape resistance to movement: a review. Landscape Ecology, 27(6), 777–797. https://doi.org/10.1007/s10980-012-9737-0