The importance of each patch

Parts 1 and 2 measured the connectivity of the whole forest. Conservation acts on pieces: this wood is to be protected, that one may be cleared for a road, this gap is where a hedge would be planted. In this part you ask the question planners ask, which patches and which links matter most for the connectivity of the whole?, and answer it the way Conefor does, by removing each patch in turn and measuring what is lost. You will split each patch's importance into three roles (its own habitat, the movement it sends and receives, and its role as a stepping stone between others), rank the links between patches, and compare the result with the centralities of network science. At the end you will have maps and tables that name the forests a species of 2 km cannot afford to lose, and the small woods that matter far more than their size suggests.

The idea#

Importance as the loss without it#

The simplest way to know what a patch does for a network is to take it away. Urban and Keitt (2001) removed patches one at a time from a landscape graph and watched how its connectivity fell; Pascual-Hortal and Saura (2006) made it the standard way to rank patches, with the indices of part 2: the importance of patch \(k\) is the percentage of the index lost when \(k\) is removed (with its links).

Three roles of a patch#

Saura and Rubio (2010) showed that dPC is a sum of three parts, each a different way a patch serves the network:

• dPCintra: the habitat inside the patch, \(a_k^2\), over PC's numerator. Only the patch's area matters; a large forest has a large intra fraction even if it is isolated.
• dPCflux: the movement starting or ending in the patch, \(\sum_{j \ne k} a_k a_j p^*_{kj}\) counted both ways. It is large for large patches that are well connected to other large patches.
• dPCconnector: what the patch contributes as a stepping stone between other patches, that is, the loss of the best paths between other pairs that went through \(k\) and must now take a worse route (or none). It does not depend on the patch's own area at all: a small wood in the right place can have a large connector fraction.

In the toy landscape of part 2, patch 1 (the largest) has dPC = 60.7%: 33.7% intra, 27.0% flux and no connector role. Patch 3 (the smallest, 20 ha) has dPC = 13.4%, of which 1.2% as a connector: it relays movement between patch 4 and the others.

Centralities#

Network science ranks nodes by their centrality: the degree (number of links), betweenness (how many shortest paths between other nodes pass through it), closeness (how few steps it is from all others) or the eigenvector centrality (connected to well-connected nodes). They are fast and need no areas. Bodin and Saura (2010) compared them with dPC and its fractions on real landscapes and found that betweenness and the connector fraction agree on the main stepping stones but not on all: centralities count steps and ignore how much habitat lies at each end, while dPCconnector weighs paths by probability and area. Terraflux gives both, so that you can see where they disagree.

When to use it#

• Prioritising patches for protection: the patches with the highest dPC are those whose loss would cost the most reachable habitat. Many regional plans rank forests or wetlands this way, from Conefor's tables.
• Finding stepping stones: patches with a high dPCconnector compared with their area, which an area-based ranking would miss.
• Assessing a project: the dPC of the patches a road or a mine would destroy is the share of the network's connectivity at stake, in a currency comparable between projects.
• Choosing where to restore a link: the links with the highest importance, or, for new links, conn.link_change (the gain of adding or improving one).

What it is not for: dPC removes one patch at a time, so it does not say what happens if two neighbouring patches are both lost (they may each look dispensable because the other remains). For sets of patches, remove them together (build the graph without them and compare its ECA) or use a spatial prioritization (spatial prioritization).

What you need#

The habitat graph of part 1 (clc_100m.tif, forests 23–25, edge-to-edge distances) and one dispersal distance: importance is computed for one kernel at a time (a list of distances gives a row per patch and distance). Here, 2 km with a probability of one half: in part 2 it is the distance at which the three fractions of the network are equal, a species of the middle range.

Build the model#

1. The importance of every patch#

Add conn.node_importance and connect graph.graph to its graph. Set distance = 2000 and probability = 0.5. Its table has a row per patch: id, distance, and for each index of part 2 the share lost without the patch (dA, dNC, dNL, dH, dCCP, dLCP, dIIC, dF, dAWF, dPC), the fractions dIICintra, dIICflux, dIICconnector, dPCintra, dPCflux, dPCconnector, and varPC, the variation of PC itself. metrics limits the list (["A", "PC"]), and BC, BCIIC, BCPC add Conefor's betweenness indices.

2. Paint them on the map#

conn.to_map paints a column of the table on the patches. Add four, all with graph.graph as graph and importance.table as table, and set their column to dPC, dPCintra, dPCflux and dPCconnector.

3. Links and centralities#

conn.link_importance removes each link in turn: its table has from, to, distance and dPC_removal, the share of PC lost. conn.centrality gives, per patch, degree, eigen, close, BWC (betweenness) and cluster, computed on the links shorter than distance (here also 2 km).

format = "terraflux/1"

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

[[nodes]]
id = "graph"
op = "conn.graph_from_raster"
inputs = { map = "land.raster" }
params = { habitat = [23, 24, 25], area_unit = "ha", distance = "edge", landscape_area = 250000 }
note = "The forest patches and the gaps between them."

[[nodes]]
id = "importance"
op = "conn.node_importance"
inputs = { graph = "graph.graph" }
params = { distance = 2000, probability = 0.5 }
note = "dPC and its fractions, per patch."

[[nodes]]
id = "dpc"
op = "conn.to_map"
inputs = { graph = "graph.graph", table = "importance.table" }
params = { column = "dPC" }

[[nodes]]
id = "connector"
op = "conn.to_map"
inputs = { graph = "graph.graph", table = "importance.table" }
params = { column = "dPCconnector" }
note = "The stepping-stone role only."

[[nodes]]
id = "links"
op = "conn.link_importance"
inputs = { graph = "graph.graph" }
params = { distance = 2000, probability = 0.5 }
note = "The share of PC lost without each link."

[[nodes]]
id = "centrality"
op = "conn.centrality"
inputs = { graph = "graph.graph" }
params = { distance = 2000 }
note = "Degree, betweenness... on the links up to 2 km."

In the application, sort a table by a column with a click on its header: sorting importance by dPC gives the ranking, by dPCconnector the stepping stones. To keep the tables, add outputs to the model and run it from the command line (terraflux run importance.toml --out results writes results/importance.csv), or connect a vector.write after conn.to_vector when the patches are polygons.

Read the results#

Which forests matter most#

Enlarge

The importance of the 154 patches, summed, is 216%: dPC values overlap (the flux between two patches is lost when either is removed, so it counts in both), and they are compared with each other, never added. The ranking at 2 km:

Three kinds of patches appear:

• Large reservoirs (55, 110, 8): most of their importance is intra and flux. They hold habitat and exchange dispersers with their neighbours. Losing one would cost a fifth to a quarter of the network.
• Hubs (47, 30, 96): forests of 2,400 to 2,900 ha in the north with a large flux and a large connector fraction. Patch 47 is the most important connector of the landscape: it ties the north-west to the woods of the north-centre.
• Stepping stones (65, 100, 58, 81): small woods (46 to 406 ha) whose dPC is almost entirely connector. Patch 81, of 46 ha, ranks 92^nd by area and 16^th by importance. These are the patches an area-based ranking would sacrifice first.

Importance against area#

Enlarge

The rank correlation (Spearman's) between area and dPC is 0.84: area explains most of the ranking, which is why large patches are a safe first priority. It is 1.00 for dPCintra (it is area), 0.90 for dPCflux, but only 0.56 for dPCconnector. The patches off the band are the reason to compute dPC at all.

Which links matter most#

Enlarge

Of the 2,457 links (pairs with a probability above pij_min), only 286 would cost any PC if cut: the rest are redundant, as the best path between their patches goes another way. The most important is the link between 30 and 47 (5.6% of PC, a gap of 1.1 km), then 96–100 (3.4%, 300 m), and a run of short links along the northern chain: 47–58 (2.9%), 65–77 and 58–65 (2.8% each), 77–81 (2.8%). Read with the patches, the result is a corridor of small woods that holds the north together: protecting it means protecting the woods and the gaps between them (hedges, field margins, the farmland a species crosses).

Betweenness and connector#

On the links up to 2 km, betweenness ranks 47 first (2,819), then 30, 65, 110 and 96; the rank correlation with dPCconnector is 0.72. They agree on the hubs. They disagree on patches such as 77 (47 ha): its betweenness is 0, because its neighbours 65 and 81 are 1.8 km apart, within 2 km of each other, and the shortest path in steps skips it; but its dPCconnector is 2.8%, because the path through it (gaps of 300 and 361 m) is far likelier than the direct jump (\(0.5^{300/2000} \times 0.5^{361/2000} = 0.79\) against \(0.5^{1800/2000} = 0.54\)). This is the difference Bodin and Saura (2010) described: when the question is about dispersal, prefer the probabilities.

Settings that change the answer#

• The dispersal distance. At 250 m almost all importance is intra (part 2: the fractions), and the ranking is the ranking by area. The connector role appears between 1 and 5 km, and its winners change with the distance: compute dPC at two or three distances for a group of species, and look for patches important at all of them.
• The probability. A larger probability at the same distance makes every move likelier, so fewer patches are needed as stepping stones.
• The attribute. With habitat quality as the attribute (attribute, weighted = true in conn.graph_from_raster), a large but poor patch loses importance to a small, good one.
• The distance measure. Least-cost distances change which links are short; part 4 shows that the stepping stones of the north lose most of their role when the farmland between them is costly.

Common pitfalls#

• Adding dPC values. The dPC of two patches is not the loss if both are removed. Remove them together.
• Ranking by dPC without the fractions. The top of the list is mostly large patches, which everyone already values. The connector fraction is where the analysis adds information.
• One distance for all species. A stepping stone for a species of 2 km may be irrelevant for one of 500 m.
• The border. Patches near the edge of the study area lose neighbours outside it and look less important.
• Ties at zero. Many patches have dPCconnector = 0: they relay nothing. A ranking by connector should start with those above zero.

Going further#

• Part 4: least-cost links: the importance of the patches when the matrix is not uniform.
• Part 5: the MSPA network: the importance of the bridges and cores of a forest's structure.
• Part 6: ProtConn: the importance of each reserve for the connectivity of a protected network.
• Costs and corridors: where to restore a corridor between two forests.
• The operators: conn.node_importance, conn.link_importance, conn.link_change, conn.centrality, conn.focal_nodes (importance within a neighbourhood of each patch), conn.to_map, conn.to_vector.

References#

• Bodin, Ö., & Saura, S. (2010). Ranking individual habitat patches as connectivity providers: integrating network analysis and patch removal experiments. Ecological Modelling, 221(19), 2393–2405. https://doi.org/10.1016/j.ecolmodel.2010.06.017
• Pascual-Hortal, L., & Saura, S. (2006). Comparison and development of new graph-based landscape connectivity indices: towards the priorization of habitat patches and corridors for conservation. Landscape Ecology, 21(7), 959–967. https://doi.org/10.1007/s10980-006-0013-z
• Saura, S., & Rubio, L. (2010). A common currency for the different ways in which patches and links can contribute to habitat availability and connectivity in the landscape. Ecography, 33(3), 523–537. https://doi.org/10.1111/j.1600-0587.2009.05760.x
• Saura, S., Vogt, P., Velázquez, J., Hernando, A., & Tejera, R. (2011). Key structural forest connectors can be identified by combining landscape spatial pattern and network analyses. Forest Ecology and Management, 262(2), 150–160. https://doi.org/10.1016/j.foreco.2011.03.017
• Urban, D., & Keitt, T. (2001). Landscape connectivity: a graph-theoretic perspective. Ecology, 82(5), 1205–1218. https://doi.org/10.1890/0012-9658(2001)082[1205:LCAGTP]2.0.CO;2