Chapter 7

Habitat networks¶

Step back far enough from a landscape and the habitat patches become dots, and the possibility of moving between them becomes lines. That picture is a graph, the object of a branch of mathematics with two centuries of results about paths, components and flows. Since the turn of the century, graphs have become the most widely used framework to measure landscape connectivity and to decide which patches and links to protect or restore (Urban & Keitt, 2001; Saura & Pascual-Hortal, 2007).

In this chapter

graphs for ecologists: nodes, links, paths and components;
habitat availability: why connectivity should be measured as reachable habitat;
the integral index of connectivity (IIC), the probability of connectivity (PC) and the equivalent connected area (EC), computed by hand on a small network;
the importance of every patch and link, and what its three fractions mean;
choosing distances, and protected-area connectivity (ProtConn).

Graphs for ecologists¶

A habitat graph has two ingredients (Urban & Keitt, 2001; Galpern et al., 2011):

Nodes: the habitat patches, each with an attribute, usually its area, sometimes its quality-weighted area or carrying capacity.
Links: the connections between pairs of patches, weighted by a distance: edge to edge, centre to centre, or a cost distance through a resistance surface (chapter 6).

In a binary graph, two patches are linked when their distance is below a threshold (the dispersal distance of the species). A path is a chain of links, and a component is a group of patches all joined by paths: a cluster of habitat an organism could, in principle, traverse by stepping from patch to patch. In a probabilistic graph, every link carries a probability of direct dispersal, derived from a kernel (chapter 6), and the connection between two patches is the best product of probabilities along any path.

The strength of the approach is abstraction: once the graph is built, connectivity questions become questions about the graph, quick to answer even for thousands of patches, and easy to repeat with a patch or link removed or added.

Measuring connectivity as reachable habitat¶

Early graph indices counted structural properties: the number of components, the size of the largest one, the length of the longest path. They have a serious flaw: they can improve when habitat is lost. Remove a small isolated patch and the number of components falls, which reads as "more connected". Pascual-Hortal and Saura (2006) proposed instead to measure habitat availability or reachability: the amount of habitat an organism can reach, counting the habitat of the patch it is in and the habitat of every other patch it can get to. This single idea integrates the amount of habitat and its connectivity, consistent with the lessons of chapter 3, and has the desirable property that losing habitat never increases connectivity.

The integral index of connectivity¶

IIC = Σ_i Σ_j a_i a_j / (1 + nl_ij) / A_L² a_i, a_j: attributes (areas) of patches i and j; nl_ij: number of links in the shortest path between them (0 when i = j; pairs in different components contribute nothing); A_L: area of the landscape. The sum runs over all ordered pairs, including i = j.

IIC works on a binary graph: two patches either are or are not connected, and pairs further apart in the network count less.

The probability of connectivity¶

PC = Σ_i Σ_j a_i a_j p^∗_ij / A_L² p^∗_ij: the maximum product probability of all paths between i and j (1 when i = j), from direct probabilities p_ij = e^{−k d_ij}.

PC is the probability that two points placed at random in the landscape fall in habitat areas that are reachable from each other (Saura & Pascual-Hortal, 2007). Its numerator, PCnum, is the reachable habitat. Because PC depends on the area of the whole landscape, its values are often tiny and hard to read. The equivalent connected area is more intuitive (Saura et al., 2011a):

EC(PC) = √PCnum The area of a single patch that would provide the same amount of reachable habitat as the whole network. It lies between the area of the largest patch and the total habitat area.

Try it: PC by hand¶

Four patches with areas in hectares and edge-to-edge distances in metres, and a species with a 50% chance of crossing 1 km:

import numpy as np
import pandas as pd
import landschaft as ls

areas = np.array([100.0, 50.0, 20.0, 80.0])
d = np.array([
    [0,    800,  1500, 5000],
    [800,  0,    600,  4000],
    [1500, 600,  0,    3000],
    [5000, 4000, 3000, 0],
], dtype=float)

k = -np.log(0.5) / 1000.0
p = np.exp(-k * d)                       # direct dispersal probabilities

The best path between two patches may go through others (1 → 2 → 3 is more probable than 1 → 3 directly). The maximum product probability is a shortest-path problem on −ln p, solved here with the Floyd–Warshall algorithm:

cost = -np.log(p)
for m in range(len(areas)):                      # allow paths through patch m
    cost = np.minimum(cost, cost[:, [m]] + cost[[m], :])
p_star = np.exp(-cost)

pc_num = (areas[:, None] * areas[None, :] * p_star).sum()
landscape_area = 10_000.0                        # hectares
{"PCnum": pc_num, "PC": pc_num / landscape_area**2, "EC(PC)": np.sqrt(pc_num)}

toy = ls.HabitatGraph(areas, d, landscape_area=landscape_area)
toy.overall(["PC"], distance=1000, probability=0.5)[["PCnum", "PC", "EC(PC)"]]

The library agrees. EC(PC) = 172 ha: the network of 250 ha behaves like a single patch of 172 ha, larger than the largest patch (100 ha) because patches 1, 2 and 3 are close enough to share much of their habitat, smaller than the total because the links are imperfect and patch 4 is far away.

Which patches matter?¶

The importance of a patch is the percentage of connectivity lost when it is removed (Keitt et al., 1997; Saura & Pascual-Hortal, 2007):

dPC_k = 100 × (PC − PC_{remove k}) / PC

def pc_numerator(areas, cost):
    for m in range(len(areas)):
        cost = np.minimum(cost, cost[:, [m]] + cost[[m], :])
    return (areas[:, None] * areas[None, :] * np.exp(-cost)).sum()


full = pc_numerator(areas, -np.log(p))
by_hand = []
for k_removed in range(len(areas)):
    keep = np.arange(len(areas)) != k_removed
    rest = pc_numerator(areas[keep], -np.log(p[np.ix_(keep, keep)]))
    by_hand.append(100 * (full - rest) / full)

importance = toy.node_importance("PC", distance=1000, probability=0.5)
importance.assign(dPC_by_hand=by_hand)[["id", "dPC", "dPC_by_hand", "dPCintra", "dPCflux", "dPCconnector"]]

Saura and Rubio (2010) showed that dPC splits into three fractions, which are three different ways in which a patch contributes:

Fraction	Contribution	Depends on
dPCintra	the habitat inside the patch	its area only
dPCflux	the dispersal flow starting or ending in the patch	its area and position
dPCconnector	its role as a stepping stone on the best paths between other patches	its position only

In the toy network, patch 1, the largest, matters mostly through its own area (dPCintra). Patch 3, the smallest, has the largest connector fraction: the best paths from patches 1 and 2 to the distant patch 4 go through it. Patch 2 is also a connector, on the best path from 1 to 3; patches 1 and 4 lie at the ends of the chain and connect nobody.

On a real landscape¶

Build the forest network of the example map, with edge-to-edge distances:

landcover = ls.load_raster(ls.example_data("clc"))
forest_net = ls.HabitatGraph.from_raster(
    landcover, habitat=[23, 24, 25], area_unit="ha",
    distance="edge", max_distance=10_000,
    landscape_area=landcover.size * 25 * 25 / 1e4,
)
total_forest = forest_net.habitat_area                  # 51 538 ha

overall = forest_net.overall(["NC", "PC"], distance=[250, 500, 1000, 2000, 5000, 10_000], probability=0.5)
overall.assign(**{"EC as % of forest": 100 * overall["EC(PC)"] / total_forest})[
    ["distance", "NC", "EC(PC)", "EC as % of forest", "PCstep(%)"]].round(1)

For a species that crosses 250 m of farmland half of the time, the 158 forest patches form 110 components and the reachable forest is less than 30% of the total; for one that crosses 5 km, two components and two thirds. PCstep(%) is the share of connectivity that depends on stepping stones: it grows from 4% to more than half. The same forest is, for different species, a fragmented set of woodlots or a well-connected network.

Rank the patches at 2 km:

patch_area = dict(zip(forest_net.ids.tolist(), forest_net.attributes.tolist()))
ranking = forest_net.node_importance("PC", distance=2000, probability=0.5)
ranking["area_ha"] = ranking["id"].map(patch_area)
ranking.sort_values("dPCconnector", ascending=False)[
    ["id", "area_ha", "dPC", "dPCintra", "dPCflux", "dPCconnector"]].head().round(2)

Sorted by connector fraction, the list shows patches that an area-based ranking would overlook: patch 84, of 47 ha (smaller than the median forest patch), carries 3.5% of the connectivity of the whole network, almost all of it as a stepping stone. Losing it would cost more connectivity than losing any of eight patches more than ten times its size. Maps of these fractions (forest_net.to_raster(ranking, "dPCconnector")) are among the most useful products for conservation planning.

link_importance does the same for links: the loss of connectivity if a link were cut, e.g. by a new road. centrality returns betweenness and other network centralities, which rank patches by how many shortest paths cross them (Bodin & Saura, 2010).

Choosing distances and probabilities¶

The distance and probability parameters describe the species:

The usual choice is the median dispersal distance with probability 0.5, or a maximum dispersal distance with a small probability such as 0.05. They define the same exponential kernel in different ways.
When no single species is targeted, analyse a range of distances (for example 250 m to 10 km, as above) to represent a range of species.
Use edge-to-edge distances for patches that dispersers leave from anywhere along their edge, centroid distances for point-like nodes, and least-cost or commute-time distances when the matrix is heterogeneous (distance="least-cost", "commute-time").
Set max_distance above the largest distance with a non-negligible probability, since pairs beyond it are left out.
Patches cut by the border of the study area lose part of their neighbours; where possible, include a buffer around the study area.

Protected-area connectivity¶

Protected areas are more effective as a network than as isolated units. The ProtConn indicator is the percentage of a region covered by protected land that is also connected (Saura et al., 2017, 2018). It is based on the same equivalent-connected-area logic, with protected areas as nodes and the region as the landscape, and it splits the unconnected remainder into fractions (protected areas that are isolated, connected only through unprotected land, or through other regions). ProtConn is used to report progress towards international targets that ask for protected-area systems to be well connected, from the Aichi Target 11 (17% of land by 2020) to Target 3 of the Kunming–Montreal Global Biodiversity Framework (30% by 2030). See ls.conn.protconn and the habitat networks tutorial.

Structural networks from MSPA¶

MSPA (chapter 4) already contains a graph: cores are nodes and bridges are links. ls.gtn_components and ls.gtn_importance analyse that structural network directly on the map, and ls.conefor_inputs exports it as node and distance files. Combining MSPA's structural connectors with graph indices is an effective way to identify key corridors (Saura et al., 2011b).

Key points

A habitat graph abstracts patches as nodes with an attribute and connections as links with a distance or probability.
Measuring connectivity as reachable habitat (IIC, PC) integrates habitat amount and connectivity, and never rewards habitat loss.
EC(PC) is the size of a single patch equivalent to the whole network, between the largest patch and the total habitat.
The importance of a patch (dPC) has three fractions: intra (its area), flux (its area and position) and connector (its position as a stepping stone). Small patches can be critical connectors.
Distances and probabilities describe the species; analyse a range when in doubt.

Check your understanding¶

Why can the number of components be a misleading connectivity index?
A network has EC(PC) equal to the area of its largest patch. What does that mean?
A patch has a high dPCconnector and a negligible dPCintra. What would you recommend for it, and why?

Answers

Because removing an isolated patch reduces the number of components, which reads as better connectivity although habitat was lost.
The other patches add (almost) nothing reachable: they are too small or too isolated from each other and from the largest patch, so the network is effectively a single patch.
Protect it, and consider it for restoration or enlargement: it adds little habitat itself but holds the network together as a stepping stone, so its loss would disconnect larger patches.

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.
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.
Keitt, T. H., Urban, D. L., & Milne, B. T. (1997). Detecting critical scales in fragmented landscapes. Conservation Ecology, 1(1), 4.
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, 959–967.
Saura, S., Bastin, L., Battistella, L., Mandrici, A., & Dubois, G. (2017). Protected areas in the world's ecoregions: how well connected are they? Ecological Indicators, 76, 144–158.
Saura, S., Bertzky, B., Bastin, L., Battistella, L., Mandrici, A., & Dubois, G. (2018). Protected area connectivity: shortfalls in global targets and country-level priorities. Biological Conservation, 219, 53–67.
Saura, S., Estreguil, C., Mouton, C., & Rodríguez-Freire, M. (2011a). Network analysis to assess landscape connectivity trends: application to European forests (1990–2000). Ecological Indicators, 11(2), 407–416.
Saura, S., & Pascual-Hortal, L. (2007). A new habitat availability index to integrate connectivity in landscape conservation planning: comparison with existing indices and application to a case study. Landscape and Urban Planning, 83(2–3), 91–103.
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.
Saura, S., & Torné, J. (2009). Conefor Sensinode 2.2: a software package for quantifying the importance of habitat patches for landscape connectivity. Environmental Modelling & Software, 24(1), 135–139.
Saura, S., Vogt, P., Velázquez, J., Hernando, A., & Tejera, R. (2011b). Key structural forest connectors can be identified by combining landscape spatial pattern and network analyses. Forest Ecology and Management, 262(2), 150–160.
Urban, D., & Keitt, T. (2001). Landscape connectivity: a graph-theoretic perspective. Ecology, 82(5), 1205–1218.