ekokrati / docs / Connectivity theory

Habitat connectivity theory

1. The graph representation of a landscape

Habitat networks are modelled as graphs — mathematical structures consisting of nodes and edges. In ekokrati.graph:

  • Nodes represent habitat patches. Each node carries the patch area $a_i$, which serves as a proxy for the population size a patch can support.
  • Edges represent dispersal routes between patches. Each edge carries the probability $p_{ij}$ that an individual successfully disperses from patch $i$ to patch $j$.

This is the patch-matrix model: the landscape is reduced to two types of land — habitat (the patches) and inhospitable matrix (everything else). Dispersal through the matrix is characterised by the dispersal kernel alone; the matrix is treated as homogeneous.

What the graph captures:

  • Patch size and its effect on population capacity
  • Inter-patch distances and how they reduce dispersal probability
  • The spatial arrangement of patches and which pairs can be reached from which

What the graph abstracts away:

  • Patch shape and internal structure
  • Variation in habitat quality within patches (area serves as the quality surrogate)
  • Heterogeneity within the matrix (see Dispersal kernel for how the kernel can reflect this)

The simplification is deliberate. It makes the problem tractable for landscapes with hundreds of patches while retaining the ecological information that drives conservation decisions: which patches are large, which are well-connected, and which sit at critical junctions in the network.

2. The Integral Index of Connectivity (IIC)

IIC uses a binary edge model: two patches are either connected or not, depending on whether the dispersal distance falls within a user-specified threshold.

$$ \text{IIC} = \frac{1}{A_L^2} \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_i \cdot a_j}{1 + nl_{ij}} $$

where:

  • $A_L$ is the total landscape area
  • $a_i$, $a_j$ are patch areas
  • $nl_{ij}$ is the number of links on the minimum-link path between $i$ and $j$ ($nl_{ii} = 0$; pairs with no path do not contribute to the sum)

The denominator $1 + nl_{ij}$ penalises indirect connections: directly adjacent patches ($nl = 1$) contribute $a_i a_j / 2$; pairs two steps apart contribute $a_i a_j / 3$; and so on. Disconnected pairs contribute nothing.

Range: IIC ∈ [0, 1]. IIC = 1 only when the entire landscape is a single patch. For a fully connected landscape with multiple patches, IIC < 1 because indirect connections are penalised. IIC → 0 as the landscape becomes more fragmented and patches become more isolated.

When to use IIC: IIC is appropriate when the ecological question is topological — whether patches are reachable at all, not how probable that dispersal event is. It is faster to compute than PC (no maximum-probability path search) and is useful as a coarse connectivity screen or when dispersal data are unavailable.

3. The Probability of Connectivity (PC)

PC extends IIC from binary to probabilistic edges. Instead of asking whether two patches are connected, it asks: what is the probability that dispersal between them succeeds along the best available route?

$$ \text{PC} = \frac{1}{A_L^2} \sum_{i=1}^{n} \sum_{j=1}^{n} a_i \cdot a_j \cdot p^*_{ij} $$

where $p^*_{ij}$ is the maximum-product path probability — the highest dispersal probability achievable over any path from $i$ to $j$, computed as the product of individual edge probabilities along that path. By convention, $p^*_{ii} = 1$ (a patch is trivially connected to itself); isolated patches contribute only through their self-term.

Why maximum-product, not sum? Summing all path probabilities would double-count the same dispersal event represented by different routes. The maximum-product path represents the single most probable dispersal route, which is the ecologically relevant quantity: an individual uses one path, and the best available path determines whether connectivity is maintained.

Range: PC ∈ [0, 1]. PC → 1 only when the entire landscape is one patch (all $p^*_{ij} = 1$). PC → 0 when all patches are isolated. The square root of PC, scaled by $A_L$, gives EC(PC) — see EC(PC).

Relationship to IIC: PC is the probabilistic generalisation of IIC. When all edges are binary ($p_{ij} \in \{0, 1\}$) and paths are evaluated by number of links rather than probability product, PC reduces to IIC. In practice, PC is more sensitive to the quality of connections, not just their existence.

4. EC(PC) — Equivalent Connected Area

See EC(PC) for the full treatment.

5. Assumptions and limitations

Centroid-to-centroid vs edge-to-edge distances. ekokrati.graph computes inter-patch distances as edge-to-edge (minimum distance between patch boundaries) by default. Centroid-to-centroid distances are available as an alternative. Edge-to-edge distances are more appropriate for species that disperse from habitat boundaries; centroid distances better represent round-trip movement between patch centres. The choice affects dPC rankings on landscapes where patches vary greatly in size.

The patch-matrix model. Treating the matrix as homogeneous ignores barriers (roads, rivers) and corridors (hedgerows, riparian strips) that structured movement. This is the main source of model uncertainty. The dispersal kernel partially compensates by representing average landscape resistance through the parameterisation of $\theta$, but it cannot represent spatially explicit barriers.

Scale dependence. PC and dPC are not absolute properties of a landscape — they are properties of a landscape under a specified dispersal scenario. The same landscape looks very well connected to a wide-ranging raptor and highly fragmented to a sedentary invertebrate. Always report the dispersal parameters alongside any connectivity metric.

Patch area as a habitat quality proxy. The model assumes that habitat quality is proportional to patch area: a 10 ha patch supports roughly ten times the population of a 1 ha patch. Where habitat quality varies substantially — for example, a small patch of high-quality ancient woodland adjacent to a large patch of degraded secondary habitat — area-based results should be interpreted carefully. A custom value_field (planned) will allow patch-level quality weights to replace raw area.

Key references

  • Pascual-Hortal, L. & Saura, S. (2006). Comparison and development of new graph-based landscape connectivity indices. Landscape Ecology, 21(7), 959–967.
  • Saura, S. & Pascual-Hortal, L. (2007). A new habitat availability index to integrate connectivity in landscape conservation planning. Landscape and Urban Planning, 83(2–3), 91–103.
  • Urban, D. & Keitt, T. (2001). Landscape connectivity: a graph-theoretic perspective. Ecology, 82(5), 1205–1218.
  • 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.