Dispersal kernel

The dispersal kernel describes how the probability of successful dispersal declines with the distance between two habitat patches. It translates geographic separation into the edge weights that drive every connectivity metric ekokrati.graph computes.

1. The negative-exponential kernel

ekokrati.graph uses the negative-exponential (or exponential decay) kernel:

$$ p(d) = e^{-\theta \cdot d} $$

where $d$ is the distance between two patches and $\theta > 0$ is the decay coefficient. The probability of crossing a gap of zero metres is 1; it declines monotonically toward zero as $d$ increases.

Why exponential? The exponential kernel has strong empirical support across many taxa and is the foundation of Hanski's Incidence Function Model and Saura & Pascual-Hortal's original PC formulation. Its single parameter makes it identifiable from the minimal data typically available for conservation planning (one observed or assumed dispersal distance and a probability). Alternatives such as fat-tailed kernels (e.g. inverse-power law) or two-part kernels better describe species with occasional long-distance dispersal events, but require additional data to parameterise reliably. The two-parameter and piecewise-linear extensions described in the Dispersal-kernels roadmap are planned for a future release.

The $\theta$ parameter and mean dispersal distance. For the negative-exponential kernel, the mean dispersal distance is $1/\theta$. A steep kernel (large $\theta$, small mean distance) describes sedentary species; a shallow kernel (small $\theta$, large mean distance) describes wide-ranging species.

2. The (probability, distance) parameterisation

Ecologists rarely know $\theta$ directly. Instead, they know — or can estimate — the probability that an individual successfully disperses between patches separated by a particular ecologically meaningful distance. ekokrati.graph uses this natural parameterisation:

Specify one analysis distance $d_\text{ref}$ (metres) and the dispersal probability $p$ that you believe applies at that distance.

$\theta$ is then derived automatically:

$$ \theta = \frac{-\ln(p)}{d_\text{ref}} $$

For example: "I think there is a 50% chance that a dormouse can disperse 1 000 m" gives $\theta = \ln(2)/1000 \approx 0.000693 \text{ m}^{-1}$.

This follows the Conefor convention (Saura & Pascual-Hortal 2007) and matches the parameterisation used in most published connectivity studies, making results directly comparable.

Important consequence. When you run ekokrati.graph with multiple analysis distances (e.g. 500 m, 1 000 m, 3 000 m), a different $\theta$ is computed for each distance. "50% probability at 500 m" describes a different species than "50% probability at 3 000 m". This is intentional: the distances list is a spectrum of dispersal scenarios, not one species at multiple scales.

3. What the distances list means

Each entry in the distances list defines a complete, independent connectivity analysis under a different dispersal scenario:

The resulting EC(PC) and dPC values describe the landscape's connectivity for that species type. Plotting EC(PC) against distance gives a distance profile — a species-spectrum fingerprint of the landscape that shows how connectivity changes as dispersal ability increases.

A landscape that is well-connected at 300 m but drops sharply by 3 000 m has strong local structure (many small patches close together) but poor landscape- scale connectivity (few links between clusters). A nearly flat profile means the landscape is uniformly well- or poorly-connected regardless of species mobility.

Comparison across distances within one landscape is meaningful. Comparison across landscapes using the same distance list is also valid, provided the same probability is used.

4. Min. edge probability (PC heuristic)

The minimum edge probability (also called the PC heuristic) is a computational threshold, not a dispersal parameter. Any pair of patches whose best dispersal path has probability below this floor is treated as effectively disconnected and excluded from the PC and dPC computation.

Ecologically it can be read as: "ignore dispersal routes with less than X% chance of success".

The default of 0.05 (5%) matches the Conefor convention. A path contributing less than 5% of maximum-probability connectivity has negligible effect on dPC rankings and EC(PC) values while potentially requiring significant computation time to evaluate. Lowering the threshold (e.g. 0.01) captures rarer long-distance connections; raising it (e.g. 0.10) gives a more conservative, higher-confidence connectivity picture.

See PC heuristic for the technical detail and guidance on when to adjust the default.

5. Landscape resistance and the homogeneous-matrix assumption

The exponential kernel over Euclidean distance implicitly assumes that the matrix between patches is homogeneous — all land between patches is equally difficult to cross. This is a deliberate simplification, not an oversight.

The alternative — parameterising a resistance surface for the matrix land cover — requires substantial calibration effort, and the empirical literature shows that patch importance rankings (which patches are most critical for conservation) are largely robust to this choice for most landscapes and taxa. The main exception is strong linear barriers (motorways, large rivers) where Euclidean distance overestimates connectivity across the barrier.

ekokrati.graph addresses that specific case through barrier layers (the fullscreen editor's "remove edges crossing barrier layer" tool) rather than a full resistance surface. See Landscape resistance — design rationale for the full analysis.

Key references

• 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.
• Hanski, I. (1994). A practical model of metapopulation dynamics. Journal of Animal Ecology, 63(1), 151–162.
• Nathan, R. et al. (2012). Mechanisms of long-distance seed dispersal. Trends in Ecology & Evolution, 23(11), 638–647.