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.
2.1 The same on-ramp for every kernel form
The argument above is not special to the exponential: you can state a (distance, probability) point for any shape, and both the analysis form and the kernel explorer accept one for every form, under a single convention:
The point pins the scale; the shape stays yours.
Each form has one distance-like parameter that stretches the curve without changing its character — that is what the point solves for. Shape and amplitude ($b$, $s$, $A$) are the ecology of the tail: how fat, how steep, what fraction of individuals ever leaves. A single point cannot identify them, so they are held at the values you set. Inventing them would fabricate information your observation does not carry.
| Kind | Solved for | Closed form (from $p(d)=y$) |
|---|---|---|
exponential |
$\theta$ | $\theta = -\ln y / d$ |
exp_power |
$a$ | $a = d \,/\, (-\ln(y/A))^{1/b}$ |
twodt |
$a$ | $a = d \,/\, \sqrt{y^{-1/b} - 1}$ |
sigmoid (Hill) |
$d_{50}$ | $d_{50} = d \,/\, (A/y - 1)^{1/b}$ |
sigmoid (logistic) |
$d_{50}$ | $d_{50} = d - s\ln(A/y - 1)$ |
piecewise |
— | the point is inserted as a breakpoint |
Every solution is exact, so the calibrated curve passes through your point by construction. Two cases are rejected rather than fudged: a probability at or above the amplitude $A$ (the curve never reaches it, so no scale exists), and a logistic point implying $d_{50} \le 0$ (the drop would sit before the origin).
2b. Kernel parameter reference
Every form, its parameters, and what "distance" means for it. The kernel explorer
(/graph/dispersal-models) renders this same table live, and the analysis form
shows the "what it does" column under each input — all three are generated from
one metadata source, so they cannot drift apart.
These notes describe what each knob does to the curve, which is a mathematical fact. What value a given species takes is an ecological judgement you make: the tool offers shapes and consequences, never an authoritative parameter (ADR-015).
Negative exponential — $p(d) = e^{-\theta d}$
Distance means: your reference distance is where $p$ equals the probability you gave; half the dispersal reach sits inside $\ln 2/\theta$.
| Parameter | Unit | What it does |
|---|---|---|
theta |
1/m | Decay rate. Larger $\theta$ falls off faster; the median distance is $\ln2/\theta$ ($\theta = 0.000693 \rightarrow 1\,000$ m). |
Exponential-power — $p(d) = A\,e^{-(d/a)^b}$
Distance means: $a$ is the characteristic range — where the curve has fallen to about 37% of $A$ (exactly so when $b = 1$).
| Parameter | Unit | What it does |
|---|---|---|
A |
probability | Probability at distance zero — the ceiling. Below 1 when only part of the population ever attempts to leave. |
a |
m | Characteristic range: stretches the curve without changing its shape. Solved for by a calibration point. |
b |
— | Tail weight. $b<1$ fat-tailed (rare long jumps matter), $b=1$ plain exponential, $b=2$ Gaussian shoulder with a short tail. |
2Dt — $p(d) = (1 + d^2/a^2)^{-b}$
Distance means: $a$ is the core range: inside it the curve is flat-ish, beyond it decay is algebraic (a power law, not exponential).
| Parameter | Unit | What it does |
|---|---|---|
a |
m | Core range — the width of the plateau before algebraic decay takes over. Solved for by a calibration point. |
b |
— | Tail weight. Smaller $b$ = heavier tail (more rare long-distance events); the tail falls off as $d^{-2b}$. |
Sigmoid — Hill $A/(1+(d/d_{50})^b)$ · logistic $A/(1+e^{(d-d_{50})/s})$
Distance means: $d_{50}$ is the half-probability distance — where $p$ has dropped to $A/2$. Connectivity is near-full well inside it, near-zero well outside.
| Parameter | Unit | What it does |
|---|---|---|
A |
probability | Probability on the plateau — the ceiling the curve never exceeds. |
d50 |
m | Where $p$ falls to half the plateau. Solved for by a calibration point. |
b (Hill) |
— | How abrupt the drop is. $b\approx1$ is a gentle slide; $b\gtrsim10$ approaches a hard distance threshold. |
s (logistic) |
m | Width of the drop: $p$ goes from ~73% to ~27% of $A$ across roughly $2s$. Small $s$ approaches a hard cut-off. |
Piecewise-linear — linear interpolation between breakpoints
Distance means: exactly what you type. Each breakpoint is a probability you assert at a distance, joined by straight lines, and zero past the last one.
| Parameter | Unit | What it does |
|---|---|---|
breakpoints |
m, probability | Ascending distances with the probability at each. A calibration point is inserted as a breakpoint rather than solving for anything. |
3. What the distances list means
Each entry in the distances list defines a complete, independent connectivity analysis under a different dispersal scenario:
| Distance | Dispersal probability | Implied species |
|---|---|---|
| 300 m | 0.5 | Sedentary forest specialist |
| 1 000 m | 0.5 | Moderately mobile meadow butterfly |
| 3 000 m | 0.5 | Wide-ranging large mammal |
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.