Inputs
Example data table
These sample scenarios illustrate how connectivity and migration influence spread speed.
| Scenario | r (1/day) | m (1/day) | k | L (km) | s | b | Speed c (km/day) |
|---|---|---|---|---|---|---|---|
| Baseline | 0.25 | 0.05 | 4 | 2 | 1.00 | 0.00 | ≈ 1.41 |
| Higher migration | 0.25 | 0.10 | 4 | 2 | 1.00 | 0.00 | ≈ 2.00 |
| More connections | 0.25 | 0.05 | 8 | 2 | 1.00 | 0.00 | ≈ 2.00 |
| Lower suitability | 0.25 | 0.05 | 4 | 2 | 0.50 | 0.00 | ≈ 1.00 |
Formula used
This calculator uses a simple diffusion-style approximation for a patch network, then applies a classic reaction–diffusion wave-speed result.
- Effective growth: reff = s · r
- Effective diffusion: Deff ~ m · (k/2) · L²
- Spread speed: c ~ 2 · √(reff · Deff)
- Bias adjustment: cadj = c · (1 + b)
- Travel time: t ~ R / cadj
How to use this calculator
- Choose time and distance units matching your dataset.
- Enter r from early growth in a typical patch.
- Enter m as the fraction moving between patches per time.
- Set k and L to represent network connectivity and spacing.
- Use s to model interventions or habitat quality effects.
- Optionally set b to represent directional corridors or flow.
- Enter R to estimate how long the spread front takes.
- Press Calculate to view results above the form.
Professional overview
1) What a metapopulation spread rate represents
In a metapopulation, local patches host growth while movement connects them. A spread rate summarizes how fast a colonization front advances through space when many patches interact. Here, speed is reported in your chosen distance per time unit, helping compare scenarios consistently.
2) Growth sets the amplification clock
The early-stage growth rate r (for example, 0.25 per day) controls how quickly a newly seeded patch builds enough abundance to export movers. A scaling factor s (0–1) reduces growth for limited resources, control measures, or unsuitable habitat, producing reff = s·r.
3) Migration and connectivity create effective diffusion
Migration m acts like a step frequency, while average degree k captures how many neighboring patches receive those movers. With mean spacing L, a random-walk approximation gives Deff ~ m·(k/2)·L². Doubling m or k roughly doubles diffusion, raising speed by about √2.
4) Why the square-root speed law matters
Reaction–diffusion theory predicts c ~ 2·√(reff·Deff). Because of the square root, modest improvements in growth or movement often yield diminishing returns: quadrupling diffusion is needed to double speed. This is useful when prioritizing interventions under budget constraints.
5) Interpreting the sample numbers
Using r=0.25/day, m=0.05/day, k=4, and L=2 km gives Deff ~ 0.4 km²/day and c ~ 1.41 km/day. If migration increases to 0.10/day, diffusion doubles and the predicted speed rises to about 2.0 km/day. These values match the example table and help sanity-check inputs.
6) Directional bias for corridors and flows
Some landscapes have preferred directions: rivers, road networks, winds, or currents. The bias factor b (0–1) applies a conservative multiplier cadj = c·(1+b). For b=0.3, speeds increase by 30%. Treat this as a first-order correction, not a replacement for detailed transport modeling.
7) Travel-time planning across a target region
Enter a region length R to estimate arrival time t ~ R/cadj. For example, if c is 2 km/day and R is 50 km, the model predicts about 25 days to traverse that distance. This supports operational timelines, monitoring design, and sensitivity checks.
8) Limits and when to refine the model
The calculator assumes early exponential growth, roughly local movement, and similar patch spacing. Rare long-distance jumps, strong heterogeneity, seasonal forcing, and network bottlenecks can change front behavior. Use the outputs as screening estimates, then validate with simulations or field data when decisions are high-stakes.
FAQs
1) What does r mean in this tool?
r is the early-phase per-time growth rate within a typical patch. It should reflect the initial exponential increase
before saturation. Use the time unit selector to match your r estimate.
2) How should I choose the migration rate m?
m represents the fraction leaving a patch per time step. If 5% of individuals disperse daily on average, use 0.05 per
day. For weekly surveys, convert to a weekly rate.
3) What is k and why does it matter?
k is the average number of connected neighboring patches. Higher k means more receiving routes, increasing effective
mixing. In this approximation, diffusion scales roughly linearly with k.
4) Is L the same as a map distance?
L is a representative spacing between connected patches, not necessarily straight-line distance. Use the mean edge
length of your connectivity graph, or a typical movement distance between interacting patches.
5) What does the scaling factor s capture?
s reduces growth to represent interventions, lower habitat quality, or limited resources. Setting s=0.6 means only 60%
of the baseline growth contributes to spread, lowering speed through r_eff.
6) When should I use directional bias b?
Use b when a consistent corridor or flow accelerates spread in one direction. Keep b conservative (0–0.3) unless you
have strong evidence. It multiplies speed by (1+b).
7) Are results valid for long-range dispersal events?
Not fully. The diffusion approximation is best for mostly local movement. Rare long jumps can speed up spread beyond
this estimate. Consider a fat-tailed kernel or explicit simulation when long-range events are important.
Use results wisely, validate assumptions, and cite sources responsibly.