Table 1.
Heuristic interpretation of the nondimensional parameters in Eq 1.
Fig 1.
An example of the qualitative dynamics of the 1D model from t = 0 to t = 10.
In dimensional terms, this corresponds to an invasion over a domain approximately 10km square after about 100 years (for a discussion of the dynamics of the invasion process see [36]). Parameter values are as in [36]. Values of A near one represent dense forest near carrying capacity, while values near zero represent largely uninvaded areas.
Fig 2.
The state of an invasion on a 2D domain at t = 10.
Dimensional in this and following figures are as in the previous figure.
Fig 3.
Random variates are created to form a grid of size N + 2 × N + 2 (white, hollow dots).
At each of the N × N inner grey dots ρ0 is calculated by averaging the value of α across the nine neighbouring points on the lattice. At some xi,j = (xi, yj) the neighbouring points are shown larger and in dark grey.
Fig 4.
A typical resolution of the random field.
Fig 5.
The state of A at t = 7 (left) and t = 10 (right) for the values of ρ0 given in Fig 4.
Fig 6.
The proportion of points (xi, yj) on the lattice with A(xi, yj, 10) > T for varying T, with results for a constant value of ρ0 (red squares) and results for randomly generated values of ρ0 (black circles).
Fig 7.
Histogram of the distribution of final signed distances for varying ρ0 (left). Histogram of the distribution of final signed distances for constant ρ0 (right).
Fig 8.
Evolution of the mean of the distance from the origin over an invasion for varying ρ0 (solid line) and constant ρ0 = 1 (dashed line).
For the constant case Δx was set to 10−2 to eliminate discretisation artefacts (results were otherwise similar for Δx = 10−1).
Fig 9.
The mean of distance of ‘invaded’ points from the initial condition for varying ρ0 (black solid lines) and a linear fit of the mean distance (red solid lines).
The corresponding data for constant ρ0 is shown with dashed lines.
Fig 10.
Averaged evolution of the mean distance of invaded cells to the initial condition across one thousand iterations.
Table 2.
Combinations of distributions and parameters considered (note that truncnorm(a, b, c, d) refers to a truncated normal distribution with bounds a and b and with μ = c, σ2 = d).
Fig 11.
Mean distances from invaded cells to the initial condition over time (left) and proportion of points invaded at t = 10 for differing thresholds (right) for ρ0 ∼ S(X1) (solid black line), ρ0 ∼ S(X2) (dashed black line), ρ0 ∼ S(X3) (dotted black line) and ρ0 = 1 (dashed red line).
Fig 12.
Representative states for ρ0 ∼ S(X1) (top left), ρ0 ∼ S(X2) (top right) and ρ0 ∼ S(X3) (bottom) at t = 10.
See bottom left for the scale for all three states.
Fig 13.
Mean distances from invaded cells to the initial condition over time (left) and proportion of points invaded at t = 10 for differing thresholds (right) for κ ∼ S(X1) (solid black line), κ ∼ S(X2) (dashed black line), κ ∼ S(X3) (dotted black line) and κ = 1 (dashed red line).
Fig 14.
Representative states for κ ∼ S(X1) (top left), κ ∼ S(X2) (top right) and κ ∼ S(X3) (bottom) at t = 10.
See bottom left for the scale for all three states (note that this is different to that of the previous two figures).
Fig 15.
Mean distances from invaded cells to the initial condition over time (left) and proportion of points invaded at t = 10 for differing thresholds (right) for γ ∼ S(X1) (solid black line), γ ∼ S(X2) (dashed black line), γ ∼ S(X3) (dotted black line) and γ = 1 (dashed red line).
Fig 16.
Representative states for γ ∼ S(X1) (top left), γ ∼ S(X2) (top right) and γ ∼ S(X3) (bottom) at t = 10.
See bottom left for the scale for all three states.
Table 3.
Spatial spread rates for each of the combinations of parameters and distributions considered (rounded to two decimal places).
Table 4.
Times of the onset of spatial spread for each of the combinations of parameters and distributions considered (rounded to two decimal places).
Fig 17.
The distribution of A at t = 10 for a model with an advective term (ν ≈ (1, 0)) and ρ0 ∼ S(X1).
Note that the initial condition in this case is a circular ‘blob’ of width 0.4 rather than a thin line as in previous plots.
Fig 18.
An aerial photo of an invasive population (note that the origin of this population appears to have been in the upper right-hand corner of this figure) from near Lake Pukaki, New Zealand.
Further details and earlier images of this site can be found in [41]. Image sourced from the LINZ Data Service and licensed by The Canterbury Aerial Imagery (CAI) consortium for reuse under a CC BY 4.0 license.