Figure 1.
Examples of the landscape patterns used for simulations of range advance.
Nine example landscapes generated from fractals are shown, along with the idealized “regular” and “cross” landscapes. All landscapes have the same overall amount of habitat: 1%. In total we used 10 randomly generated fractals for each of 11 levels of fractal dimension. The patchy landscapes consist of clusters of varying sizes, reminiscent of the pattern of many fragmented natural habitats. The channeled landscapes are the negative image of the patchy landscapes (i.e. they are the gaps left between clusters of non-habitat). Similar patterns may exist in nature for habitats associated with rivers or with ecotones. The patchy landscapes with stepping stones represent the kind of pattern that could be achieved with deliberate habitat re-creation (0.9% pre-existing patches and 0.1% stepping stones along the multiple shortest paths, see methods).
Figure 2.
The value of a “stepping stone” of habitat as a function of its location.
given a source patch at (0, 0) and target patch at (0, 2). Speed is defined as the probability of colonizing the target at all (not going extinct) divided by the mean time until the target is colonized. (a) Speed vs. location of the stepping stone shown in two dimensions (x, y) with one parameter set: colonization parameters α = 1, g = 1 and extinction probability μ = 0.2. (b) Speed vs. location of the stepping stone shown in one dimension, along the straight line between the source patch and the target patch, and the effect of varying the colonization kernel α = [1,2,4] shown with [black,red,blue] and g = [0.5,1,2] shown with [dotted,plain,dashed] lines where μ = 0.2. (c) the effect of varying the extinction probability μ = [0.05,0.2,0.4] shown with [dotted,plain,dashed] lines, with α = [1,2,4] shown with [black,red,blue] and g = 1.
Figure 3.
The trade-off between conductivity (good for speed of advance) and aggregation (good for steady-state occupancy).
(a) Rank correlation between landscape metrics of conductivity and aggregation. Red points are from patchy landscapes, blue from channeled landscapes and orange from patchy landscapes with stepping stones. Aggregation increases with fractal dimension within each family of landscape (cf fig. 1). Large black cross represents the cross landscape and square represents the regular landscape. (b) The occupancy of landscapes in simulations before range advance started, at the end of the 200 time-step “burn-in”, against aggregation with symbols as in (a). (c–d) The speed of advance into the unoccupied landscape in cells per time step, against the conductivity (c) and aggregation (d) metrics, with symbols as in (a). Each point represents one simulation run. Metapopulation parameters were a mean dispersal distance of 8 cells, fecundity of 100 (propagules produced by an occupied cell) and per-cell extinction rate of 0.2.