Spatial heterogeneities of human-mediated dispersal vectors accelerate the range expansion of invaders with source–destination-mediated dispersal

Rapid range expansions of invasive species are a major threat to ecosystems. Understanding how invasive species increase their habitat ranges and how environmental factors, including intensity of human activities, influence dispersal processes is an important issue in invasion biology, especially for invasive species management. We have investigated how spatially heterogeneous factors influence range expansion of an invasive species by focusing on long-distance dispersal, which is frequently assisted by human activities. We have developed models varying two underlying processes of a dispersal event. These events are described by source and destination functions that determine spatial variations in dispersal frequency and the probability of being a dispersal destination. Using these models, we investigated how spatially heterogeneous long-distance dispersal influences range expansion. We found that: (1) spatial variations in the destination function slow down late population dynamics, (2) spatial variations in the source function increase the stochasticity of early population dynamics, and (3) the speed of early population dynamics changes when both the source and the destination functions are spatially heterogeneous and positively correlated. These results suggest an importance of spatial heterogeneity factors in controlling long-distance dispersal when predicting the future spread of invasive species.

 by rasterizing a population into a 2,048 2,048  square grid for each time step. A large area size results in slower population dynamics with less temporal fluctuations; however, note that the asymptoticgrowth rate and the spatial factor are non-dimensional and invariant of the area size and the spatial unit.
If otherwise specified, we independently generate the vector distribution ( , ) h x y with randomly selected parameters from [1,5]   and [ 5,1]   − − before each realization, rasterize the distribution into a 512 512  square grid, and expand its size by four. The resolution of the grid is lower than the population grid described above (i.e., a grid cell of the vector distribution covers a 16-times bigger area than that of a grid cell of the population grid), due to the large computational demand for determining colony locations. This resolution reduction results in loss of the fine-scale spatial variance of background intensity distribution, which may introduce a bias in the resultant population dynamics. However, for several selected parameter combinations, we qualitatively got the same time courses with and without resolution reduction (not included). Therefore, we consider that resolution reduction does not affect our conclusion.
We implemented colony removal in the present models as follows; at the end of each time step, we (1) rasterize the whole population on a grid again, (2) break the rasterized grid into patches of connected cells that are covered by colonies, (3) count an area size of each patch, and (4) remove all colonies on a patch if its size exceeds the given threshold representing the size needed to detect a colony.

SI 2. Spatial distribution of human-mediated dispersal vectors
To extract the general properties of the dynamics, we need a randomly generated vector distribution ( , ) 0 h x y  . To get such random distributions, on the present study, we adopted an algorithm that have been proposed as a model of ecological spatial structures 1-3 that is fundamentally the same algorithm as a classic approximation of fractional Brownian motions.
The algorithm realizes a spatial distribution of intensities, namely ( , ) M x y , by applying inverse discrete Fourier transform (IDFT) on randomized 2-dimensional phases with given power-spectraldensity distribution. The resultant ( , ) M x y will be isotropic surface with a given autocorrelation structure 3 . Note that we need to keep antisymmetric structure of randomized phases to get real-valued surface 1

SI 3. Independent source and destination functions and estimated asymptotic growth rate
In the main text, we assume that the source and destination functions are (1) spatially heterogeneous but derived from an identical vector distribution or (2) only one of them is spatially heterogeneous. Here, we relax this assumption for more generality of our argument.
We determined two vector distributions s ( , ) h xy and d ( , ) h xy independently using the algorithm described at SI 2. Parameters  and  are independently chosen from [1,5]  relationships between the asymptotic growth rate and the spatial factor we are discussing. Figure S2: Estimated asymptotic growth rates (blue circles) are almost linear function of cubic root of the spatial factor h F . A blue line indicates theoretical values. The source and the destination functions are determined by using independently generated vector distributions. Note that the spatial factor can be below 1 because the source function ( , ) xy  is not equal to ( , ) S x y  as in the source-destinationmediated-dispersal models in the main text.