Figure 1.
Realisation of three cellular lattice landscapes (512 × 512) with the same proportion of suitable habitat (73728 cells ca 28%) but generated using different neutral models (after [33]).
Black cells denote suitable habitat. Figure A depicts a landscape generated by selecting the relevant proportion of cells at random and assigning suitability to these cells. Figure B is a landscape generated by hierarchical curdling using three stages. In the first stage, the landscape is divided into 6464 size tiles and then 75% of these tiles are selected as containing suitable habitat. The second stage of the process involves dividing the selected tiles from the first stage and dividing them into 8
8 tiles. 75% of the 8
8 tiles are selected in each of the coarser resolution tiles selected in the previous stage. The final stage selects 50% of the cells contained in the selected tiles of the previous stage and assigns them as suitable habitat. Figure C shows a landscape generated by fractional Brownian motion with a Hurst exponent of 0.2 using the algorithm outlined in this manuscript. The continuous output from this process is divided into suitable and non-suitable habitat by ranking the cell values and selecting the top 73728 as suitable.
Figure 2.
Three realisations of landscapes (256 × 256 cells) created by the spectral synthesis algorithm for fractional Brownian motion.
Figure A is generated using a Hurst exponent of 0.1, figure B uses a Hurst exponent of 0.5, and finally, 0.9 is used as the Hurst exponent in the generating mechanism for figure C.
Figure 3.
Illustration of neighbourhood definitions used in construction of the spatial autocorrelograms used in this study.
Figures A–D display networks of cells connected with links running at angles of ,
,
, and
clockwise from the
-axis respectively. The boundaries of the network in each spatial dimension are wrapped around to form a torus. Red lines denote the network links between the cell centres.
Figure 4.
One realisation of landscapes generated using each of the two algorithms described in this paper with accompanying directional autocorrelograms.
Figures A and B are realisations of a landscape generated by employing algorithms 1 [24] and 2 (adapted from [23]) respectively. Both landscapes have been generated using a Hurst exponent of
. Figures C and D are the accompanying directional autocorrelograms to figures A and B respectively with Moran's
calculated over each of the four networks shown in figure 3. Blue triangles represent values of Moran's
that are significantly larger (
) than what would be expected in a random landscape Red inverted triangles denote values of Moran's
significantly smaller (
) than what would be expected in a random landscape.
Figure 5.
Series of box plots showing the difference in autocorrelation measured using Moran's along lines parallel to the two cardinal axes.
Figure A displays the results for landscapes generated by employing both synthesis algorithms with a Hurst exponent of (high heterogeneity). Figure B displays similar results for landscapes generated with a Hurst exponent of
(intermediate heterogeneity) whilst figure C shows results for landscapes generated using a Hurst exponent of
(low heterogeneity). The dashed red line shows the location of no difference between the autocorrelation measured in each axis direction (where
Moran's
is zero) and represents the expected median for a series of isotropic landscapes. Lighter coloured boxes show the inter-quartile range of the results from a synthesis algorithm with a median of
Moran's
closer to zero at the respective distance class than the alternative algorithm. Conversely, darker coloured boxes indicate that the magnitude of the median value of
for a given synthesis algorithm exceeds that exhibited by landscapes generated using the alternative. The notches of the box plots extend from the median to
multiplied by the inter-quartile range divided by the square root of the sample size (in this case 100) representing a rough 95% confidence interval for the median based on asymptotic normality [35]. The box whiskers extend to the full range of data values.
Figure 6.
Series of box plots showing the difference in autocorrelation measured using Moran's along lines parallel to the two intercardinal axes.
Figure A displays the results for landscapes generated by employing both synthesis algorithms with a Hurst exponent of (high heterogeneity). Figure B displays similar results for landscapes generated with a Hurst exponent of
(intermediate heterogeneity) whilst figure C shows results for landscapes generated using a Hurst exponent of
(low heterogeneity). The dashed red line shows the location of no difference between the autocorrelation measured in each axis direction (where
Moran's
is zero) and represents the expected median for a series of isotropic landscapes. Lighter coloured boxes show the inter-quartile range of the results from a synthesis algorithm with a median of
Moran's
closer to zero at the respective distance class than the alternative algorithm. Conversely, darker coloured boxes indicate that the magnitude of the median value of
for a given synthesis algorithm exceeds that exhibited by landscapes generated using the alternative. The notches of the box plots extend from the median to
multiplied by the inter-quartile range divided by the square root of the sample size (in this case 100) representing a rough 95% confidence interval for the median based on asymptotic normality [35]. The box whiskers extend to the full range of data values.
Figure 7.
Three samples, taken at times 0, 10, and 20, from two time series of landscapes (128 ×128 ×128) generated using the spectral synthesis algorithm described in this paper.
Both landscapes are generated using Hurst exponents of 0.9 in the spatial dimension but figure A uses a temporal Hurst of 0.1 and figure B uses a temporal Hurst of 0.9.
Figure 8.
Combining an environmental gradient surface with local environmental heterogeneity to provide a composite landscape (see [31]).
Figure A illustrates a simple gradient landscape (256256) with each cell value set to one minus
multiplied by the absolute distance (in number of cells) of the y-coordinate of the cell centre from the middle of the landscape. This provides a piecewise linear gradient with values bounded between zero and one symmetrically decreasing from the centre of the landscape. Figure B illustrates a fractal landscape (256
256), generated using the spectral synthesis algorithm described in this paper with a Hurst exponent of 0.5, and normalised so that all values fall between the range of zero and one. Figure C shows a simple combination of these landscapes, 0.5 multiplied by the values of gradient landscape plus 0.5 multiplied by the values of the fractal landscape, to produce another landscape with values still bounded between zero and one.