Landscape as a Model: The Importance of Geometry
[['The focus of this study is spatially explicit predictive models designed to support decisionmaking ( e . g . , population establishment and spread , climate change , and flood risk ) , which should have reliable , probabilistic , and mappable results .', 'In cases in which there are few relevant validation data ( e . g . , nonnative species and climate change ) , the model cannot be calibrated statistically , and it is therefore important that biases and uncertainties are dealt with explicitly so that confidence can be placed in the results .', 'Uncertainty may surround all components of a model ( e . g . , input data and processes ) , but bias by definition usually results from the way that processes are implemented in the model .', 'In this study , we explored how spatial structure can be a source of bias , and present an approach that allows uncertain landscape data to be incorporated into model output with minimal bias .', 'There are many different landscape models in the literature ( see [1] for a recent and comprehensive list ) , all of which allow a process ( population ) model to interrogate explicit locations or regions of space , and choosing the most appropriate landscape model for the study in hand is important [1] .', 'We focus on the use of cells in a mosaic-based model [2] to represent processes in space , which requires the subdivision of space into a tessellation of discrete , internally homogeneous patches within which a process occurs .', 'Although this is an elegant , abstract concept , the use of cells to represent uncertain spatial processes is often desirable in real-world applications .', 'First , some information is better represented by an areal unit than by a point location ( e . g . , water ) , whereas other information is considered to conceptually occupy an area of real space defined by its boundary ( e . g . , an animal social group ) .', 'Second , the limited understanding of many of these systems requires us to model at the scale for which most is known ( e . g . , the behaviour of individuals within a social group of animals is often not well-understood , whereas the size , productivity , or spatial description of the whole population may be simpler to study and is well-described ) .', 'Third , the raw data used to describe the landscape ( e . g . , satellite and aerial photography ) are subject to errors and uncertainties .', 'By modelling processes at scales significantly larger than that of the underlying data , these problems become statistically tractable ( e . g . , Land Cover 2000 [3] ) .', 'Although in many such models , the attribute values of cells are directly calculable from habitat or geographical data ( e . g . , vegetation type ) , here we use conceptually abstract and attribute-free cells in order to consider only geometry ( specifically , shape ) and neighbourhood ( number and arrangement of adjacent , interacting patches ) in a homogeneous landscape .', 'In this study , we have used population modelling concepts to demonstrate the potential for bias in cell-to-cell movement of information ( e . g . , individuals ) resulting from the geometry of a mosaic virtual landscape .', 'Population models predominately use raster virtual landscapes , and the description of home ranges or social groups with single squares ( e . g . , [4–6] ) or a square arrangement ( e . g . , [7] ) is not unusual .', 'Cell-to-cell movement is implemented using either von Neumann ( e . g . , [8] ) or Moore ( e . g . , 6 , 9–11] ) neighbourhoods ( four or eight neighbours , respectively ) , often with some directional component [12–14] .', 'Some studies note that the geometry of the virtual landscape has the potential to affect simulation results [15 , 16] , and the interaction strengths of orthogonal and diagonal neighbours in rasters are sometimes weighted using an appropriate algorithm [16 , 17] .', 'Landscape permeability is sometimes defined using raster cells ( e . g . , [18] ) , although some authors have suggested using multiscaled rasters to represent patchy landscapes ( e . g . , [19] ) .', 'Other studies use hexagonal geometry for both spatial analysis [20 , 21] and modelling [15 , 22–25] because the strengths of all neighbourhood interactions are equal .', 'Irregular ( variable shape and size ) geometry has been used extensively in population modelling , but usually to parameterize or display discrete , spatially disparate habitat patches with explicit connectivity based on the distance between patches [26–28] or vector-based movement rules [29–31] , and studies such as that by Ovaskainen [32] exemplify this approach .', 'A few models use tessellated irregular shapes across a whole landscape , and implement cell-to-cell movement as part of the simulation [33 , 34] , whereas Dunn and Majer [35] suggest that Voronoi ( Dirichlet ) cells are a convenient way to represent multiply scaled data , but they do not go into detail about dispersal mechanisms .', 'However , no attempt has been made to specifically test how geometries other than rasters may affect movement in a mosaic landscape , and an explicit consideration of geometry does not appear to be an integral part of most population modelling studies .', 'The representation of real-world landscapes is complex , with the description of features represented as discrete objects subject to both qualitative and quantitative variability , uncertainty , or both [36] .', 'Uncertainty within virtual landscapes is already considered in some disciplines ( e . g . , [37 , 38] ) .', 'We suggest that all spatially explicit population models should consider how uncertainty in landscape representation may affect model output , just as sensitivity analyses on process model parameters have become standard practice .', 'Clearly this is entirely dependent on the nature of the study undertaken ( data , scale , structure , and discipline ) , so we cannot begin to describe how individual studies in diverse disciplines should address this issue .', 'However , it seems inevitable that population modellers will adopt a probabilistic approach to spatial studies ( which can be easily implemented through the use of alternative landscapes in successive runs of the model ) , and we provide a mechanism by which minimally biased landscapes can be created .', 'We created landscape mosaics with raster , hexagonal , and irregular geometries with which to model and compare the cell-to-cell exchange of information .', 'We are not aware of any other study that directly compares the potential for systematic bias in the movement of information across the spectrum of possible geometries of mosaic virtual landscapes .', 'We highlight how the geometry of cells in a raster virtual landscape affects both qualitative and quantitative aspects of spatial representation of irregular shapes , but leave the attributional representation of features ( e . g . , heterogeneous habitat [39 , 40] ) and subsequent impacts on movement or process [41 , 42] to another study .', 'We believe that these concepts are generally applicable across a broad range of spatially explicit modelling disciplines .'], ['Cells in the raster and hexagonal virtual landscapes had a fixed number of neighbours ( Table 1 ) .', 'Cells in the Dirichlet landscape had a mean of exactly six neighbours , though there was variation about this value within individual landscape instances .', 'The CGD virtual landscapes all resembled pixelated versions of the Dirichlet landscape .', 'However , both the visual and mathematical approximation improved as the resolution of the underlying raster was increased , as demonstrated by both the mean and standard deviation of the number of neighbours ( Table 1 ) .', 'Cells in the aggregate map had approximately six neighbours , and were a range of shapes because the sequential building rules meant that growing cells were often geometrically constrained by neighbours .', 'Of the geometries tested in this study , the mean number of neighbours of a cell was six , or its approximation , with the exception of the rasters .', 'There was variation in the distribution of cell sizes within the irregular virtual landscapes ( Table 1 ) .', 'We measured and compared all possible unique cell-to-cell step lengths ( measured between centre-of-mass centroids ) in five landscapes: the three regular landscapes , and single instances of the Dirichlet and the CGD4 landscapes ( Figure 3 ) .', 'In the von Neumann and hexagonal landscapes , only one step length was ever possible , with lengths 1 km and 1 . 074 km , respectively .', 'In the Moore landscape , two steps were equally probable , with lengths 1 km and 1 . 41 km producing a mean step of 1 . 21 km per landscape .', 'Step lengths in the Dirichlet landscape were gamma distributed ( Figure 3 ) with a mean of 1 . 095 km , which is close to that found in the hexagonal landscape; the step lengths of each cell in the CDG4 landscape were similarly distributed with a mean of 1 . 18 km , though the distribution was less smooth as a result of the finite distribution of cell shapes and hence step lengths ( Figure 3 ) .', 'We measured the land area from a number of raster depictions of a fine-scale vector description of a real-world object ( United Kingdom ( UK ) coastline ) .', 'Rasters were created at a range of scales with a variable origin ( shifted successively by 10% of the resolution west and south ) .', 'A further ten rasters , at a resolution of 10 km , were created with a fixed origin but with the orientation of the grid rotated successively by 9° .', 'The qualitative form of a raster representation of the UK differed with a change in origin or orientation of the grid ( Figure 8 ) .', 'Small objects such as islands appeared or disappeared , became connected or disconnected from the mainland or each other , or changed their shape radically .', 'This would have clear effects on any model involving terrestrial movement .', 'Although we have only shown this at one scale , these undesirable effects are fractal , and would be present as a possible bias at all scales , and would worsen at larger resolutions [44] .', 'The mean area reported by the raster representations of the UK decreased with increasing scale ( Figure 9 ) , though as a fractal property , would show bias at all scales .', 'The best mean estimate ( 99 . 9% of the vector original ) was derived from the finest scale raster ( 1 km ) , but mean estimate of national area fell to as low as 98 . 5% at a 100 km resolution .', 'In addition , the variance around these mean figures was considerable and also increased with scale .', 'The worst performing rasters ( two out of ten instances at a 100 km resolution ) showed an area of only 89 . 6% , an alarming loss of 10 . 4% of the British land surface .', 'The estimates of national area produced by iteration across multiple rotations of a raster grid at 10 km resolution had a mean of 100%; individual instances ranged from 99 . 3% to 100 . 6% of the vector original , suggesting that at this scale , the orientation of the raster had only a small quantitative effect on area .', 'All forms of raster representation have biases , variant with scale .', 'In contrast , an irregular geometry ( specifically a vector representation ) can be subdivided into as many randomized vector cells as necessary .', 'Any estimate of gross area , length , or geographic property is perfect ( zero bias ) .', 'When multiple instances of virtual landscapes are required , we recommend the use of irregular geometry to avoid introducing bias to representation of its extent .'], ['Simple random walks in virtual landscapes with regular geometries can produce enormous qualitative biases in the direction and extent of movement and hence bias the distribution of populations in space .', 'For all our investigations ( accessibility , random movement , and directed random movement ) , the regular geometries performed poorly at some or all scales for measures of both distance and direction .', 'The nature and strength of the bias was , in part , a function of the length of the movement and the resolution of the landscape , with the potential for different biases to worsen after both short-distance and long-distance movement .', 'In an ideal homogeneous virtual landscape , the distance travelled by an individual ( from its origin ) after random movement should be independent of the direction travelled at each step .', 'Regular grids all restrict the direction of movement to the same few angles at every step , so the final positions of individuals cannot be independent of the grid structure .', 'Even if the scope of the movement neighbourhood is extended to include more distant cells ( e . g . , to include the 16 cells adjacent to the eight immediate neighbours in the Moore neighbourhood ) , so that single steps may include jumps over the immediate neighbours , a regular geometry restricts the available directions to some degree .', 'In comparison , the direction of neighbouring cells in a single , irregular virtual landscape is not set by the geometry , and when enough multiple irregular virtual landscapes are considered together , available directions assume a uniform circular distribution .', 'We demonstrated that , in the same way that regular landscapes restrict the available directions for movement , they severely restrict cell-to-cell step lengths , and in part , this may explain a component of the bias in movement .', 'The hexagonal landscape is the optimum arrangement of circles packed in space using a single iteration .', 'Dirichlet landscapes emulate circular cells over many iterations ( Figure 10 ) , and the mean step length closely approximates that of the hexagonal landscape , even in a single iteration .', 'Although the step lengths in the CGD landscapes followed the same general distribution as the vector Dirichlet landscape , individual step lengths were more or less frequent than expected .', 'We suggest that although the results of random walks appeared similar in all the irregular landscapes , a smooth distribution of step lengths is less likely to be a source of bias , and therefore , at fine scales in particular , we recommend the vector Dirichlet landscape over the CGD irregular landscapes .', 'We propose that the average number of neighbours per cell is a useful metric for quantifying the potential for landscape geometry to introduce bias in movement .', 'A mean close to six neighbours appears to be the ideal; all the irregular landscapes have this property .', 'Although the hexagonal landscape has six neighbours per cell , the variance is zero , and therefore directional movement must be restricted; we therefore extend the metric to include a nonzero variance .', 'The square geometries fail in both respects .', 'Because we have identified the potential for bias in movement in regular geometries , we believe that studies need to show ( not assume ) that movement in their own spatial models produces no bias .', 'Bias due to geometry may not be apparent in models with complex rules for cell to adjacent cell movement , e . g . , dependence on heterogeneous landscape quality ( e . g . , habitat preference [6] and permeability [47] ) .', 'Where sources of geometrical bias have been identified , individual studies have attempted to compensate with a variety of approaches [16 , 17] .', 'Not only is this compensation dependent on the spatial and temporal scale of the model for which it is designed , it adds more potential for bias to the model ( albeit pulling in the opposite direction ) and may adversely interact with other components of the model .', 'There is no way to be certain that biases masked at one scale will not produce artefacts when a predictive model is extrapolated in time or space .', 'In contrast , there is no potential for bias in both the direction and distance of movement of individuals across virtual landscapes with irregular geometry .', 'Some of the deficiencies of cell-to-cell movement across regular landscape geometries identified here might be overcome by iteration of rasters , specifically by randomizing origin and orientation .', 'However , this process can change the quality of the representation of the whole landscape , and it is this that we concentrate on here .', 'The dependence on scale in the adequate representation of complex shapes with rasters has been well-discussed elsewhere [44 , 48–52] .', 'Our representation of the UK coastline , and the measurement of its area , although not novel , allowed us to focus specifically on the qualitative and quantitative effects of using multiple regular geometries to represent a complete extent .', 'We used the UK landmass as an example of a real-world object that has an irregular extent whose boundaries could not be defensibly redefined as a regular shape for a national-scale model .', 'The representation of real-world features with regular geometries must always be an approximation [50] whose adequacy can only be measured by model output .', 'We have shown that at any finite scale , a feature represented by a single virtual landscape with regular geometry will always show qualitative and quantitative errors .', 'Some of these errors can be extreme , and the modeller choosing a single instance of a regular geometry with which to represent the landscape has no way of knowing how adequate it is without testing several representations .', 'Biases in the mean output could be reduced through iteration of regular geometries ( with random origin and orientation ) and a careful choice of scale , but the interpretation of results would only be acceptable where the bias was quantified .', 'In contrast , any landscape subject to a spatial modelling study can be split up into irregularly shaped cells with no detrimental qualitative or quantitative effect on the representation of the whole .', 'If accuracy in feature representation is important , the superior alternative to the raster is the irregular and vector-based mosaic .', 'The problems of feature representation by regular geometries are compounded by the connection between scale and structure; the form and significance of any feature alters as soon as the scale is changed .', 'This is especially problematic where interacting processes occur at different scales in the same landscape .', 'Because scale has such a strong effect on the properties of model output , it is a pity that studies suggesting quantitative methods for determining the most appropriate scale have not been pursued ( but see [53] for a generic approach , [54 , 55] for more specific applications ) .', 'In the absence of quantitative rules , modellers have to rely on common sense or experience to choose scale , and therefore should clearly demonstrate that their choice is appropriate by showing that the bias produced by the model at that scale is acceptable .', 'The choice of resolution in many raster-based models to date is often either derived from technical data ( e . g . , satellite imagery ) ( e . g . , [39 , 56] ) or chosen to be the nearest integer measurement of an apparently relevant process ( usually biological; e . g . , [4 , 6 , 57] ) .', 'We have shown that the potential for bias is always present in regular virtual landscapes even at high resolutions , and the impact of that bias is a function of scale .', 'If only one raster landscape is used in a model , its origin and orientation appear arbitrary yet are usually unchallenged ( this is apparent from the lack of ability to rotate rasters away from a north–south orientation in some GIS packages ) .', 'If the resolution of the raster is high enough , the representation of features will be little changed by origin and orientation , but model processes may be affected .', 'However , if the data or the process suggests modelling at a low resolution , using a raster must sorely compromise the representation of the landscape .', 'Using irregular mosaic landscapes solves two problems .', 'First , the interaction of bias with scale is removed from the process model ( c . f . , directional bias in movement in this study ) .', 'Second , the quality of the spatial representation of available data no longer depends solely on resolution; landscapes formed from vector-based , irregular cells can remain faithful to the available data at any scale and in any single instance .', 'The modeller is thus free to set a scale appropriate to other model processes .', 'Lindenmayer , Fischer , and Hobbs [1] emphasize that the ability to choose one of a number of landscape models is important in fauna research .', 'We suggest that , in the mosaic-based landscape paradigm , the automatic use of multiple landscape models should be widespread .', 'Real landscapes comprise irregular and complex shapes [58] that can only be well-described using irregular cells .', 'There will always be quantitative uncertainty in the location and boundary shape of cells and qualitative errors in their internal description even if irregular virtual landscapes are created deterministically from underlying habitat patch data rather than randomly ( e . g . , Dirichlet polygons ) .', 'The only way to explore and incorporate the uncertainty associated with landscape representation is to model with multiple , alternative virtual landscapes and present results as probabilistic maps ( e . g . , [59] ) .', 'This applies equally to regular and irregular mosaic landscapes .', 'The computational demands of running irregular models such as the ones in this study are not necessarily more than those of a raster model .', 'Because the movement rules are so simple , all that is required to implement movement is a list of cell IDs and associated neighbours .', 'All the investigations in this study were run in Python , only using the GIS for creating and displaying landscapes .', 'We admit that preparation of an irregular landscape set requires some extra work; however , if it is accepted that multiple virtual landscapes are necessary ( e . g . , different random centroids for Dirichlet tessellations , and different origins and orientations for rasters ) , the effort in preparing irregular and regular landscapes becomes almost indistinguishable .', 'The irregular landscapes also appear to yield greater returns , since they are scale-independent and bias-free , and can represent features within the landscape as well as the available data allow .', 'It is worth pointing out that the ideal vector implementation of an irregular mosaic landscape can be approximated very easily by a raster aggregate; the use of models such as the CGD ( Figure 1E–1G ) brings the benefits of unbiased movement , although the best representation of edges and features is only achieved by using a very high resolution raster .', 'We have shown that all single instances of irregular mosaics have similar structure ( number of neighbours ) and statistical properties ( e . g . , step length distribution ) , which results in scale-free and similarly unbiased movement of information ( populations ) .', 'However , the structure and statistical properties of regular mosaics differ from each other and from that of the irregular mosaics , resulting in biased movement that cannot be easily compared .', 'We suggest that two spatial models using irregular virtual landscapes of any scale may be more easily compared than those using regular geometry , which must have both the same scale and the same structure , and therefore the same bias .', 'We believe that the use of irregular geometry to form virtual landscapes may bring many additional benefits .', 'We illustrate these with examples from population modelling , in which information has an integer form ( e . g . , individuals ) , but the principles should be applicable in many disciplines , including those concerned with the movement of infinitely divisible information ( e . g . , water ) .', 'The area and shape of cells in an irregular geometry can vary across the extent of the virtual landscape so that regions requiring a detailed spatial description ( complex habitat patches , linear features such as rivers , etc . ) can be represented either with a greater density of irregular cells or exclusively with a single cell .', 'This property of irregular landscapes has also been identified by Dunn and Majer [35] and is analogous to the raster approach of Tischendorf [19] .', 'In turn , this permits an improvement in the description of cell attributes and reduces their uncertainty .', 'This is especially important where cell attributes have the ability to affect the spatial output of the model ( e . g . , landscape connectivity , patch permeability , and population persistence [60] ) .', 'We have validated how the aggregation of real-world features into irregular cells can provide a sufficiently irregular virtual landscape to avoid bias ( i . e . , aggregate map , Figure 1H ) .', 'Another useful benefit of irregular geometry is illustrated by considering a fixed point in space .', 'A single irregular cell containing this point is obviously not circular , but cells taken from sufficient iterations of the virtual landscape will approximate a circular kernel around the point ( Figure 10 ) .', 'This concept is useful both in describing individual behaviour ( i . e . , zones of perception ) or group dynamics ( i . e . , social interaction and density dependence ) .', 'Most dispersal kernels in continuous space implicitly define movement to nearest neighbours as the most frequent , with vector movement resulting in individuals moving preferentially to sites that are close [61] .', 'By using irregular cells , such kernels can be tuned with biological and geographical realism ( e . g . , interaction groups are bounded by major roads or coastline , perception zones do not include impenetrable habitat , and movement cannot cross rivers; see Figure 10 ) .', 'Finally , we observe that the geometry and size of real-world processes and objects ( such as home ranges , social group territories , habitat patches , and habitat quality ) are irregular and variable ( [58]; specific examples include the spatial arrangement of subpopulations of rabbits [62] , badgers [63] , and coyotes [64] ) .', 'There is a significant body of literature in identifying habitat patches in real-world landscapes ( e . g . , [65–67] ) .', 'Their subsequent representation with single cells of regular geometries is inappropriate .', 'In addition , if there are few data on how cells are formed in the real landscape , the most defensible way of expressing them in a model is through multiple , randomly generated ( irregular ) mosaic landscapes .', 'This study was prompted by a desire to construct a universal framework within which uncertainties in landscape description could be included explicitly in model function and results , and in which movement could be modelled as simply as possible in an unbiased , generic , and flexible manner .', 'A single virtual landscape formed with regular geometry has a huge potential for bias .', 'In contrast , this study has shown that even a single virtual landscape formed with irregular geometry has no potential to bias the direction or distance of movement of information ( e . g . , individuals ) , and although the defensible use of a random property ( in this case , geometry ) requires multiple instances , the variation between irregular landscapes is small .', 'Any representation using a regular geometry is at best a good approximation .', 'Even when multiple instances of regular geometries with random origins and rotations are measured , the mean output still has the potential for bias , and the variation between instances is large .', 'The representation of an extent by an irregular geometry shows no error .', 'We recommend the use of irregular geometry and multiple random instances in creating any virtual landscape , which eliminates bias in the movement of information and the representation of real-world extents .', 'Both are specifically recommended for models that are designed to help make decisions , so that the probabilistic output encompasses the uncertainty in both population processes and spatial representation .', 'As this is the first study recommending the use of irregular geometries , and we have not covered issues of internal representation ( e . g . , small features and heterogeneous landscape quality ) , it is difficult to state unequivocally that they are completely superior to regular geometries , but the results presented here suggest that they should be the first choice for modelling virtual landscapes .'], ['Our virtual landscapes were created to have a mean cell area of 1 km2 across the study area , which was a 25 , 361 km2 area of southeast England .', 'The two rasters used an arbitrarily chosen origin of ( 0 , 0 ) on the British National Grid ( BNG ) ; the placement of the hexagonal landscape was entirely arbitrary .', 'The Dirichlet landscape was formed by a vector tessellation using the ArcInfo Thiessen on 25 , 361 random points drawn from a uniform distribution within the study area .', 'The coarse-grain landscapes were created from three raster landscapes ( origin at 0 , 0 BNG ) , with 500-m , 333-m , and 250-m resolutions .', 'Within each raster , all squares were associated with , and then aggregated to , the nearest of 25 , 361 randomly chosen squares ( see Figure 11 and pseudocode at the end of this section ) .', 'The aggregate map was created from Land Cover 2000 [3] , a fine-scale vector habitat coverage .', 'A random habitat patch was selected and neighbours absorbed in turn until a total area of 1 km2 ± 5% was achieved .', 'The neighbour with the next-closest centroid was always chosen with the aim of maximizing circularity; other joining rules are possible , e . g . , habitat similarity , but the geometric optimization was chosen for its simplicity .', 'The neighbourhood of any focal cell in the hexagonal and irregular landscapes was defined as all cells in the landscape with a boundary ( or point on the boundary ) shared with the focal cell .', 'The interior of a virtual landscape was defined as all cells that were more than 1 km from its boundary .', 'The number of neighbours ( Ne ) and area ( Ae ) of interior cells were measured across all virtual landscapes and all instances on irregular landscapes .', 'The minimum , mean , maximum , and standard deviation of Ne and Ae were calculated for each virtual landscape .', 'Probes were developed to quantify the effects of landscape geometry on cell-to-cell movements , and are based on the following generalized approach .', 'In a discrete landscape , individuals move from one cell e into any neighbouring cell with probability 1/Ne , where Ne is the number of neighbours of e .', 'We compared the performance of the probes against the simplest vector random walk .', 'This unbiased benchmark moved an individual a fixed distance ( 1 km ) from its starting position ( x , y ) at a randomly chosen angle in ( 0 , 2π ) .', 'Movement probes always consisted of 10 , 000 independent individuals .', 'Irregular virtual landscapes used 1 , 000 individuals across ten randomized instances , with figures and statistics presented as the sum of the instances ( see Figure 10 ) .', 'For the accessibility probe , we calculated the minimum number of steps required for an individual in any cell in the landscape to access the origin ( cell containing BNG 448500 , 104500; southwest corner ) in all virtual landscapes .', 'We mapped the limit of the region accessible from the origin with 100 steps and measured the mean of the minimum number of steps required to reach a range of distances from the origin at 10° angles .', 'The random movement probe started in a cell containing the origin ( BNG 524500 , 179500; centre of virtual landscape ) .', 'Vector random walks started at the origin .', 'We used four sets of parameters ( probability of movement , p , number of time steps , t ) to simulate a variety of forced and unforced , short and long walks: ( p = 1 , t = 5 ) , ( p = 1 , t = 100 ) , ( p = 0 . 5 , t = 10 ) , and ( p = 0 . 5 , t = 200 ) .', 'Direct comparison of population displacements can only be made between probes where the product of p and t is equal , thus unforced walks last twice as long as equivalent forced walks .', 'The population distribution also was calculated for directed random walks , in which neighbouring cells were divided into “backward” ( toward the origin ) and “forward” ( around or away from the origin ) neighbours .', 'Fully directed ( no backward movement ) and semidirected ( 10% possibility of movement backward ) random walks were simulate
['Introduction', 'Results', 'Discussion', 'Methods']
['In all models , but especially in those used to predict uncertain processes ( e . g . , climate change and nonnative species establishment ) , it is important to identify and remove any sources of bias that may confound results .', 'This is critical in models designed to help support decisionmaking .', 'The geometry used to represent virtual landscapes in spatially explicit models is a potential source of bias .', 'The majority of spatial models use regular square geometry , although regular hexagonal landscapes have also been used .', 'However , there are other ways in which space can be represented in spatially explicit models .', 'For the first time , we explicitly compare the range of alternative geometries available to the modeller , and present a mechanism by which uncertainty in the representation of landscapes can be incorporated .', 'We test how geometry can affect cell-to-cell movement across homogeneous virtual landscapes and compare regular geometries with a suite of irregular mosaics .', 'We show that regular geometries have the potential to systematically bias the direction and distance of movement , whereas even individual instances of landscapes with irregular geometry do not .', 'We also examine how geometry can affect the gross representation of real-world landscapes , and again show that individual instances of regular geometries will always create qualitative and quantitative errors .', 'These can be reduced by the use of multiple randomized instances , though this still creates scale-dependent biases .', 'In contrast , virtual landscapes formed using irregular geometries can represent complex real-world landscapes without error .', 'We found that the potential for bias caused by regular geometries can be effectively eliminated by subdividing virtual landscapes using irregular geometry .', 'The use of irregular geometry appears to offer spatial modellers other potential advantages , which are as yet underdeveloped .', 'We recommend their use in all spatially explicit models , but especially for predictive models that are used in decisionmaking .']
['Many different areas of science try to simulate and predict ( model ) how processes act across virtual landscapes .', 'Sometimes these models are abstract , but often they are based on real-world landscapes and are used to make real-world planning or management decisions .', 'We considered two separate issues: how movement occurs across landscapes and how uncertainty in spatial data can be represented in the model .', 'Most studies represent the landscape using regular geometries ( e . g . , squares and hexagons ) , but we generated landscapes of irregular shapes .', 'We tested and compared how the shapes that make up a landscape affected cell-to-cell movement across it .', 'All of the virtual landscapes formed with regular geometries had the potential to bias the direction and distance of movement .', 'Those formed with irregular geometry did not .', 'We have also shown that describing whole real-world landscapes with regular geometries will lead to errors and bias , whereas virtual landscapes formed with irregular geometries are free from both .', 'We recommend the use of multiple versions of virtual landscapes formed using irregular geometries for all spatially explicit models as a way of minimizing this source of bias and error; this is especially relevant in predictive models ( e . g . , climate change ) that are difficult to test and are designed to help make decisions .']
['none', 'ecology', 'science policy']