## Introduction

The extension of ecological models into the spatially explicit realm presents one of the most rewarding but also one of the most challenging aspects of model development. Traditionally, ecological models have focused on describing interactions between individuals in terms of the mean density of individuals in a population. Models derived from this so-called mean field assumption have provided many new insights in ecological theory but, without the inclusion of local interactions between individuals, the lack of spatial structure in these models can produce very different conclusions on crucial phenomena such as invasion speed and species coexistence than their spatially explicit counterparts (Ovaskainen & Cornell 2006; Murrell 2010).

Whilst the spatial element can, in some cases, represent a substantial leap in complexity, it can often elucidate the mechanisms of otherwise confusing observations. For example, the addition of spatial structure in models of predator–prey dynamics in Murrell (2005) and Kondoh (2003) have shown that equilibrium prey densities are negatively linked to the spatial covariance of the antagonists, which can increase when prey fecundity is increased. This extension thus provides an alternative spatial explanation for the ‘paradox of enrichment’ of Rosenzweig (1971). Moreover, some core principles of the theory of competition, such as the assertion that a high ratio of intraspecific to interspecific competition provides community stability (appearing in many text books such as Putman & Wratten 1984), have been shown to be incomplete when interrogated with models able to explicitly describe and simulate the spatial aggregation of conspecifics (Neuhauser & Pacala 1999; Murrell 2010). In applied ecology, spatially explicit models are also commonly used to describe the spatial arrangement of populations and dispersal of individuals, and have shown themselves invaluable in the context of reserve selection strategies and responses to climate change (for example Moilanen *et al.* 2005; Willis *et al.* 2009).

One of the crucial elements of a spatially explicit model is the specification on how this space is represented. Indeed, Murrell (2005) postulates that one of the reasons why the findings of Wilson, de Roos & McCauley (1993) appear to contradict the demonstration in Murrell (2005) that increased prey movement reduces the equilibrium population size is that the study of Wilson, de Roos & McCauley (1993) represents space as a discrete lattice of environments with each patch able to support a maximum of one individual. This type of stochastic cellular automaton is one commonly employed in ecological models (see Silvertown *et al.* 1992; Jeltsch *et al.* 1996; Mustin *et al.* 2009, for more examples), although other variants where populations of more than one individual (as implemented in Travis & Dytham 2002), or communities of more than one species (as implemented in Travis, Brooker & Dytham 2005), can inhabit a single cell are also used.

Whilst lattice models have the potential to provide many novel ecological insights (Nakamaru 2006), with some authors exalting these methods as a ‘paradigm’ (Hogeweg 1988), their simplification of spatial structure can lead to a number of biases in the interpretation of their output. No more so is this bias shown so prominently than in the methodologies employed to model dispersal through these habitats. The most basic simplification of dispersal, often denoted ‘stepping-stone’ dispersal or sometimes ‘nearest-neighbour’ dispersal (Kimura & Weiss 1964), defines movement as a local process where individuals can only move to adjacent lattice cells with some given probability, usually uniformly selected amongst the neighbourhood of cells (although see Topping *et al.* 2003; Wiegand *et al.* 2004, for other weighting methods). For rectangular lattices, different concepts of the neighbourhood are employed (see Milne *et al.* 1996): ‘Moore neighbourhoods’ define the eight neighbouring cells in the horizontal, vertical and diagonal directions as potential destinations for dispersing individuals (Topping *et al.* 2003; Wiegand *et al.* 2004, for example), whilst ‘von Neumann’ neighbourhoods consider only the four cardinally adjacent cells as potential destinations for dispersing individuals (Söndgerath & Schröder 2002, for example). However, Holland *et al.* (2007) show that both neighbourhood definitions can exhibit unnatural artefacts, both in terms of the spatial densities observed when considering multiple realisations of such defined dispersal events and the maximum traversable distance after a set number of time steps.

In continuous space, the probability density function of dispersal distances of a motile individual (or propagule in sessile organisms) from the point of origin is often referred to as the distance distribution (Nathan & Muller-Landau 2000), the circular distribution (Wilson 1993) or the distance pdf (Cousens, Dytham & Law 2008). These distributions describe the probability of the magnitude of a movement event but not its direction. In a one-dimensional world, the distance distribution is the folded equivalent of a displacement distribution, where displacement also accounts for the direction of movement and can therefore be negative. We can extend these one-dimensional descriptions of displacement into the spatial domain by describing dispersal in terms of its polar coordinates from the point of origin. For models with descriptions of dispersal in continuous space, there are a number of distributions of spatial displacement available to the investigator (see Clark *et al.* 1999; Cousens, Dytham & Law 2008). This is not the case for discrete lattice-based dispersal. Outside the simple stepping-stone models of dispersal there is a dearth of appropriate models for the calculation of cell-to-cell dispersal probabilities. To avoid confusion, the term ‘dispersal kernel’ will hereafter refer to the probability density function of displacement and not the probability density function of dispersal distance.

To address some of the deficiencies of stepping-stone models of dispersal, Chesson & Lee (2005) describe a number of families of integer-valued displacement distributions for use in lattice models of arbitrary dimensionality. These distributions have the flexibility to allocate non-zero probabilities of dispersal to cells beyond the nearest neighbours and hence can potentially provide a mechanism of dispersal not too dissimilar to their continuous counterparts. The distributions described in Chesson & Lee (2005) also exhibit a number of desirable qualities that make their development a significant step forward for incorporating more realistic dispersal in cell-based studies. Firstly, most of the distributions described in Chesson & Lee (2005) have functional forms that are closed under convolution. This means that when iterating the dispersal forward a number of time periods, total displacement is simply a re-parametrisation of the one-step displacement distribution. More generally, this means that we are able to parametrise the displacement distribution as a function of time. Secondly, each of the displacement kernels have a parameter controlling the kurtosis of the probability distribution and allowing flexibility in specification of the probability weight of the tails of the distribution. This is particularly useful for helping to include the effects of long-distance dispersal that often requires a ‘fat-tailed’ displacement distribution (Hovestadt, Messner & Poethke 2001; Petrovskii, Morozov & Li 2008). Finally, the displacement distributions of Chesson & Lee (2005) also exhibit asymptotic radial symmetry, which ameliorates some of the artefacts of lattice-based dispersal described by Holland *et al.* (2007).

Field data such as telemetry or seed shadow data are often used to parametrise continuous models of dispersal (see Greene *et al.* 2004), but such data are rarely applied so explicitly in the parametrisation of lattice dispersal, nor are such data collected in such a way as to be applicable in these settings. Whilst Chesson & Lee (2005) provide models of lattice dispersal with desirable mathematical properties, the underlying theoretical basis of these models is the mixture of a random quantity of stepping-stone dispersal sub-stages, requiring that individuals disperse cardinally with respect to the artificial geometry placed upon them within each of these dispersal sub-stages. On a two-dimensional grid, this means that although an individual can disperse further than the nearest neighbours, the final dispersal of the entire time step is comprised of a number of stepping-stone dispersal sub-steps, with each dispersal sub-step limited to movement within a von Neumann neighbourhood. It is difficult to see the theoretical link between such models and those that are commonly fitted to dispersal data. We adopt here a different approach and instead describe a general method for the approximation of continuous displacement distributions on lattices of arbitrary resolution. We use this methodology to derive approximate cell-to-cell transition probabilities for commonly employed models of continuous dispersal and describe how this method can be extended to allow for common boundary conditions and irregularly shaped source and destination patches.

For convenience, all notation used in this paper is summarised in Table 1.

Description | Support | |
---|---|---|

r | Dispersal distance | |

θ_{1} | Angle of dispersal measured in an anticlockwise direction from the positive x-axis | |

θ_{2} | Angle of dispersal measured in a clockwise direction from the positive y-axis | |

Step function defined in Eqn 2 | ||

j | Potential source coordinates (j_{x},j_{y}) | |

k | Potential destination coordinates (k_{x},k_{y}) | |

g_{·}(r,θ_{2}) | Two-dimensional dispersal kernel described in terms of polar displacement | |

c_{·}(j_{x},j_{y},k_{x},k_{y}) | Two-dimensional dispersal kernel described in terms of the source and destination Cartesian coordinates (see Eqn 6) | |

a_{x}a_{y} | The width and height of the simulation arena respectively | |

J | Source cell bounded by j_{x1} and j_{x2} on the x-axis and j_{y1} and j_{y2} on the y-axis | |

K | Destination cell bounded by k_{x1} and k_{x2} on the x-axis and k_{y1} and k_{y2} on the y-axis | |

K^{(i1,i2)} | Translation of the destination cell bounded by [k_{x1} + i_{1}a_{x}] and [k_{x2} + i_{1}a_{x}] on the x-axis and by [k_{y1} + i_{2}a_{y}] and [k_{y2} + i_{2}a_{y}] on the y-axis | |

Probability of moving from cell J to cell K, calculated according to the approximation method in the superscript brackets (CC denotes centroid-to-centroid, AC area-to-centroid, CA centroid-to-area, and AA area-to-area dispersal; Eqns 5, 9, 7, and 8 respectively) | ||

corrected for the incorporation of restricting boundary conditions (see equation S2-2) | ||

corrected for the incorporation of periodic boundary conditions (see equations S2-3, S2-4, and S2-5) | ||

J | A source patch consisting of N_{J} cells | |

K | A destination patch consisting of N_{K} cells | |

Probability of moving from patch J to patch K calculated using the underlying cell transition probabilities, , according to Eqns 12 and 13 | ||

P^{′′}^{(·)} | A transition matrix with each element, , containing the probability of moving to cell K from cell J with periodic boundary correction applied | |

A vector with each element, , containing the probability that an individual resides within cell J at time t according to the relevant approximation method | ||

w_{tJ} | Probability that an individual resides within cell J at time t under a continuous Gaussian diffusion process (see Eqn 18) | |

w^{′′}_{tJ} | w_{tJ} with periodic boundary correction applied (see Eqn 19) |