A discrete space model for continuous space dispersal processes

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Abstract

The simulation of dispersal processes in landscapes over large spatial extents is challenging because of the large difference in geographical scale between overwhelmingly dominant localised dispersal events, and rare long-distance dispersal events which typically drive overall rates of spread. While localised dispersal may point to high resolution individual level models, long-distance dispersal events are likely to involve much coarser grid-based models. In this paper we propose a discrete space (i.e., grid-based) model for dispersal processes in continuous space. We start by illustrating the behaviour of continuous space walks when their movement is discretised to a grid. The importance of short time period cell-to-cell moves which return a walk to its previous grid cell location is identified. A conceptual model which uses a Markov chain buffer phase between cells to replicate the observed behaviour of discretised continuous space walks is proposed. Analysis of the Markov chain shows that it can be parameterised using just two parameters in addition to the dispersal kernel. An algorithm for implementation of the proposed model is presented. Empirical results demonstrate that the proposed mechanism produces good matches to continuous space dispersal processes with both exponential and heavy-tailed dispersal kernels.

Introduction

The movement of organisms in spatially heterogeneous landscapes, and the role of such movement in driving fundamental ecological processes is receiving growing attention from ecologists (Holyoak et al., 2008, Nathan et al., 2008, Schick et al., 2008). Dispersal, whether active or passive, is fundamental, to such ‘movement ecology’, but modelling dispersal processes poses some difficult scaling-related challenges. At one level, the processes underpinning movement, such as micro-turbulence in the dispersal of seeds by wind or the behavioural decisions made by foraging animals, are extremely fine-grained in space and time. These fine-scale processes, however, may drive ecological dynamics over large spatial extents and over long periods of time (Levey et al., 2008, Perry and Enright, 2006). Integrating processes across multiple scales make trade-offs between the spatio-temporal extent of models of dispersal processes and the grain over which they operate inevitable (see Wiens, 1989). While it is possible to have a coarse-grained model that considers a broad extent, or a fine-grained model that considers a small(er) extent, it is not usually possible, or even necessarily desirable, to combine broad extent with fine-grained detail.

Nevertheless, advances in computer power and software design have enabled individual-based models (IBMs) to operate over ever increasing extents. For example, when individual-based ‘forest gap’ models first appeared in the 1970s they considered only a few hundred individuals over an area of approximately 0.1 ha (see Perry and Enright, 2006). Current state-of-the-art forest gap models can consider more than 2 × 107 individuals over areas up to 40 km2 (e.g., Chave, 2001). In theory at least, as computer power continues to increase it will be possible to represent more and more individuals over larger and larger areas. However, the implementation and analysis of such models is bound to become ever more difficult, and computational upper-bounds to the number of individuals that can be represented will remain. Furthermore, even highly detailed models will most likely be presented in summary forms, using population totals of individuals on a grid cell-by-grid cell basis, raising questions about how critical it is to actually run the models at such high levels of detail. Thus, there remains a need to develop techniques that will, at least partially, help to resolve the problems associated with the grain-extent trade-off faced by modellers in ways other than the ‘brute-force’ simulation of more individuals over larger areas.

As an example of a situation where such methods would be useful, consider the spread of an invasive plant species in a large urban area such as Auckland (New Zealand). The greater Auckland area covers an area of around 100 km × 100 km meaning that even comparatively coarse grid cells of 100 m × 100 m will result in a grid of 1 × 106 cells. However, most plant species disperse their seeds locally, with seeds only rarely travelling more than a few meters from the parent plant. Although occasional long-distance dispersal (LDD) events are very significant in the invasion process, typical seed dispersal kernels are strongly leptokurtic (i.e., heavy-tailed) (Greene and Calogeropoulos, 2002). The use of individual-based models to explore such landscape-level vegetation dynamics poses challenges of spatial scaling. In particular, a key question is at what spatial grain (resolution) modelling and analysis of seed dispersal should focus. While many seed dispersal studies have considered local dynamics (i.e. distances of less than ≈ 100 m) it is increasingly clear that understanding the invasion process requires considering dispersal within the wider landscape (Levey et al., 2008, Levin et al., 2003, Nathan, 2006).

This leads to the difficult computational problem of dealing with high resolutions (to capture local dispersal from individual plants) over large spatial extents (to capture LDD and broad-scale landscape heterogeneity). Promising new approximation methods that reduce the computational expense of simulating dispersal of seeds from millions of individuals in IBMs have begun to appear (e.g., Govindarajan et al., 2007), but such methods do not solve the problem of representing individual entities over large spatial extents. One way to handle these problems is to use coarse grid-based representations of the environment, and simply store the number or density of individuals within each grid cell. Grid-based (lattice) models are frequently used to explore the spatio-temporal dynamics of ecological communities, in both theoretical and real settings (Wissel, 2000). In theoretically motivated lattice models dispersal often simply represents as movement of seeds to a focal cell's nearest neighbours. In addition, numerical methods have been developed that allow representation of leptokurtic dispersal kernels on integer lattices (i.e., in discrete space, see Chesson and Lee, 2005). In grid-based models, the spatial location of individuals in each grid cell is not explicitly represented. Rather the number of individuals in each cell is stored (e.g., Pearson and Dawson, 2005) making such models spatially implicit at the level of individual grid cells. In the simplest case individual grid cells are simply characterised as being either occupied or empty (e.g., McInerny et al., 2007, Travis, 2003). While grid-based models are computationally much less demanding than IBMs, it is unclear to what extent representing the landscape as a coarse grid biases predictions of the rate of spread of expanding populations. Such issues are likely to affect the simulation of any diffusive process over large spatio-temporal extents.

Our goal in this paper is to explore the feasibility of using a discrete space random walk to approximate continuous space dispersal processes. Our approach targets the type of dynamics arising from the dispersal of plant propagules rather than organisms which actively explore and forage in the environment (although it is potentially extendable to cases such as colonial insects). We first observe some statistical properties of continuous random walks when they are discretised to a grid. Based on these observations, we develop a simple conceptual model for continuous random walks across a grid, and develop an approach to parameterise it by combining results from classical geometric probability with a simple Markov chain approximation for grid cell-to-grid cell movements. We then show, based on simulation results, that our model produces overall rates of spread similar to those of a continuous space individual-based dispersal model. We conclude with advice on how our model can be parameterised for any dispersal kernel, and identify limitations and possible extensions of the model. The Markov chain-based method that we have developed allows us to model dispersal, which is typically dominated by short-distance events, on coarse grids over large spatial extents. Thus, it is applicable to a wide range of ecological problems including the migration of species in response to climate change and the spread of invasive species.

Section snippets

Continuous space random walks and the problem of discretisation

To determine the most suitable discrete space approximation for continuous space dispersal with significant long-distance dispersal (i.e. a heavy-tailed distribution of dispersal distances), we established a simple experimental framework. Simple random walks in continuous space are treated as the baseline case. While this ignores the additional complexities associated with any particular organism's dispersal—factors such as numbers of seeds, establishment, etc.—if a discrete space random walk

Results

We report results for a number of comparisons between simple continuous random walks and our discrete space approximation. In each experiment all walks originate from the centre of a unit grid cell at location (0, 0). Where real-valued coordinates are known, as for the continuous random walk baseline case, when analysis is carried out in terms of grid cell locations and grid cell-to-grid cell movements, these are determined by rounding the real-valued coordinates to the nearest integer. The

Discussion

The proposed Markov chain mechanism has good potential for producing a discrete space approximation to continuous space random walks, which matches the overall characteristics of continuous space walks with the same distribution of walk step lengths. A match is possible both in terms of (i) the overall rate of progress of the walks across space, and (ii) the behaviour of the walks with respect to movements from cell-to-cell, particularly the oscillatory behaviour seen in Fig. 2, Fig. 3. This is

Conclusions

The proposed discrete space model shows considerable promise as a basis for extensive yet detailed models of dispersal processes. For the classes of dispersal process considered here, the model is readily tuned in a simple two stage procedure. First, the desired rate of spread is chosen by setting the b parameter. Then, independently, the entry–exit time frequency distribution of the continuous space process can be matched by choosing an appropriate value of s. The relative independence of the

Acknowledgments

The authors are grateful to Jürgen Gröneveld who commented on a draft of this paper and to Mark Holmes who provided insightful comments at a critical stage. Errors and omissions which remain are of course the responsibility of the authors.

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