Families of discrete kernels for modeling dispersal
Introduction
Spatially explicit models are of increasing importance in population biology, especially in ecology where such models are relatively recent (Dieckmann et al., 2000). Key questions concern the patterning of organisms in space (Levin, 1992), the relationship of the patterning of organisms to patterns in the environment (Roughgarden, 1978), and the rate and pattern of spread of a species or allele across a landscape (Kinezaki et al., 2003, Lewis and Pacala, 2000). How organisms become patterned in space is of intrinsic interest (Klausmeier, 1999, Levin, 1992), but such patterns may also be used to draw conclusions about underlying processes. For example, spatial patterns may help distinguish different mechanisms of species coexistence (Bolker et al., 2003), or indicate dispersal distances (Ouborg et al., 1999). Spatial patterns potentially affect other processes. They may change the nature and outcomes of species interactions (Kareiva and Wennergren, 1995), potentially promoting coexistence of competitors (Bolker and Pacala, 1999, Hassell et al., 1994, Murrell and Law, 2003, Snyder and Chesson, 2003) or stabilizing host–parasitoid and predator–prey relationships (Briggs and Hoopes, 2004, Comins et al., 1992, De Roos et al., 1998).
Spatially explicit models inevitably require the use of functions called kernels, which describe dispersal in space (Snyder and Chesson, 2003) or represent interactions between individuals as functions of their distance apart (Bolker and Pacala, 1999, Snyder and Chesson, 2004). Our concern here is with dispersal kernels. In discrete time, a dispersal kernel defines for each spatial location the probability distribution of places dispersed from that location in one unit of time. The variance of this distribution defines the spatial scale of dispersal, but also important is kurtosis, which reflects the shape of the distribution. In nature, leptokurtosis is common, that is, dispersal kernels are often observed to have a sharp peak at the point of origin and a long tail. They are thus far from the Gaussian (normal) distributions often used in modeling. Of most importance, leptokurtosis has been shown to greatly increase the rate of spread of an invading organism or allele, and has been hypothesized to explain the faster than expected rates of spread sometimes found in nature (Cain et al., 1998, Kot et al., 1996, Lewis and Pacala, 2000). Moreover, recent modeling studies show that dispersal kurtosis may have important repercussions for the dynamics of spatial host–parasitoid interactions (Wilson et al., 1999) and disease (Brown and Bolker, 2004). Finally, leptokurtic models are of value in estimating dispersal characteristics from field data, giving greater precision when kurtosis is appropriately modeled (Clark et al., 1999). It is only recently, however, that suitable families of dispersal kernels, allowing broad ranges of kurtosis, have been in use. Hence, investigations of the full realistic range of kurtosis, including the extremes sometimes observed, are just beginning. This article facilitates this endeavor by providing new families of dispersal kernels for discrete space and time that allow exploration of the effects of kurtosis ranging from the Gaussian value to infinity.
In spatially explicit models, space can be represented as discrete or continuous, but integer lattices in one or two dimensions have advantages for many problems (Snyder and Chesson, 2003, Thomson and Ellner, 2003). However, models of dispersal on integer lattices are not well developed. The earliest integer-lattice models use stepping-stone dispersal: in one unit of time, only nearest neighbors of a lattice point are accessible (Kimura and Weiss, 1964, Malécot, 1969). Such models are useful for qualitative assessment of the effects of localized dispersal (Barton et al., 2002). Quantitative effects, and especially questions about the shape of the dispersal kernel, as discussed above, demand more sophisticated treatments. However, there has been very little development in the statistical literature of suitable probability distributions on integer lattices. Indeed, there is a need for parametric families of probability distributions on integer lattices in which the degree of kurtosis is a parameter so that the effects of leptokurtic dispersal can be studied in models.
Most discrete probability distributions of concern to statisticians are restricted to the nonnegative integers (Johnson et al., 1992). As a consequence, dispersal is often modeled by discretizing distributions on continuous Euclidean space for use on a lattice (e.g. Higgins and Richardson, 1999, Ibrahim et al., 1996). However, theory for the original continuous distributions does not apply to discretizations. Indeed, such features as convolutions, moments, and the relationships between them, transfer at best approximately to discretizations, and may be especially misleading for cases where the median dispersal distance is only a few lattice points. Similar difficulties arise with the common approach of using a probability distribution for distance dispersed to define the probabilities of dispersing to multidimensional lattice points regardless of direction (e.g. Levin and Kerster, 1975, Rousset, 2000).
We develop here a class of integer lattice distributions with a special focus on their applications to modeling dispersal. These distributions are designed to be simulated readily, with properties that are easy to define and control. We provide several families of such distributions defining dispersal kernels in any number of dimensions, although serious applications in population biology rarely go beyond two. These families are defined by time randomizations of convolutions of stepping stone kernels. A counterpart of this technique was recently applied by Yamamura (2002) to create a class of leptokurtic distributions for modeling dispersal in continuous space. For the most part, the families that we derive preserve convolutions, and so multiple dispersal steps of a kernel remain in the same family. The moments of these kernels have an elegant simplicity of interpretation, and their Fourier transforms have compact forms. Calculation of the probabilities for these distributions, i.e. calculation of the kernel itself, is often more complex, but robust numerical techniques are available in general. We build these distributions from stepping-stone distributions as the basic elements. In general, however, they have infinite tails and their properties vary from discrete approximation of multivariate normality to strong leptokurtosis suitable for representing rare long-distance dispersal.
To facilitate understanding, a list of notation is provided as Table 1.
Section snippets
Foundations
Given a probability mass function , for some random variable on the d-dimensional integer lattice , a dispersal kernel can be defined as the function of two variables . This function gives the probability of dispersing in one unit of time from lattice point to lattice point . Such kernels are translationally invariant because the dispersal probabilities depend only on the displacement , not separately on the point of origin . Because of the direct
Fourier transforms
For analytical modeling purposes, one of the most useful features of a kernel is its Fourier transform. The Fourier transform is unique to a kernel and can be used to calculate the kernel, as discussed below. However, the Fourier transform is more useful than the kernel itself when modeling population dynamics that are linear (Lande, 1991), or approximately linear (Lande et al., 1999, Roughgarden, 1977, Roughgarden, 1978, Snyder and Chesson, 2003). For studies of invasions, the rate of spread
Distributions for N and U
We have just seen how a discrete distribution on the nonnegative integers for N, or alternatively a positive continuous distribution for U, yields a dispersal kernel. Our task now is to choose N and U distributions that confer desirable properties on the kernel. Because n convolutions of a kernel represent n steps of that kernel, interpretability of the family is aided if convolutions of a kernel remain in the family. We also seek ways of changing the spatial scale of dispersal without changing
Moments
The first few moments of the these new kernels are easily derived in the cases where they are finite. The th order moments of can be expressed in terms of the nth and lower order moments of . The th order moments of are finite if and only if the nth order moments of N are finite (Appendix C). In all cases, because the stepping-stone kernel is symmetric about 0, the components are symmetric about 0. Hence, whenever . Note also that because the stepping-stone steps
Formulae for calculating the probabilities
The actual probabilities for a kernel are needed for various numerical calculations, for example finding expected values of nonlinear functions that are not polynomials, exponential or trigonometric functions. However, to use these kernels for simulation, the probabilities are not needed. Instead one generates values of , by generating values of N, and adding up N independently generated values of the individual stepping-stone steps, . Some potential distributions for N (e.g. negative
Discussion
Because measured dispersal in nature often deviates markedly from Gaussian, families of dispersal kernels that allow various departures from the Gaussian case are needed for spatial population studies. Such families of kernels are available for continuous space (Clark et al., 1999, Yamamura, 2002). However, models where populations are distributed on integer lattices have an important theoretical role, and so families of dispersal kernels for integer lattices, which we provide here, are needed
Acknowledgments
Thanks are due to two anonymous reviewers for comments that have improved the manuscript.
This work was supported by NSF grant DEB-9981926 (PC), NSF DEB 9602226 (Research Training Grant in Nonlinear Dynamics in Biology, CTL) and UCD Graduate Studies Dissertation Year Fellowship (CTL).
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Present address: Department of Biological Sciences, Stanford University, Stanford, CA 94305, USA.