## Introduction

Information about seed dispersal and recruitment is crucial for understanding the genetic structure of plant populations, plant invasions and, in some cases, species coexistence (reviewed in Nathan & Muller-Landau 2000). Further, the recruitment subroutine is an essential part of stand dynamics simulators being developed by foresters to predict stand density and volume (LePage *et al*. 2000). Nonetheless, empirical delineation of seed and seedling dispersal curves within forests has been a difficult task because the individual dispersal curves of conspecific trees usually overlap. There are a number of methods available for determining individual dispersal curves (Greene & Calogeropoulos 2002), but by far the most economical is the inverse modelling approach pioneered by Ribbens *et al*. (1994). Under this approach, maximum likelihood methods are used to estimate the terms of the dispersal function, given the spatial distribution and sizes of potential parent trees around each sample location.

The inverse modelling approach has now been used in a number of different studies, but with disagreement among practitioners over the most appropriate functional form of the dispersal curve. Ribbens *et al*. (1994) used a two-parameter Weibull function (sometimes referred to as the exponential family; Clark *et al*. 1999). Clark *et al*. (1999) proposed a composite dispersal function (the ‘2Dt’ function) that was exponential in shape, but with a normally distributed variable for the scale parameter. They argued that this function was a better descriptor of dispersal curves than the two-parameter Weibull used by Ribbens *et al*. (1994). Stoyan & Wagner (2001) claimed the lognormal was superior to the Weibull. Meanwhile, other authors (e.g. LePage *et al*. 2000 for the Weibull, Tanaka *et al*. 1998 for the lognormal) have simply adopted one or another of these functions, intuiting, perhaps, that they will perform about equally well. Nonetheless, the choice of the function is critical; as noted by Nathan & Muller-Landau (2000), some functions have far tails that are too thin to permit metapopulation persistence (let alone a migrational velocity sufficient to explain the Holocene record).

There is general agreement on the basic expression for the dispersal curve:

where *Q _{Dx}* is the deposition density (seedlings m

^{−2}) at distance

*x*from a source producing

*Q*seeds,

*B*is the basal area of a tree (or some other allometrically related size measure),

*ƒ*

_{1}(

*x*) is the seed dispersal kernel,

*ƒ*

*s*

_{g(x,y)}is the density-dependent mortality in Cartesian space (

*x*,

*y*) after abscission but prior to germination, and

*a*

_{1}and

*b*are empirical coefficients. The coefficient

*a*

_{1}translates tree size (basal area) into seed production (

*Q*) as

*Q*=

*a*

_{1}

*B*

^{b}

*.*Analogously, we can imagine

*Q*asexual buds along the roots of species such as

*Fagus*or

*Populus*that reproduce via root sprouts. The function

*s*

_{g(x,y)}is problematic as it represents the density response of granivores to the density of all the conspecific seeds as well as allospecific seeds, and the preferences and densities of the granivore species themselves. In equation 1, the surviving seed densities are subsequently reduced further by two types of losses: density-independent mortality at germination, and shortly thereafter, conditioned by seedbed-related mortality (

*s*

_{s(x,y)}) and density-dependent herbivory (

*ƒ*

*s*

_{h(x,y)}) on the germinants and seedlings. There is no longer any doubt regarding the importance of density-dependent mortality (e.g. Harms

*et al*. 2000) at both the seed and seedling stage, and thus the shape and scale of the realized recruitment kernel (

*ƒ*(

*x*)) may well look quite different from the original seed dispersal kernel (

*ƒ*

_{1}(

*x*)). To date no study of inverse modelling has yet tried to parameterize this complete density-dependent function. Instead, modellers have used simplified versions:

(e.g. Ribbens *et al*. 1994), where *ƒ*(*x*) is now the realized kernel with the original seed dispersal curve (*sensu*Nathan & Muller-Landau 2000) modified by predation, or, in a slightly more complex form:

(LePage *et al*. 2000), with the seedbed-generated survivorship term now explicit.

It is this latter form (equation 2b) that we will use in the subsequent analyses.

### the alternative dispersal kernels (*f*(*x*))

We examine three dispersal kernels for tree seeds or seedlings. Each is a two-parameter distribution: the modified Weibull (as used by Ribbens *et al*. 1994), the lognormal (introduced by Greene & Johnson 1989), and the 2Dt (as proposed by Clark *et al*. 1999) (Fig. 1 and Table 1). In each case we will deal with *density* per distance (i.e. the kernel divided by 2π*x*). We restrict ourselves here to closed-form expressions, and thus ignore mechanistic individual-trajectory models such as those of Nathan *et al*. (2001) or Tackenberg (2003) for wind or Murray (1988) for bird defaecation of fruits. Further, we ignore ‘mixed-model’ empirical formulations that require more than two parameters as advocated by Bullock & Clarke (2000) among others.

ƒ(x) | |
---|---|

Weibull | (1/N) exp(–Lx^{S}) |

2Dt | S/(πL[1 + (x^{2}/L)]^{S+1}) |

Lognormal | [1/((2π)^{1.5}Sx^{2})] exp(– (ln(x/L))^{2}/(2S^{2})) |

The Weibull can take on very different shapes depending on the value of the parameter S. When this shape parameter is 3 (as with Ribbens *et al*. 1994 or LePage *et al*. 2000) or higher, densities of recruits are relatively invariant with distance at first but then decline quite sharply (Fig. 1). The far tail is very thin. With decreasing S, the initial ‘plateau’ in density becomes less marked and the tail more extensive: S = 2 for example leads to the right half of a Gaussian distribution and S = 1 is the familiar negative exponential. Note that the modified Weibull, unlike the 2Dt or lognormal, requires a normalizer (Table 1) as it is not a true probability distribution (i.e. does not sum to 1) when the function is multiplied by 2π*x* to create recruits per annulus rather than recruits per area per annulus.

The shape of the 2Dt is affected less dramatically by changes in the shape parameter, S. With increasing S, the density of recruits in the near tail becomes greater while the far tail density becomes both thicker and more extensive. For the lognormal, increases in S thicken the far tail while pushing the modal density back towards the source tree. Meanwhile, the scale parameter, denoted as L in Table 1, merely affects such distance measures as the median or mean distance travelled.

Clark *et al*. (1999) argued that only the 2Dt had the right shape for both the near and far tails of the *ƒ*(*x*). Reasonably, they argued that post-Holocene migration velocities required an extensive far tail containing an unspecified but presumably large fraction of the Q seeds produced. They pointed out that the modified Weibull can never place a sufficiently great proportion of Q in the far tail unless the shape parameter S (Table 1) is very small (< 1.0). But in that case, the near tail would be concave near the source. They then claimed that near-tail concavity is never observed for point sources. Both Ribbens *et al*. (1994) and LePage *et al*. (2000) used a much larger value (S = 3) for the Weibull, forcing the curve to maintain convexity near the source tree, but of course producing a much foreshortened far tail. Thus, Clark *et al*. (1999) argued, the 2Dt is dramatically better than the Weibull because it can simultaneously capture the shape of the curve both ‘near and far’. Similarly, they argued that the lognormal, which places the modal deposition away from the source, cannot be a useful expression because the near tail must be convex. For empirical justification of this assertion of the near tail convexity, Clark *et al*. (1999) cite a pair of modelling studies that lack empirical data and two examples of inverse modelling. Realistically, inverse modelling cannot be used to prove the near-bole convexity because neither of the cited studies tested alternate forms that were not convex near the bole.

In a preliminary analysis, we found that the three dispersal terms can be quite similar over a limited range. For example, constraining them so that the median dispersal distance is 20 m and the 95th percentile occurs at 100 m, the three functions predict seed or seedling densities within 1.5-fold of each other from about 6 m to 200 m from a tree. Thus, at the scale at which ecologists have sampled (and this includes every data set discussed here), the three functions will tend to differ primarily in the very near tail, that is, close to the maternal parent. (According to Greene & Calogeropoulos (2002) they will also differ substantially in the far tail.)

Our purpose is to compare the three candidate dispersal terms in two ways. First, we will look at the handful of point source studies (a single tree well-isolated from other conspecifics but nonetheless deep within a forest) that exist in the literature and ask which of the three alternative dispersal functions (*ƒ*(*x*)) is a better expression for empirical data. Secondly, we will repeat Clark's comparison of the 2Dt and Weibull, but include the lognormal now, as we examine inverse modelling results from an eastern hardwood forest.