vol. 166, no. 3 the american naturalist september 2005 ൴ Mechanistic Analytical Models for Long-Distance Seed Dispersal by Wind G. G. Katul,1,2,* A. Porporato,2,† R. Nathan,3,‡ M. Siqueira,1,§ M. B. Soons,4,k D. Poggi,1,5,# H. S. Horn,6,** and S. A. Levin6,†† 1. Nicholas School of the Environment and Earth Science, Box ymptotic power-law tail has an exponent of Ϫ3/2, a limiting value 90328, Duke University, Durham, North Carolina 27708; verified by a meta-analysis for a wide variety of measured dispersal 2. Department of Civil and Environmental Engineering, Pratt kernels and larger than the exponent of the bivariate Student t-test School of Engineering, Duke University, Durham, North Carolina (2Dt). We tested WALD using three dispersal data sets on forest trees, 27708; heathland shrubs, and grassland forbs and compared WALD’s per- 3. Department of Evolution, Systematics and Ecology, Alexander formance with that of other analytical mechanistic models (revised Silberman Institute of Life Sciences, The Hebrew University of versions of the tilted Gaussian Plume model and the advection- Jerusalem, Givat Ram, Jerusalem 91904, Israel; diffusion equation), revealing fairest agreement between WALD pre- 4. Plant Ecology Group, Utrecht University, Sorbonelaan 16, 3584 dictions and measurements. Analytical mechanistic models, such as CA Utrecht, The Netherlands; WALD, combine the advantages of simplicity and mechanistic un- 5. Dipartimento di Idraulica, Trasporti e Infrastrutture Civili, derstanding and are valuable tools for modeling large-scale, long- Politecnico di Torino, Torino, Italy; term plant population dynamics. 6. Department of Ecology and Evolutionary Biology, Princeton Keywords: analytical model, canopy turbulence, long-distance seed University, Princeton, New Jersey 08544 dispersal, mechanistic dispersal models, Wald distribution, wind dispersal. Submitted September 4, 2004; Accepted April 4, 2005; Electronically published July 20, 2005 Online enhancements: appendixes. The past decade witnessed a proliferation of studies that address the importance of seed dispersal in ecological pro- cesses (Clark et al. 1999; Cain et al. 2000, 2003; Nathan and Muller-Landau 2000; Wenny 2001; Nathan et al. 2002b; Wang and Smith 2002; Levin et al. 2003). A major abstract: We introduce an analytical model, the Wald analytical emphasis in these studies is modeling seed dispersal using long-distance dispersal (WALD) model, for estimating dispersal ker- nels of wind-dispersed seeds and their escape probability from the both the phenomenological approach (Clark 1998; Tanaka canopy. The model is based on simplifications to well-established et al. 1998; Clark et al. 1999, 2001; Bullock and Clarke three-dimensional Lagrangian stochastic approaches for turbulent 2000; Nathan et al. 2000; Stoyan and Wagner 2001; Higgins scalar transport resulting in a two-parameter Wald (or inverse Gauss- et al. 2003a) and, especially for wind dispersal, the mech- ian) distribution. Unlike commonly used phenomenological models, anistic approach (Greene and Johnson 1989, 1995, 1996; WALD’s parameters can be estimated from the key factors affecting Okubo and Levin 1989; Horn et al. 2001; Nathan et al. wind dispersal—wind statistics, seed release height, and seed terminal 2001, 2002a, 2002b; Soons and Heil 2002; Tackenberg velocity—determined independently of dispersal data. WALD’s as- 2003; Tackenberg et al. 2003; Soons et al. 2004). Both modeling approaches have been shown to provide reliable * Corresponding author; e-mail: [email protected]. predictions of observed seed dispersal patterns. However, † E-mail: [email protected]. the phenomenological approach has been favored for ‡ E-mail: [email protected]. modeling dispersal in large-scale and long-term popula- § E-mail: [email protected]. tion studies (Levin et al. 1997, 2003; Clark 1998; Higgins k E-mail: [email protected]. and Richardson 1999; Chave 2000; Chave and Levin 2003) # E-mail: [email protected]. because of its inherent simplicity. Simplicity is important ** E-mail: [email protected]. for implementation in spatially explicit population models †† E-mail: [email protected]. that integrate the spatial structure of landscapes, quantify Am. Nat. 2005. Vol. 166, pp. 368–381. ᭧ 2005 by The University of Chicago. the spread of expanding populations of invasive and native 0003-0147/2005/16603-40608$15.00. All rights reserved. species, including pests, and estimate gene flow patterns. This content downloaded from 132.64.68.94 on Mon, 14 Nov 2016 09:32:41 UTC All use subject to http://about.jstor.org/terms Analytical Models for Seed Dispersal by Wind 369 Detailed mechanistic approaches, despite their advantages proaches while avoiding their major disadvantages. Ad- of being estimated independently of the dispersal data, ditionally, it will provide the means to extrapolate from being generally applicable, and providing insights into the the commonly measured dispersal distances near the underlying transport mechanism, require computer-inten- source (or near-field dispersion) to LDD or escape prob- sive simulations of wind statistics and hence are imprac- abilities from the canopy. The latter are much more dif- tical for large-scale, long-term applications. ficult to measure. Furthermore, because the parameters of Simplified mechanistic models that relate mean wind such an analytical mechanistic model—seed terminal ve- conditions and seed attributes to dispersal distances are locity, seed release height, and wind conditions—are easily based on “ballistic” models (Greene and Johnson 1989, interpretable and measurable, it provides the means for 1995, 1996; Nathan et al. 2001, 2002b; Soons and Heil estimating LDD for essentially any wind-dispersed species. 2002). These models capture the mode of the dispersal In this article, we introduce a new analytic expression curve well but fail to reproduce its tail, that is, long- derived from a simplified three-dimensional stochastic dis- distance dispersal (LDD) events (Bullock and Clarke 2000; persion model that retains the essential physics in CELC. Nathan et al. 2002b). In many ecosystems, LDD is a cru- As we explain below, this model converges to a Wald (or cially important determinant of spatial spread, gene flow, inverse Gaussian) distribution; hence, we call it the Wald and species coexistence (Levin et al. 2003). This under- analytical long-distance dispersal (WALD) model. We estimation of the tails is attributed to an underestimation compare the new model to two other analytical mecha- in uplifting and escape of seeds from the canopy, events nistic models: the tilted Gaussian plume and a solution to that play a major role in LDD (Horn et al. 2001; Nathan the advection-diffusion equation proposed by Okubo and et al. 2002a). Levin (1989). The latter two models were numerically re- To partially circumvent this problem, a coupled Eule- vised to partially account for the effect of leaf area density rian-Lagrangian closure (CELC) model (Hsieh et al. 1997, on the vertically averaged mean velocity and turbulent 2000; Katul and Albertson 1998; Katul and Chang 1999; diffusivity. For simplicity, we focus on one-dimensional Nathan et al. 2002a) has recently been applied to seed dispersal kernels (or crosswind-integrated models) and re- dispersal by wind (Nathan et al. 2002a; Soons et al. 2004). fer to dispersal kernels as the probability density function This model reproduced well the observed seed dispersal of locating a seed on the ground (or forest floor) with data collected vertically along a 45-m-high tower for five respect to a point source at a given height (i.e., “distance wind-dispersed tree species in a deciduous forest in the distribution” sensu Nathan and Muller-Landau 2000). If southeastern United States (Nathan et al. 2002a) and hor- the dispersal process is isotropic, a two-dimensional dis- izontally for four wind-dispersed herbaceous species in persal kernel (i.e., “dispersal kernel” sensu Nathan and grasslands in the Netherlands (Soons et al. 2004). In both Muller-Landau 2000) differs from its one-dimensional cases, the model confirmed that uplifting and subsequent counterpart only by 2px, where x is the distance from the seed escape from the canopy is a necessary condition for seed source. LDD. Tackenberg (2003) arrived at a similar conclusion, We test the new model against several seed dispersal using detailed turbulent velocity measurements. data sets obtained from controlled seed release experi- The CELC model is computationally expensive, requir- ments. Ideally, the model’s capacity to predict LDD should ing thousands of trajectory calculations, thereby prohib- be tested against “real” LDD data. Yet quantifying LDD iting its use in large-scale and complex ecological models. remains an unaddressed challenge (Nathan et al. 2003), Hence, what is currently lacking is a simplified dispersal and its definition is still rather vague and case specific model that retains the main mechanisms in CELC (or (Nathan 2005). We approached these difficulties in two other complex turbulent transport models) but also pre- ways. First, we compared the performance of WALD and serves the simplicity of phenomenological models. Re- some alternative models in fitting the dispersal data after cently proposed phenomenological models, such as the setting thresholds of 15 and 110 m from the source. This binomial Student t-test (2Dt; Clark et al. 1999) and the procedure examines the model’s ability (and robustness) mixed Weibull (Higgins and Richardson 1999; Higgins et to fit the low frequency