Data from: Why we do not expect dispersal probability density functions based on a single mechanism to fit real seed shadows

Bullock et al. (Journal of Ecology 105:6-19, 2017) have suggested that the
theory behind the Wald Analytical Long Distance (WALD) model for wind
dispersal from a point source needs to be re-examined. This is on the
basis that an inverse Gaussian probability density function (pdf) does not
provide the best fit to seed shadows around individual source plants known
to be dispersed by wind. We present two reasons why we would not
necessarily expect any of the standard mechanistically derived pdfs to fit
real seed shadows any better than empirical functions. Firstly, the
derivation of “off-the-shelf” pdfs such as the Gaussian, exponential and
inverse Gaussian involves only one of the processes and factors that
together generate a real seed shadow. It is implausible to expect that a
single-process model, no matter how sophisticated in detail, will capture
the behaviour of an entire, complex system, which may involve a number of
sequential random processes, or a superposition of parallel random
processes, or both. Secondly, even if there is only one process involved
and we have a perfect model for that process, the basic parameters of the
model would be difficult to pin down precisely. Moreover, these parameters
are unlikely to remain constant over a dispersal season, so that
effectively we observe the outcome of a linear combination of dispersal
events with different parameter values, constituting a form of averaging
over the parameters of the distribution. Simple examples show that
averaging a pdf over its parameters can lead to a pdf from an entirely
different class. Synthesis. The failure of the inverse Gaussian model to
fit seed shadow data is not in itself a reason to doubt the validity of
the Wald Analytical Long Distance model for movement of particles through
the air under specified environmental conditions. A greater awareness is
needed of the differences between the Wald Analytical Long Distance and
the inverse Gaussian (or Wald) and the purposes for which they are used.
The complexity of dispersing populations of seeds means that any of the
standard mechanistically derived pdfs will actually be merely empirical in
this context. Shape and flexibility of a pdf is far more important for
adequately describing data than some perceived higher status.