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

摘要

Bullock etal. (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.