In Sect. 2.9 we already pointed out that for the description of populations of individual plants as well as of plant associations, statistical models are necessary. In ecology there is a large selection of books discussing the determination of distribution values based on given measurements. Additionally, fundamental procedures are found in the standard works on botany [68, 211]. In the following paragraphs, once again the modeling of distributions will be focused on, therefore only descriptive models are discussed.
Generally, we differentiate between regular and random distributions. The latter can be evenly distributed or can exhibit clustering. While regular distributions usually only come about through human influence, random distributions is the usual case in nature. Such distributions form their characteristics through the propagation mechanisms of the plants, through their interactions, and through the fight for resources, as well as through local conditions.
A popular form of random distribution is the Poisson distribution. If a regular measuring grid is laid over such a plant distribution, then the probable number of plants i is distributed per partial area according to the function
p(i) = ^em, (3.11)
i!
where m is the mean per partial area. The probability of finding more than the average of the plants per partial area decreases exponentially. if, additionally, such a distribution is translation invariant, then one speaks of a uniform distribution (see Fig. 3.7a). For Poisson distributions the average value is equal to the variance m = s2.
Patterns of this kind develop, for example, if plants distribute seeds by air and the seeds hit the ground at random. However, this applies only if the plants can grow arbitrarily close together, an assumption, that does not apply to many plants. Usually a plant prevents the growth of related species around itself, typically within a circular range.
The resulting mathematical distribution is called a Poisson disc distribution. in such a distribution, around each dot a circular disc with a fixed minimum radius can be positioned, without any other dot of the distribution lying on its surface. Such a distribution can be seen in Fig.3.7b. In a Poisson disk distribution, the variance is strongly reduced and depends on the circle radius and dot density. The distribution can, on the one hand, be produced by successive inserting of new dots at random, where for each newly inserted dot it must be tested whether it lies in an existing circle (dart throwing); on the other hand, through iterative changing of a Poisson distribution. Details are given in Sect. 8.1.
For clustered distributions we do not have a description basis in the form of a universal mathematical model. In many cases, however, a negative binomial distribution sufficiently describes the situation. Here we can apply















with m the mean and к the aggregation coefficient. If к decreases, the dimension of the cluster expands. Here we can apply
m ~k к .
all other terms can be determined recursively using


If we have a natural plant population with an average value of m and a standard deviation of s, the aggregate coefficient can be determined using


where CD = s2/m is called the dispersion coefficient. The dispersion coefficient gives a reference point for the form of the distribution. If CD < 1, then it is a regular distribution, CD = 1 characterizes a Poisson distribution, and CD > 1 a distribution with cluster points.
This last form of distribution usually develops with plants, that spread over roots and therefore produce their successors in direct proximity. A common example is populations of clover.