Appendix B. Dynamics within congeneric pairs.
Here we discuss the shape of the fractional abundance distribution in the context of general ecological expectations for two ecologically and evolutionarily similar species.
The extreme case of limiting similarity
Consider a niche (in some sense) that is equally good for two congeners and neither has any advantage, stabilised by intra- greater than inter-specific competition – two species neutrality rather than many species neutrality. With N sites and any site having equal probability of being occupied by species a or by species b, the probability of having N – m of one species and m of the other is given by the binomial coefficients (divided by 2N). If we have a large number of samples of N sites with two equivalent species, the most probable fractional abundance is 0.5, the distribution tailing away towards extreme disparity. The distribution gets narrower as N increases.
Transient coexistence (a version of competitive exclusion)
Suppose species a is occupying a certain large number of sites N. A similar congeneric species is added and outcompetes it. Every time an individual of species a dies it has a high probability of being replaced by an individual of species b. The probability of any individual of species a dying is the same small constant each year – the population of a dies away exponentially. Then early on the proportion of a is close to 1, but this does not last long – only one half life in this simple model – after which the proportion of b is greater and this lasts almost forever (forever with an infinite population or a continuous distribution). If for any congeneric pair we measure at a randomly sampled time we are very likely to find the winner present at a very high fractional abundance.
The shape is easily calculated. As a function of time the fractional abundance is e-kt up to one half life and thereafter 1 e-kt. If we cannot pick out two species after n half lives, then the distribution of fractional abundances r randomly sampled is
1/(1 r) + (1/n)(1/r 1/ (1 r))
and it is this function which is plotted in blue in Fig. B1 for the case of large n. A variant in which the proportion of dead species a replaced by a depends on the density of species a corresponds to weaker competitive advantage for the invader, species b. This makes only a qualitative difference; the symmetric function for this case is plotted in red in Fig. B1.
|FIG. B1. Expected fractional abundance distributions. (a) Interchangeable species. Distribution of fractional abundance expected for two species, identical except that the impact of intra- is greater than interspecific competition, coexisting with equal probability of occupying each of N sites. Filled red triangles show N = 50; filled blue circles show N = 20. (b) Distribution of fractional abundances for one species being driven out by an invasive congener (competitive exclusion). The calculations are based on the assumption of pairs being sampled at random times during exclusion. The blue line corresponds to large competitive advantage for the invader; the broken red line to a minor advantage. The data in this figure for congeneric species (gray lines in each part) could not (with any significant probability) be drawn from the distributions illustrated in red or blue.|
This exponential behavior is very simple, yet plausible. (We note that the resulting fractional abundance distribution is like that of the non-congeneric pairs drawn at random from the data. We would not wish to imply that this similarity of shape identifies the mechanism.) Sampling competing pairs along the road to competitive exclusion is likely to yield fractional abundances at the high end (0.9 1 with sufficient data) and the more equitable fractions we see in the real data are not likely.