Fig. 8.6. Domains with different structures: (1) 0 < p* < 1; (2) p* = 0; (3) p* = 1. The second and third domains are also the domains of the low probability of the co-existence of two species.

The measure of stability for the state with p* = 0 is equal to a = m2/ a2 , with p* = 1 to ap = m1/a1 . The line d: p1a2 = m2a1 divides the set of p1 and m2 (except the cone a0b) into two subsets, such that if p1a2 < m2a1 then the state with p* = 0 is stable and vice versa.

A state of the system depends on three parameters, for instance the coefficient of correlation, p, the ratio of mean values, x = p1 / m2, and the ratio of variance, F2 = of / a|. Note that there are few developed statistical procedures for their estimation by the observation data. Therefore, as it seems to us, the representation of the different domains of these variables will be more interesting, and we show them in the plane {p, F} for fixed x (Fig. 8.7).

In these variables p =

and the measure of stability (a*)2 is different in different domains.

In domain 1:

In domain 2:

In domain 3:

Comparing Figs. 8.6 and 8.7 we see that in the space of variables which are describing less the absolute values of Malthusian parameters as the statistical connection between them, the domain of co-existence is much larger than in the space { m1 , M2}. For instance, if the correlation between both Malthusian parameters is negative then they are always co-existing. The point A is a singular point, which belongs to all three domains. This means that in its small vicinity either both species exist or one of them is eliminated.

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