Problems and theorems of variational modeling in ecology of communities

The formulations of variation problem of ecological community, the existence and uniqueness theorems of the variation problem solutions and the stratification theorem are given in this article. For ecological application “the Gibb's theorem” is proved — it is an analogue of equivalence of maximum entropy with fixed energy level problem and minimum energy with fixed entropy level problem. Monotone increasing property of extended entropy functional is formulated and proved. In the case, when the number of limiting resources is greater than the number of species, balance equations are sufficient for finding the population size. The compatibility condition for the corresponding system is found. The property of variation problem for “close” species, i. e. species with “almost” proportional quotas are described.