Mathematical aspects of variation modelling in ecology of communities

A. P. Levich ; V. L. Alekseev ; V. A. Nikulin

Matematičeskoe modelirovanie, Tome 6 (1994) no. 5, pp. 55-76

Résumé

Variation problem is examined which arises as the result of the extremal principles application in the ecology of the communities. The theorems of existence and uniqueness of the variation problem solutions are proved, the properties of invariant extremals to transformations of the parameters problem are investigated. Theorem of the stratification is proved: the space of the parameters of problem $R_+^m=\{\vec L\in R^m, L^1>0,L^2>0,\dots,L^m>0\}$ divides on $2^m-I$ nonintersecting subsets $S^J$; in each of them the extremal problem with restrictions – inequalities is equal to problem with restrictions – equalities for certain unempty part $J\subset\{1,2,\dots,m\}$ of the starting parameters of problem. Biological interpretation of these theorems and results of the numerical investigation of the concrete communities are adduced.