Geodesic
Problems and theorems of variational modeling in ecology of communities
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1035-1045 
A. P. Levich; P. V. Fursova. Problems and theorems of variational modeling in ecology of communities. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1035-1045. http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Levich A. P., Teoriya mnozhestv, yazyk teorii kategorii i ikh primenenie v teoreticheskoi biologii, Izd-vo Mosk. un-ta, M., 1982 | Zbl

[2] Levich A. P., Alekseev V. L., Nikulin V. A., “Matematicheskie aspekty variatsionnogo modelirovaniya v ekologii soobschestv”, Matematicheskoe modelirovanie, 6:5 (1994), 55–76 | MR | Zbl

[3] Levich A. P., “Variational theorems and algocoenosec functioning principles”, Ecological Modelling, 131:2–3 (2000), 207–227 | DOI

[4] Levich A. P., “Entropiya kak mera strukturirovannosti slozhnykh sistem”, Trudy seminara “Vremya, khaos i matematicheskie problemy”, Vyp. 2, Knizhnyi dom “Universitet”, M., 2001, 163–176

[5] Levich A. P., Struktura ekologicheskikh soobschestv, Izd-vo Mosk. un-ta, M., 1980

[6] Levich A. P., Alekseev V. L., “Entropiinyi ekstremalnyi printsip v ekologii soobschestv: rezultaty i obsuzhdenie”, Biofizika, 42:2 (1997), 534–541

[7] Gibbs Dzh. V., Osnovnye printsipy statisticheskoi mekhaniki, Gostekhizdat, M., 1946

[8] Galeev E. M., Tikhomirov V. M., Kratkii kurs teorii ekstremalnykh zadach, Izd-vo Mosk. un-ta, M., 1989 | Zbl

[9] Bertsekas D., Uslovnaya optimizatsiya i metody mnozhitelei Lagranzha, Radio i svyaz, M., 1987 | MR | Zbl

[10] Levich A. P., “Vremya kak izmenchivost estestvennykh sistem: sposoby kolichestvennogo opisaniya izmenenii i porozhdenie izmenenii substantsionalnymi potokami”, Konstruktsii vremeni v estestvoznanii: na puti k ponimaniyu fenomena vremeni. Chast 1. Mezhdistsiplinarnoe issledovanie, Izd-vo Mosk. un-ta, M., 1996, 235–288

[11] Fursova P. V., “Raschety vidovykh obilii i oblastei limitirovaniya v variatsionnoi modeli ekologicheskogo soobschestva”, Matematicheskoe modelirovanie (to appear)