Hierarchy of models in mathematical biology and numerically-analytical methods of its investigation (review)

Yu. M. Aponin ; E. A. Aponina

Matematičeskaâ biologiâ i bioinformatika, Tome 2 (2007) no. 2, pp. 347-360.

Résumé

The problem of intermodel relations in mathematical biology is considered. Certain mechanisms of prime models generation from complicated ones are briefly reviewed (the asymptotic decomposition, self-organization, reduction principle and others). The construction of complicated model from prescribed prime models and development of hierarchy of models are also considered. The notions of minimal, maximal and primary model are treated. The key role of analytical methods of investigation is emphasized.

Bibliographie

[1] Kalman R., Falb P., Arbib M., Ocherki po matematicheskoi teorii sistem, Mir, M., 1971, 400 pp. | MR | Zbl

[2] Teoriya sistem. Matematicheskie metody i modelirovanie, Sbornik statei, Mir, M., 1989, 384 pp. (Per. s angl.)

[3] Neimark Yu. I., Matematicheskie modeli estestvoznaniya i tekhniki, Tsikl lektsii. Vypusk 1, Izd-vo NNGU, N. Novgorod, 1994, 84 pp.

[4] Babenko K. I., Osnovy chislennogo analiza, Nauka, M., 1986, 744 pp. | MR

[5] Romanovskii Yu. M., Stepanova N. V., Chernavskii D. S., Matematicheskaya biofizika, Nauka, M., 1984, 304 pp. | MR

[6] Chernavskii D. S., Starkov N. I., Scherbakov A. V., “Dinamicheskaya model povedeniya obschestva. Sinergeticheskii podkhod k makroekonomike”, Novoe v sinergetike. Vzglyad v trete tysyacheletie, Nauka, M., 2002, 239–291

[7] Riznichenko G. Yu., Matematicheskie modeli v biofizike i ekologii, IKI, M., Izh., 2003, 184 pp.

[8] Rubin A. B., Biofizika, v. 1, Teoreticheskaya biofizika, Izd-vo MGU, M.; Izd-vo Nauka, 2004, 462 pp.

[9] Chernavskii D. S., Chernavskaya N. M., Malkov S. Yu., Malkov A. S., “Geopoliticheskie protsessy kak ob'ekt matematicheskogo modelirovaniya”, Istoriya i sinergetika: Matematicheskoe modelirovanie sotsialnoi dinamiki, KomKniga, M., 2005, 103–116

[10] Malinetskii G. G., Matematicheskie osnovy sinergetiki. Khaos, struktury, vychislitelnyi eksperiment, KomKniga, M., 2005, 312 pp.

[11] Teoreticheskaya i matematicheskaya biologiya, Sb. statei, Mir, M., 1968, 448 pp. (Per. s angl.)

[12] Fomin S. V., Matematika v biologii, Znanie, M., 1969, 48 pp.

[13] Romanovskii Yu. M., Stepanova N. V., Chernavskii D. S., Chto takoe matematicheskaya biofizika (Kineticheskie modeli v biofizike), Prosveschenie, M., 1971, 136 pp.

[14] Fomin S. V., Berkinblit M. B., Matematicheskie problemy v biologii, Nauka, M., 1973, 200 pp. | Zbl

[15] Romanovskii Yu. M., Stepanova N. V., Chernavskii D. S., Matematicheskoe modelirovanie v biofizike, Nauka, M., 1975, 344 pp. | MR

[16] Ivanitskii G. R., Krinskii V. I., Selkov E. E., Matematicheskaya biofizika kletki, Nauka, M., 1978, 308 pp.

[17] Ivanitskii G. R., Borba idei v biofizike, Znanie, M., 1982, 64 pp.

[18] Bazykin A. D., Matematicheskaya biofizika vzaimodeistvuyuschikh populyatsii, Nauka, M., 1985, 182 pp. | MR | Zbl

[19] Loskutov A. Yu., Mikhailov A. S., Vvedenie v sinergetiku, Nauka, M., 1990, 272 pp.

[20] Riznichenko G. Yu., Rubin A. B., Matematicheskie modeli biologicheskikh produktsionnykh protsessov, MGU, M., 1993, 302 pp.

[21] Bazykin A. D., Nelineinaya dinamika vzaimodeistvuyuschikh populyatsii, IKI, M., Izh., 2003, 368 pp.

[22] Riznichenko G. Yu., Rubin A. B., Biofizicheskaya dinamika produktsionnykh protsessov, IKI, M., Izh., 2004, 464 pp.

[23] Chernavskii D. S., Sinergetika i informatsiya, Nauka, M., 2001, 244 pp. | MR

[24] Malinetskii G. G., Potapov A. B., Nelineinaya dinamika i khaos. Osnovnye ponyatiya, KomKniga, M., 2006, 240 pp.

[25] Malinetskii G. G., Potapov A. B., Podlazov A. V., Nelineinaya dinamika. Podkhody, rezultaty, nadezhdy, KomKniga, M., 2006, 280 pp.

[26] Grebenikov E. A., Ryabov Yu. A., Konstruktivnye metody analiza nelineinykh sistem, Nauka, M., 1979, 432 pp. | MR | Zbl

[27] Tsurkov V. I., Dinamicheskie zadachi bolshoi razmernosti, Nauka, M., 1988, 288 pp. | MR | Zbl

[28] Molchanov A. M., Nelineinosti v biologii, ONTI PNTs RAN, Puschino, 1992, 222 pp.

[29] Bom D., Obschaya teoriya kollektivnykh peremennykh, Mir, M., 1964, 152 pp.

[30] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973, 272 pp. | MR

[31] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vyssh. Shkola, M., 1990, 208 pp. | MR

[32] Mischenko E. F., Rozov N. Kh., Differentsialnye uravneniya s malym parametrom i relaksatsionnye kolebaniya, Nauka, M., 1975, 248 pp. | MR

[33] Mischenko E. F., Kolesov Yu. S., Kolesov A. Yu., Rozov N. Kh., Periodicheskie dvizheniya i bifurkatsionnye protsessy v singulyarno vozmuschennykh sistemakh, Fizmatlit, M., 1995, 336 pp. | MR

[34] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974, 504 pp. | MR

[35] Strygin V. V., Sobolev V. A., Razdelenie dvizhenii metodom integralnykh mnogoobrazii, Nauka, M., 1988, 256 pp. | MR | Zbl

[36] Zhabotinskii A. M., Kontsentratsionnye avtokolebaniya, Nauka, M., 1974, 179 pp.

[37] Gukenkheimer Dzh., Kholms F., Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, IKI, M., Izh., 2002, 560 pp.

[38] Shoshitaishvili A. N., “Bifurkatsii topologicheskogo tipa vektornogo polya vblizi osoboi tochki”, Tr. seminarov im. I. G. Petrovskogo, 1975, 279–309 | Zbl

[39] Kapitsa S. P., Kurdyumov S. P., Malinetskii G. G., Sinergetika i prognozy buduschego, Editorial URSS, M., 2003, 288 pp.

[40] Romanov A. V., “Effektivnaya konechnaya parametrizatsiya v fazovykh prostranstvakh parabolicheskikh uravnenii”, Izv. RAN. Ser. matem., 70:5 (2006), 163–178 | MR | Zbl

[41] Galitskii V. V., “O modelirovanii produktsionnogo protsessa v rastitelnom soobschestve”, Modelirovanie biogeotsenoticheskikh protsessov, Nauka, M., 1981, 104–118

[42] Galitskii V. V., Tyuryukanov A. N., “O metodologicheskikh predposylkakh modelirovaniya v biogeotsenologii”, Modelirovanie biogeotsenoticheskikh protsessov, Nauka, M., 1981, 29–47

[43] Moiseev N. N., Matematika stavit eksperiment, Nauka, M., 1979, 224 pp. | MR

[44] Volterra V., Matematicheskaya teoriya borby za suschestvovanie, Nauka, M., 1976, 288 pp. | MR

[45] Andrianov I. V., Barantsev R. G., Manevich L. I., Asimptoticheskaya matematika i sinergetika: put k tselostnoi prostote, Editorial URSS, M., 2004, 304 pp.

[46] Uspenskii V. A., Chto takoe nestandartnyi analiz?, Nauka, M., 1987, 128 pp. | MR

[47] Albeverio S., Fenstad I., Kheeg-Kron R., Lindstrem T., Nestandartnye metody v stokhasticheskom analize i matematicheskoi fizike, Mir, M., 1990, 616 pp. | MR | Zbl

[48] Aponin Yu. M., Aponina E. A., “Nestandartnyi analiz kak yazyk matematicheskogo otobrazheniya i modelirovaniya realnosti”, VII Mezhdunarodnaya konf. serii “Nelineinyi mir”. Yazyki nauki – yazyki iskusstva, NITs Regul. i khaot. dinamika, Izhevsk, 2002, 10

[49] Khibnik A. I., Shnol E. E., Programmy dlya kachestvennogo issledovaniya differentsialnykh uravnenii. Informatsionnyi material, ONTI NTsBI AN SSSR, Puschino, 1982, 16 pp.

[50] Frisman E. Ya., Shapiro A. P., Izbrannye matematicheskie modeli divergentnoi evolyutsii populyatsii, Nauka, M., 1977, 151 pp. | MR | Zbl

[51] Skaletskaya E. I., Frisman E. Ya., Shapiro A. P., Diskretnye modeli dinamiki chislennosti populyatsii i optimizatsiya promysla, Nauka, M., 1979, 166 pp. | MR

[52] Shapiro A. P., Luppov S. P., Rekurrentnye uravneniya v teorii populyatsionnoi biologii, Nauka, M., 1983, 134 pp. | MR

[53] Nemytskii V. V., Stepanov V. V., Kachestvennaya teoriya differentsialnykh uravnenii, NITs “Regul. i khaot. dinamika”, M., Izh., 2004, 456 pp.

[54] Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G., Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka, Nauka, M., 1966, 568 pp. | MR | Zbl

[55] Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G., Teoriya bifurkatsii dinamicheskikh sistem na ploskosti, Nauka, M., 1967, 488 pp. | MR

[56] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., Teoriya bifurkatsii, Sovremennye problemy matematiki. Fundamentalnye napravleniya, 5, VINITI AN SSSR, M., 1985, 5–218 | MR

[57] Bautin N. N., Leontovich E. A., Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti, Nauka, M., 1990, 488 pp. | MR | Zbl

[58] Serebryakova N. N., “Kachestvennoe issledovanie odnoi sistemy differentsialnykh uravnenii teorii kolebanii”, PMM, 27:1 (1963), 160–166 | Zbl

[59] Aponin Yu. M., Asimptoticheskie formuly dlya predelnogo tsikla pri rozhdenii iz petli separatrisy, VINITI, Puschino, 1976, 46 pp.

[60] Aponin Yu. M., Ob analiticheskoi kharakteristike izmeneniya separatrisy i predelnogo tsikla v zavisimosti ot parametra, VINITI, Puschino, 1978, 25 pp.

[61] Aponin Yu. M., O nekotorykh asimptoticheskikh otsenkakh i vychislitelnykh algoritmakh dlya issledovaniya predelnykh tsiklov i separatris sistem dvukh obyknovennykh differentsialnykh uravnenii, Avtoreferat kand. dissertatsii, Gorkii, 1979, 16 pp.

[62] Davidenko D. F., “O novom metode chislennogo resheniya sistem nelineinykh uravnenii”, DAN SSSR, 88:4 (1953), 601–602 | MR | Zbl

[63] Lobanov A. I., Petrov I. B., Starozhilova T. K., Vychislitelnye metody dlya analiza modelei slozhnykh dinamicheskikh sistem, Ch. II: Uchebnoe posobie, FIZTEKh-POLIGRAF, Dolgoprudnyi, 2002, 155 pp.

[64] Aponin Yu. M., Aponina E. A., Izbrannye algoritmy i programmy dlya EVM MIR-2. Separatrisy sistemy dvukh differentsialnykh uravnenii, ONTI NTsBI AN SSSR, Puschino, 1976, 36 pp.

[65] Kuznetsov Yu. A., Odnomernye separatrisy sistemy obyknovennykh differentsialnykh uravnenii, zavisyaschei ot parametrov. Materialy po matematicheskomu obespecheniyu EVM, ONTI NTsBI AN SSSR, Puschino, 1983, 48 pp.

[66] Aponina E. A., Aponin Yu. M., Kreitser G. P., Shnol E. E., Izbrannye algoritmy i programmy dlya EVM MIR-2. Predelnye tsikly sistemy dvukh differentsialnykh uravnenii, ONTI NTsBI AN SSSR, Puschino, 1974, 46 pp.

[67] Balabaev N. K., Lunevskaya L. V., Dvizhenie po krivoi v $n$-mernom prostranstve. Materialy po matematicheskomu obespecheniyu EVM, ONTI NTsBI AN SSSR, Puschino, 1978, 52 pp.

[68] Neimark Yu. I., Metod tochechnykh otobrazhenii v teorii nelineinykh kolebanii, Nauka, M., 1972, 472 pp. | MR

[69] Butenin N. V., Neimark Yu. I., Fufaev N. A., Vvedenie v teoriyu nelineinykh kolebanii, Nauka, M., 1976, 384 pp. | MR

[70] Neimark Yu. I., Landa P. S., Stokhasticheskie i khaoticheskie kolebaniya, Nauka, M., 1987, 424 pp. | MR

[71] Ilyashenko Yu. S., Li Veigu, Nelokalnye bifurkatsii, MTs NMO, CheRo, M., 1999, 416 pp. | MR

[72] Berezovskaya F. S., Kreitser G. P., Izbrannye algoritmy i programmy dlya EVM MIR-2. Slozhnye osobye tochki sistemy dvukh differentsialnykh uravnenii, ONTI NTsBI AN SSSR, Puschino, 1975, 56 pp.

[73] Kreitser G. P., Izbrannye algoritmy i programmy dlya EVM MIR-2. Prostye osobye tochki sistemy dvukh differentsialnykh uravnenii, ONTI NTsBI AN SSSR, Puschino, 1976, 48 pp.

[74] Zarkhin Yu. G., Kovalenko V. N., Nakhozhdenie reshenii sistemy dvukh algebraicheskikh uravnenii. Materialy po matematicheskomu obespecheniyu EVM, ONTI NTsBI AN SSSR, Puschino, 1978, 44 pp.

[75] Khibnik A. I., Periodicheskie resheniya sistemy n differentsialnykh uravnenii. Algoritmy i programmy na FORTRANE, ONTI NTsBI AN SSSR, Puschino, 1979, 72 pp.

[76] Borisyuk R. M., Statsionarnye resheniya sistemy obyknovennykh differentsialnykh uravnenii, zavisyaschei ot parametra. Materialy po matematicheskomu obespecheniyu EVM, ONTI NTsBI AN SSSR, Puschino, 1981, 68 pp.

[77] Aponin Yu. M., Antonenko Yu. N., Aponina E. A., Kovbasnyuk O. N., Yaguzhinskii L. S., “Neelektrogennyi $K^+/H^+$ – obmen, vklyuchayuschii stadiyu obmena protonami mezhdu perenoschikom i kislotno-osnovnymi gruppirovkami fosfolipidov na poverkhnosti membrany. Sravnenie teorii i eksperimenta”, Biologicheskie membrany, 13:3 (1996), 258–270

[78] Balabaev N. K., “Metodika modelirovaniya dinamiki polimerov”, Metod molekulyarnoi dinamiki v fizicheskoi khimii, Nauka, M., 1996, 257–278 pp.

[79] Lakhno V. D., “Dinamika perenosa dyrki v nukleotidnykh posledovatelnostyakh”, Kompyutery i superkompyutery v biologii, Institut kompyuternykh issledovanii, M., Izhevsk, 2002, 137–171

[80] Lakhno V. D., “Modelirovanie pervichnykh protsessov perenosa zaryada v reaktsionnom tsentre fotosinteza”, Kompyutery i superkompyutery v biologii, Institut kompyuternykh issledovanii, M., Izhevsk, 2002, 195–208

[81] Eigen M., Samoorganizatsiya materii i evolyutsiya biologicheskikh makromolekul, Mir, M., 1973, 216 pp.

[82] Eigen M., Shuster P., Gipertsikl. Printsipy samoorganizatsii makromolekul, Mir, M., 1982, 272 pp.

[83] Redko V. G., Evolyutsionnaya kibernetika, Nauka, M., 2001, 156 pp.

[84] Gudvin B., Vremennaya organizatsiya kletki, Mir, M., 1966, 251 pp.

[85] Elkin Yu. E., Gennye seti, http://www.mathcell.ru/ru/obzors/obzor Elkin.shtml

[86] Pechurkin N. S., Populyatsionnaya mikrobiologiya, Nauka, Novosibirsk, 1978, 278 pp.

[87] Svirezhev Yu. M., Pasekov V. P., Osnovy matematicheskoi genetiki, Nauka, M., 1982, 512 pp. | MR

[88] Stanishkis Yu.-K. Yu., Optimalnoe upravlenie biotekhnologicheskimi protsessami, Mokslas, Vilnyus, 1984, 256 pp.

[89] Varfolomeev S. D., Kalyuzhnyi S. V., Biotekhnologiya: Kineticheskie osnovy mikrobiologicheskikh protsessov, Vyssh. shk., M., 1990, 296 pp.

[90] Pechurkin N. S., Brilkov A. V., Marchenkova T. V., Populyatsionnye aspekty biotekhnologii, Nauka, Novosibirsk, 1990, 173 pp.

[91] Balanter B. I., Khanin M. A., Chernavskii D. S., Vvedenie v matematicheskoe modelirovanie patologicheskikh protsessov, Meditsina, M., 1980, 264 pp.

[92] Smirnova O. A., Radiatsiya i organizm mlekopitayuschikh: modelnyi podkhod, NITs “Regulyarnaya i khaoticheskaya dinamika”, M., Izhevsk, 2006, 224 pp.

[93] Ivanov V. K., Matematicheskoe modelirovanie i optimizatsiya luchevoi terapii opukholei, Energoatomizdat, M., 1986, 144 pp. | MR

[94] Bellman R., Matematicheskie metody v meditsine, Mir, M., 1987, 200 pp. | MR | Zbl

[95] Svirezhev Yu. M., Logofet D. O., Ustoichivost biologicheskikh soobschestv, Nauka, M., 1978, 352 pp. | MR

[96] Korotaev A. V., Malkov A. S., Khalturina D. A., Zakony istorii: Matematicheskoe modelirovanie razvitiya Mir – Sistemy. Demografiya, ekonomika, kultura, KomKniga, M., 2007, 224 pp.

[97] Borodkin L. I., “Nelineinaya dinamika sotsialno-politicheskikh protsessov proshlogo: metodologicheskie problemy modelirovaniya neustoichivogo razvitiya”, Istoriya i Matematika: Analiz i modelirovanie sotsialno-istoricheskikh protsessov, KomKniga, M., 2007, 8–48

[98] Bykov V. I., Modelirovanie kriticheskikh yavlenii v khimicheskoi kinetike, KomKniga, M., 2006, 328 pp.

[99] Aponin Yu. M., Vanyakin E. N., Osipov V. V., “Matematicheskoe modelirovanie protsessov periodicheskogo kultivirovaniya mikroorganizmov s uchetom dinamiki rastvorennykh gazov”, Matematika i modelirovanie, Sbornik nauchnykh trudov, ONTI NTsBI AN SSSR, Puschino, 1990, 222–232

[100] Aponin Yu. M., Populyatsionnaya dinamika bakterialnykh plazmid v usloviyakh khemostatnogo kultivirovaniya, ONTI NTsBI AN SSSR, Puschino, 1982, 18 pp.

[101] Aponin Yu. M., Aponina E. A., Velkov V. V., Matematicheskoe modelirovanie protsessov nepreryvnogo kultivirovaniya mikroorganizmov, soderzhaschikh nestabilnye gibridnye plazmidy, ONTI NTsBI AN SSSR, Puschino, 1984, 21 pp.

[102] Aponin Yu. M., Aponina E. A., Vanyakin E. N., Matematicheskoe modelirovanie protsessov nepreryvnogo kultivirovaniya s uchetom geterogennosti mikrobnykh populyatsii, ONTI NTsBI AN SSSR, Puschino, 1989, 32 pp.