Stochastic model of dynamics of biological community in conditions of consumption by individuals of harmful food resources

N. V. Pertsev ; K. K. Loginov

Matematičeskaâ biologiâ i bioinformatika, Tome 6 (2011) no. 1, pp. 1-13.

Résumé

The stochastic model describing dynamics of biological community in conditions of consumption by individuals of food resources, containing harmful substances is presented. For construction the model we used stochastic analogue of Lotka–Volterra model in the form of multivariate nonlinear birth and death process, added by the differential equations for quantity of food resources. Decrease in number of populations of community is described by death of individuals not only due to their competition and a self-capping, but also due to consumption of the food resources containing harmful substances. The recurrent equations specifying rules of variation of number of populations and quantity of food resources in an inhabitancy of individuals are constructed. On the basis of Monte-Carlo technique the algorithm for simulation the dynamics of number of populations and quantities of food resources is developed. Results of computing experiments on studying dynamics of two competing populations which individuals consume the complete set from four food resources containing two harmful substances are resulted.

Bibliographie

[1] Hallam T. G., Clark C. E., Lassiter R. R., “Effects of toxicants on populations: A qualitative approach. I. Equilibrium environment exposure”, Ecol. Model., 18 (1983), 291–304 | DOI | Zbl

[2] Hallam T. G., Clark C. E., Jordan G. S., “Effects of toxicants on populations: A qualitative approach. II. First order kinetics”, J. Math. Biol., 18 (1983), 25–37 | DOI | MR | Zbl

[3] Hallam T. G., De Luna J. L., “Effects of toxicants on populations: A qualitative approach. III. Environmental and food chain pathways”, J. Theoret. Biol., 109 (1984), 411–429 | DOI

[4] Ma Z., Gui G., Wang W., “Persistence and extinction of a population in a polluted environment”, Math. Biosci., 101 (1990), 75–97 | DOI | MR | Zbl

[5] Freedman H. I., Shukla J. B., “Models for the effect of toxicant in single-species and Predator-prey systems”, J. Math. Biol., 30 (1991), 15–30 | DOI | MR | Zbl

[6] Krestin S. V., Rozenberg G. S., “Ob odnom mekhanizme “tsveteniya vody” v vodokhranilische ravninnogo tipa”, Biofizika, 41:3 (1996), 650–654

[7] Dubey B., “Modelling the effect of toxicant on forestry resources”, Indian J. Appl. Math., 28 (1997), 1–12 | MR | Zbl

[8] Dubey B., Hussain J., “Modelling the interaction of two biological species in a polluted environment”, J. Math. Anal. Appl., 246 (2000), 58–79 | DOI | MR | Zbl

[9] Jinxiao P., Zhen J., Zhien M., “Thresholds of Survival for an $n$-Dimensional Volterra Mutualistic System in a Polluted Environment”, J. of Math. Anal. Appl., 252 (2000), 519–531 | DOI | MR | Zbl

[10] Xiao Y., Chen L., “Effects of toxicants on a stage-structured population growth model”, Appl. Math. and Comput., 123 (2001), 63–73 | DOI | MR | Zbl

[11] Mukherjee D., “Persistence and global stability of a population in a polluted environment with delay”, J. of Biol. Sys., 10:3 (2002), 225–232 | DOI | Zbl

[12] Liu B., Chen L., Zhang Y., “The effects of impulsive toxicant input on a population in a polluted environment”, J. of Biol. Sys., 11:3 (2003), 265–274 | DOI | Zbl

[13] Dubey B., Hussain J., “Nonlinear models for the survival of two competing species dependent on resource in industrial environments”, Nonlinear Analysis: Real World Applications, 4 (2003), 21–44 | DOI | MR | Zbl

[14] Pichugina A. N., “Integrodifferentsialnaya model populyatsii, podverzhennoi vozdeistviyu vrednykh veschestv”, Sib. zhurnal ind. matem., 7:4(20) (2004), 130–140 | MR | Zbl

[15] Karelina R. O., Pertsev N. V., “Postroenie dvustoronnikh otsenok dlya reshenii nekotorykh sistem differentsialnykh uravnenii s posledeistviem”, Sib. zhurnal ind. matem., 8:4(24) (2005), 60–72 | MR | Zbl

[16] Naresh R., Sundar S., Shukla J., “Modelling the effect of an intermediate toxic product formed by uptake of a toxicant on plant biomass”, Appl. Math. and Comput., 182 (2006), 151–160 | DOI | MR | Zbl

[17] Feng Z., Liu R., DeAngelis D. J., “Plant-herbivore interactions mediated by plant toxicity”, Theor. Popul. Biol., 73 (2008), 449–459 | Zbl

[18] Li Z., Chen F., “Extinction in periodic competitive stage-structured Lotka–Volterra model with effects of toxic substances”, J. of Comput. and Appl. Math., 231 (2009), 143–153 | DOI | MR | Zbl

[19] Pertsev N. V., Tsaregorodtseva G. E., Pichugina A. N., “Analiz ustoichivosti polozhenii ravnovesiya odnoi modeli populyatsionnoi dinamiki”, Vest. Om. un-ta, 2009, no. 2, 46–49

[20] Pertsev N. V., Tsaregorodtseva G. E., “Matematicheskaya model dinamiki populyatsii, razvivayuscheisya v usloviyakh vozdeistviya vrednykh veschestv”, Sib. zhurnal industr. matematiki, 13:1(41) (2010), 109–120 | Zbl

[21] Loginov K. K., “Vychislitelnye aspekty imitatsionnogo modelirovaniya dinamiki konkuriruyuschikh populyatsii v usloviyakh vozdeistviya toksichnykh veschestv”, Stokhasticheskie modeli v biologii i predelnye algebry – Stochastic models in biology and limit algebras, Trudy konferentsii (Om. Filial In-ta matematiki im. S. L. Soboleva SO RAN, 2–7 avgusta 2010 g.), Izd-vo Om. gos. un-ta, Omsk, 2010, 54–55

[22] Pichugin B. Yu., Pertsev N. V., Loginov K. K., “Stokhasticheskaya model dinamiki populyatsii, razvivayuschikhsya v usloviyakh vozdeistviya toksichnykh veschestv”, Matematicheskaya biologiya i bioinformatika, Doklady III Mezhdunarodnoi konferentsii (g. Puschino, 10–15 oktyabrya 2010 g.), ed. V. D. Lakhno, OOO “MAKS Press”, M., 2010, 208–209

[23] Barucha-Rid A. T., Elementy teorii markovskikh sluchainykh protsessov i ikh prilozheniya, Nauka, M., 1969, 512 pp.

[24] Sobol I. M., Chislennye metody Monte-Karlo, Nauka, M., 1973, 311 pp. | MR

[25] Ermakov S. M., Mikhailov G. A., Kurs statisticheskogo modelirovaniya, Nauka, M., 1976, 319 pp. | MR | Zbl

[26] Kalinkin A. V., “Markovskie vetvyaschiesya protsessy s vzaimodeistviem”, Uspekhi matematicheskikh nauk, 57:2(344) (2002), 23–84 | MR | Zbl

[27] Kalinkin A. V., Lange A. M., Mastikhin A. V., Shaposhnikov A. A., “Chislennye metody Monte-Karlo dlya modelirovaniya skhem vzaimodeistviya pri diskretnykh sostoyaniyakh”, Vestnik MGTU. Seriya Estestvennye nauki, 2005, no. 2, 53–74

[28] Kalinkin A. V., Lange A. M., “Veroyatnostnyi analog modeli konkurentsii G. F. Gauze”, Stokhasticheskie modeli v biologii i predelnye algebry – Stochastic models in biology and limit algebras, Trudy konferentsii (Om. Filial In-ta matematiki im. S. L. Soboleva SO RAN, 2–7 avgusta 2010 g.), Izd-vo Om. gos. un-ta, Omsk, 2010, 40–43

[29] Marchenko M. A., “Kompleks programm MONC dlya raspredelennykh vychislenii metodom Monte-Karlo”, Sib. zhurnal vych. matematiki, 7:1 (2004), 43–55 | Zbl

[30] Pertsev N. V., Pichugin B. Yu., “Primenenie metoda Monte-Karlo dlya modelirovaniya dinamiki soobschestv vzaimodeistvuyuschikh individuumov”, Vestnik Voronezhskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Vychislitelnye i informatsionno-telekommunikatsionnye sistemy, 2:5 (2006), 70–77