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Control of environmental pollution to conserve a population. (English) Zbl 0998.92042

Summary: The model analyzed here represents the dynamics of a population in a polluted environment. Here the net growth rate of the population depends on the concentration of the pollutant in the organism and environment. From Theorem 1, we can observe that the persistence or extinction of the population is very much dependent on \(u(t)\), the input of the pollutants into the environment. In Theorem 3, it is shown that it is possible to guarantee the persistence of the population by regulating \(u(t)\). Here, effort is used as control to regulate \(u(t)\). Apart from making the population persistent, it is also possible to control the asymptotic value of the population. This is illustrated through Theorem 4. It is assumed that the total consumption in the environment is constant and no effort is made to reduce the consumption to regulate \(u(t)\). However, we can also consider the consumption as a dynamic variable and hence study the trade off between consumption and conservation.

MSC:

92D40 Ecology
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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[1] Fleming, R. N.; Pantell, R. H., The conflict between consumption and pollution, IEEE Trans. Systems Man Cybernet., 204-208 (1974) · Zbl 0297.90019
[2] Por Francis, Dov, Lessepian Migration. Lessepian Migration, Ecological Studies 23 (1978), Springer: Springer New York
[3] Freedman, H. I.; Shukla, J. B., Models of the effect of toxicant in single species and prey-predator systems, J. Math. Biol., 30, 1, 15-30 (1991) · Zbl 0825.92125
[4] Hallam, T. G.; Clark, C. E., Nonautonomous logistic equation as models of populations in a deteriorating environment, J. Theor. Biol., 93, 303-311 (1982)
[5] Hallam, T. G.; Clark, C. E.; Jordan, G. S., Effects of toxicants on population: a qualitative approach II. First order kinetics, J. Math. Biol., 18, 25-37 (1983) · Zbl 0548.92019
[6] Hallam, T. G.; Clark, C. E.; Lassiter, R. R., Effects of toxicants on populations: a qualitative approach I. Equilibrium environmental exposure, Ecol. Model., 18, 291-304 (1983) · Zbl 0548.92018
[7] Hallam, T. G.; de Luna, J. T., Effects of toxicants on populations: a qualitative approach III. Environmental and food chain path ways, J. Theor. Biol., 109, 411-429 (1984)
[8] Hass, C. N., Application of predator-prey models to disinfection, J. Water Pollut. Control. Fed., 53, 378-386 (1981)
[9] Hauping, L.; Zhien, Ma, The threshold of survival system of two species in polluted environment, J. Math. Biol., 30, 1, 49-62 (1991) · Zbl 0745.92028
[10] Jensen, A. L.; Marshal, J. S., Application of a surplus production model to asses environmental impacts on exploited populations of Daphnia pluex in the laboratory, Environ. Pollut. Ser. A, 28, 273-280 (1982)
[11] Lakshmikantham, V.; Leela, S., Differential and integral inequalities, Vol. 1 (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[12] Lave, L. B.; Seskin, E. P., Does air pollution shorten lives, statistical and mathematical aspects of pollution problems, (Pratt, J. W., Statistics: Text Books and Monographs, Vol. 6 (1974), Marcel Dekker: Marcel Dekker New York)
[13] de Luna, J. T.; Hallam, T. G., Effects of toxicants on populations: a qualitative approach IV. Resource-consumer-toxicant models, Ecol. Model., 35, 249-273 (1987)
[14] Zhien, Ma; Baojun, Song; Hallam, T. G., The threshold of survival for systems in a fluctuating environment, Bull. Math. Biol., 51, 3, 311-324 (1989) · Zbl 0676.92010
[15] Zhien, Ma; Guirong, Cui; Wendi, Wang, Persistence and extinction of a population in a polluted environment, Math. Biosci., 101, 75-97 (1990) · Zbl 0714.92027
[16] S.A. Nelson, The problem of oil pollution of the sea, in: Advances in Marine Biology, Academic Press, London, 1970, pp. 215-306.; S.A. Nelson, The problem of oil pollution of the sea, in: Advances in Marine Biology, Academic Press, London, 1970, pp. 215-306.
[17] Otto, K., Diseases of Marine Animals, Vol. 1 (1980), Wiley: Wiley New York
[18] Shukla, J. B.; Freedman, H. I.; Pal, V. N.; Misra, O. P.; Agarwal, M.; Shukla, A., Degradation and subsequent regeneration of a forestry resource: a mathematical model, Ecol. Model., 44, 219-229 (1989)
[19] Woodman, J. N.; Cowling, E. B., Airborne chemicals and forest health, Environ. Sci. Technol., 21, 120-126 (1987)
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