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The threshold of survival for system of two species in polluted environment. (English) Zbl 0745.92028
The authors modify the classical Volterra model for two interacting species by subtracting time dependent, dynamically modeled, terms from the inherent growth rates of the species that account for the toxic effects of an environmental pollutant. Necessary and sufficient conditions for the persistence (in the mean) and for the extinction of each individual species are proved for predator-prey, competitive, and cooperative interactions that have a stable, coexistence equilibrium in the absence of the polluting toxin.
These results are used to obtain persistence (in the mean) and extinction criteria based upon the parameters of the toxin dynamics (in particular, the exogenous input rate of toxin into the environment).

MSC:
92D40 Ecology
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[1] Hallam, T. G., Clark, C. E., Lassiter, R. R.: Effects of toxicants on populations: a qualitative approach I. Equilibrium environmental exposure. Ecol. Modelling 18, 291-304 (1983) · Zbl 0548.92018 · doi:10.1016/0304-3800(83)90019-4
[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, 25-37 (1983) · Zbl 0548.92019
[3] Hallam, T. G., De Luna, J. T.: Effects of toxicants on populations: a qualitative approach III. Environmental and food chain pathways. J. Theor. Biol. 109, 411-429 (1984) · doi:10.1016/S0022-5193(84)80090-9
[4] Hallam, T. G., Ma Zhien: Persistence in population models with demographic fluctuations. J. Math. Biol. 24, 327-339 (1986) · Zbl 0606.92022
[5] De Luna, J. T., Hallam, T. G.: Effects of toxicants on populations: a qualitative approach IV. Resource-Consumer-Toxicant models. Ecol. Modelling 35, 249-273 (1987) · doi:10.1016/0304-3800(87)90115-3
[6] Ma Zhien, Song Bao-Jun, Hallam, T. G.: The threshold of survival for systems in a fluctuating environment. Bull. Math. Biol. 51(3), 311-323 (1989) · Zbl 0676.92010
[7] Ma Zhien, Hallam, T. G.: Effects of parameter fluctuations on community survival. Math. Biosci. 86, 35-49 (1987) · Zbl 0631.92019 · doi:10.1016/0025-5564(87)90062-9
[8] Ma Zhien, Wang Wendi: Asymptotical behavior of Lotka-Volterra models. In: Mathematical Modelling and dynamic behavior in biosystems, pp. 44-52. Depart. Math. Xi’an Jiaotong University 1988
[9] Kirlinger, G.: Permanence of some ecological systems with several predator and one prey species. J. Math. Biol. 26, 217-232 (1988) · Zbl 0713.92025 · doi:10.1007/BF00277734
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