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The threshold of survival for systems in a fluctuating environment. (English) Zbl 0676.92010
Consider the following systems in a fluctuating environment: \[ (1)\quad dx/dt=(x/g(x))[r_ 0-r_ 1c_ T(t)-xf(x)],\quad t\in {\mathbb{R}}^ 1_+; \] \[ (2)\quad dx/dt=xF(r(t),x),\quad x\in {\mathbb{R}}^ 1_+\quad (Kolmogorov\quad model); \] \[ (3)\quad da/dt=f(t)-\beta ax,\quad dx/dt=x[r_ 0-r_ 1c_ T(t)-kxa^{-1}];\quad (Leslie\quad resource- consumer\quad model); \] \[ (4)\quad da/dt=f(t)-\omega x[1-\exp (-\eta a/x)],\quad dx/dt=x[r_ 0-r_ 1c_ T(t)-\gamma \omega \exp (-\eta a/x)],\quad (Gallopin\quad resource-consumer\quad model). \] The authors demonstrated the existence of a survival threshold for situations where demographic parameters are fluctuating, generally, in a nonperiodic manner. In general, the survival threshold is determined by a relationship between mean stress measure in organisms, the ratio of the population intrinsic growth rate and stress response rate. Five theorems are proved.
Reviewer: Bingxi Li

92D40 Ecology
92D25 Population dynamics (general)
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[1] Barber, M. C., L. A. Suarez and R. R. Lassiter 1988. ”Kinetic Exchange of Nonpolar Organic Pollutants by Fish.”Envir. Tox. Chem., in press.
[2] Dickson, K. L., A. W. Maki and J. Cairns Jr. 1982.Modeling the Fate of Chemicals in the Aquatic Environment, Ann Arbor, MI: Ann Arbor.
[3] Freedman, H. I. and P. Waltman. 1985. ”Persistence in a Model of Three Competitive Populations,”Math. Biosci 73, 89–101. · Zbl 0584.92018 · doi:10.1016/0025-5564(85)90078-1
[4] Gallopin, G. C. 1971. ”A Generalized Model of Resource-Population System: I. General Properties. II. Stability Analysis.Oecologia 7, 382–413;7, 414–432. · doi:10.1007/BF00345861
[5] Hallam, T. G. 1986. ”Population Dynamics in Homogeneous Environment.” InMathematical Ecology: An Introduction, T. G. Hallam and S. A. Levin (Eds), pp. 61–94, Springer, New York.
[6] –, Clark, C. E. and G. S. Jordan. 1983. ”Effects of Toxicants on Populations: A Qualitative Approach. II. First Order Kinetics.”J. math Biol. 18, 25–37. · Zbl 0548.92019 · doi:10.1007/BF00275908
[7] – and J. L. deLuna. 1984. ”Effects of Toxicants on Populations: A Qualitative Approach. III. Environmental and Food Chain Pathways.J. theor. Biol. 109, 411–429. · doi:10.1016/S0022-5193(84)80090-9
[8] –, R. R. Lassiter, J. Li and W. McKinney. 1988.Physiologically Structured Population Models in Risk Assessment. Biomathematics and Related Computational Problems, L. Ricciardi (Ed.), Amsterdam: Riedel. · Zbl 0647.92022
[9] – and Z. Ma 1986. ”Persistence in Population Models with Demographic Fluctuations.J. math. Biol. 24, 327–339. · Zbl 0606.92022 · doi:10.1007/BF00275641
[10] – and Z. Ma 1987. ”On Density and Extinction in Continuous Population Models.J. math. Biol. 25, 191–201. · Zbl 0641.92011 · doi:10.1007/BF00276389
[11] Hutchinson, G. E. 1978.An Introduction to Population Ecology. Yale: New Haven. · Zbl 0414.92026
[12] Leslie, P. H. 1984. ”Some Further Notes on the Use of Matrices in Population Mathematics.”Biometrika 35, 213–245. · Zbl 0034.23303 · doi:10.1093/biomet/35.3-4.213
[13] Ma, Z. and T. G. Hallam. 1987. ”Effects of Parameter Fluctuations on Community Survival.Math. Biosci. 86, 35–49. · Zbl 0631.92019 · doi:10.1016/0025-5564(87)90062-9
[14] Moriarty, F. 1987.Ecotoxicology. New York: Academic Press.
[15] Saxena, J. and F. Fisher (Eds), 1981.Harard Assessment of Chemicals. Current Developments, Vol. 1. New York: Academic Press.
[16] Schultz, T. W., G. W. Holcombe and G. L. Phipps, 1986. ”Relationships of Quantitative Structure-activity to Comparative Toxicity of Selected Phenols in thePimephales promelas andTetrahymena pyriformis Test Systems.”Ecotoxicol. Environ. Saf 12, 146–153. · doi:10.1016/0147-6513(86)90051-5
[17] Sheehan, P. J., P. R. Miller, G. C. Butler and P. Bourdeau (Eds). 1984.Effects of Pollutants at the Ecosystem Level. New York: Wiley.
[18] Smith, F. E. 1963. ”Population Dynamics inDaphnia magna and a New Model for Population Growth.Ecology 44, 651–663. · doi:10.2307/1933011
[19] Veith, G. D., D. J. Call and L. T. Brooke. 1983. ”Structure Toxicity Relationships for the Fathead Minnow,Pimephales promelas: Narcotic Industrial Chemicals.Can. J. Fish. Aquat. Sci. 40, 681–698. · doi:10.1139/f83-096
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