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Predator-prey systems with group defence: The paradox of enrichment revisited. (English) Zbl 0612.92017
The authors discuss survival and extinction of predators in models of predator-prey systems exhibiting group defence of the prey. Such phenomena were less studied before. They show that if there is no mutual interference among predators, enrichment of the prey can lead to extinction in a closed ecosystem and thereby provide more support for Rosenzweig’s warning that is well-known and controverted.
But if there is mutual interference among predators, the model is uniformly persistent just as in previus work. Numerical examples are also indicated.
The models of this paper can be carefully studied with a specific response function. Recently, Zhang Faqin obtained conditions on existence and uniqueness of limit cycles and persistence and extinction for this model with generally chosen parameters (to appear in J. Biomath., Vol. 3).
Reviewer: Chen Lansun

92D40 Ecology
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[1] Andrews, J. F. 1968. ”A Mathematical Model for the Continuous Culture of Microorganisms Utilizing Inhibitory Substrates”.Biotechnol. Bioengng 10, 707–723. · doi:10.1002/bit.260100602
[2] Aris, R. and A. E. Humphrey. 1977. ”Dynamics of a Chemostat in which Two Organisms Compete for a Common Substrate”.Biotechnol. Bioengng 10, 1375–1386. · doi:10.1002/bit.260190910
[3] Beddington, J. R. 1975. ”Mutual Interference Between Parasites of Predators and Its Effect on Searching Efficiency.”J. Anim. Ecol. 44, 331–340. · doi:10.2307/3866
[4] Boon, B. and H. Landelout. 1962. ”Kinetics of Nitrite Oxidation by Nitrobacter Winogradski.”Biochem. J. 85, 440–447.
[5] Bush, A. W. and A. E. Cook. 1976. ”The Effect of Time Delay and Growth Rate Inhibition in the Bacterial Treatment of Wastewater.”J. theor. Biol. 63, 385–395. · doi:10.1016/0022-5193(76)90041-2
[6] Butler, G. J., H. I. Freedman and P. E. Waltman. 1986. ”Uniformly Persistent Systems.”Proc. Am. math. Soc. 96, 425–430. · Zbl 0603.34043 · doi:10.1090/S0002-9939-1986-0822433-4
[7] Erbe, L. H. and H. I. Freedman. 1985. ”Modeling Persistence and Mutual Interference among Subpopulations of Ecological Communities.”Bull. math. Biol. 47, 295–304. · Zbl 0571.92026 · doi:10.1007/BF02460038
[8] Freedman, H. I. 1976. ”Graphical Stability, Enrichment, and Pest Control by a Natural Enemy.”Mathl Biosci. 31, 207–225. · Zbl 0373.92023 · doi:10.1016/0025-5564(76)90080-8
[9] –. 1979. ”Stability Analysis of a Predator-Prey System with Mutual Interference and Density-dependent Death Rates.”Bull. math. Biol. 41, 167–178. · Zbl 0387.92016 · doi:10.1007/BF02547925
[10] –. 1980.Deterministic Mathematical Models in Population Ecology. New York: Marcel Dekker. · Zbl 0448.92023
[11] – and V. S. H. Rao. 1983. ”The Trade-off Between Mutual Interference and Time Lags in Predator-Prey Systems.”Bull. math. Biol. 45, 991–1004. · Zbl 0535.92024 · doi:10.1007/BF02458826
[12] Gilpin, M. E. 1972. ”Enriched Predator-Prey Systems: Theoretical Stability.”Science 177, 902–904. · doi:10.1126/science.177.4052.902
[13] Hassell, M. P. 1971. ”Mutual Interference between Searching Insect Parasites.”J. Anim. Ecol. 40, 473–486. · doi:10.2307/3256
[14] Holling, C. S. 1965. ”The Functional Response of Predators to Prey Density and its Role in Mimicry and Population Regulation.”Mem. ent. Soc. Can. 45, 3–60.
[15] Holmes, J. C. and W. M. Bethel. 1972. ”Modification of Intermediate Host Behaviour by Parasites.”Zool. J. Linn. Soc., Suppl. 1 51, 123–149.
[16] Huffaker, C. B., K. P. Shea, S. G. Herman. 1963. ”Experimental Studies on Predator: Complex Dispersion and Levels of Food in an Acarine Predator-Prey Interaction.”Hilgardia 34, 305–329. · doi:10.3733/hilg.v34n09p305
[17] Luckinbill, L. S. 1973. ”Coexistence in Laboratory Populations ofParamecium Aurelia and Its PredatorDidinium Nasutum.”Ecology 54, 1320–1327. · doi:10.2307/1934194
[18] McAllister, C. D., R. J. Lebrasseur and T. R. Parsons. 1972. ”Stability of Enriched Aquatic Ecosystems.”Science 175, 562–564. · doi:10.1126/science.175.4021.562
[19] May, R. M. 1972. ”Limit Cycles in Predator-Prey Communities.”Science 177, 900–902. · doi:10.1126/science.177.4052.900
[20] Riebesell, J. F. 1974. ”Paradox of Enrichment in Competitive Systems.”Ecology 55, 183–187. · doi:10.2307/1934634
[21] Rogers, D. J. and M. P. Hassell. 1974. ”General Models for Insect Parasite and Predator Searching Behaviour: Interference.”J. Anim. Ecol. 43, 239–253. · doi:10.2307/3170
[22] Rosenzweig, M. L. 1971. ”Raradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time.”Science 171, 385–387. · doi:10.1126/science.171.3969.385
[23] –. 1972a. ”Reply to McAllisteret al.”Science 175, 564–565.
[24] –. 1972b. ”Reply to Gilpin.”Science 177, 904. · doi:10.1126/science.177.4052.904
[25] – and W. M. Schaffer. 1978. ”Homage to the Red Queen II. Coevolutionary Response to Enrichment of Exploitation Ecosystems.”Theor. Pop. Biol. 14, 158–163. · Zbl 0383.92020 · doi:10.1016/0040-5809(78)90009-6
[26] Schaffer, W. M. and M. L. Rosenzweig. 1978. ”Homage to the Red Queen I. Coevolution of Predators and their Victims.”Theor. Pop. Biol. 14 135–157. · Zbl 0383.92019 · doi:10.1016/0040-5809(78)90008-4
[27] Tener, J. S. 1965.Muskoxen. Ottawa: Queen’s Printer.
[28] Yang, R. D. and A. E. Humphrey. 1975. ”Dynamics and Steady State Studies of Phenol Biodegeneration in Pure and Mixed Cultures.”Biotechnol. Bioengng 17, 1211–1235. · doi:10.1002/bit.260170809
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