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Hopf bifurcation in three-species food chain models with group defense. (English) Zbl 0761.92039
Summary: Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.

34C23Bifurcation (ODE)
34D99Stability theory of ODE
Full Text: DOI
[1] Boon, B.; Laudelout, H.: Kinetics of intrite oxydation. Nitrobacter winogrodski biochem. J. 85, 440-447 (1962)
[2] Chow, S. N.; Hale, J. K.: Methods of bifurcation. (1982) · Zbl 0487.47039
[3] Cooke, K. L.; Grossman, Z.: Discrete delay, distributed delay and stability switches. J. math. Anal. appl. 86, 592-627 (1982) · Zbl 0492.34064
[4] Cooke, K. L.; Den Driessche, P. Van: On zeros of some transcendental equations. Funkcial. ekvac. 29, 77-90 (1986) · Zbl 0603.34069
[5] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. (1977) · Zbl 0363.92014
[6] Erbe, L. H.; Freedman, H. I.; Rao, V. S. H.: Three species food chain models with mutual interference and time delays. Math. biosci. 80, 57-80 (1986) · Zbl 0592.92024
[7] Freedman, H. I.: Deterministic mathematical models in population ecology. (1986) · Zbl 0448.92023
[8] Freedman, H. I.; Gopalsamy, K.: Nonoccurrence of stability switching in systems with discrete delays. Canad. math. Bull. 31, 52-58 (1988) · Zbl 0607.34062
[9] Freedman, H. I.; Quan, H.: Interactions leading to persistence in predator-prey systems with group defense. Bull. math. Biol. 50, 517-530 (1988) · Zbl 0673.92011
[10] Freedman, H. I.; Rao, V. S. H.: The trade-off between mutual interference and time lags in predator-prey systems. Bull. math. Biol. 45, 991-1004 (1983) · Zbl 0535.92024
[11] Freedman, H. I.; Rao, V. S. H.: Stability criteria for a system involving two time delays. SIAM J. Appl. math. 46, 552-560 (1986) · Zbl 0624.34066
[12] Freedman, H. I.; So, J. W. H.: Global stability and persistence of simple food chains. Math. biosci. 76, 69-86 (1985) · Zbl 0572.92025
[13] Freedman, H. I.; Wolkowicz, G.: Predator-prey systems with group defense: the paradox of enrichment revised. Bull. math. Biol. 48, 493-508 (1986) · Zbl 0612.92017
[14] Gilpin, M. E.: Enriched predator-prey systems: theoretical stability. Science 177, 902-904 (1972)
[15] Gopalsamy, K.: Delayed responses and stability in two-species systems. J. austral. Math. soc. Ser. B 25, 473-500 (1984) · Zbl 0552.92016
[16] Gopalsamy, K.; Aggarwala, B. D.: Limit cycles in two-species competition with time delays. J. austral. Math. soc. Ser. B 22, 148-160 (1980) · Zbl 0458.92014
[17] Hale, J. K.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[18] Hassard, B. D.: A code for Hopf bifurcation analysis of autonomous delay-differential systems. Oscillation, bifurcation and chaos, CMS conference Proceedings, 447-463 (1986)
[19] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation. (1981) · Zbl 0474.34002
[20] Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. entomology. Soc. Canada 45, 3-60 (1965)
[21] Holmes, J. C.; Bethel, W. M.: Modification of intermediate host behavior by parasites. Zoolog. J. Linear. soc. Suppl. 1 51, 123-149 (1972)
[22] Huffaker, C. B.; Shea, K. P.; Herman, S. G.: Experimental studies on predation: complex dispersion and levels of food in an alarine predator-prey interaction. Hilgardia 34, 305-429 (1963)
[23] Luckinbill, L. S.: Coexistence in laboratory populations of paramecium aurelia and its predator. Dudinium nasutium ecology 54, 1320-1327 (1973)
[24] Macdonald, N.: Time lays in biological models. (1978) · Zbl 0403.92020
[25] Marsden, J. E.; Mckracken, M.: The Hopf bifurcation and its applications. (1976)
[26] May, R. M.: Limit cycles in predator-prey communities. Science 177, 900-902 (1972)
[27] Mcallister, C. D.; Lebrasseur, R. J.; Parsons, T. R.: Stability of enriched aquatic ecosystems. Science 175, 564-565 (1972)
[28] Mischaikow, K.; Wolkowicz, G.: A connection matrix approach illustrated by means of a predator-prey model involving group defense. Mathematical ecology, 682-716 (1988) · Zbl 0687.92018
[29] Mischaikov, K.; Wolkowicz, G.: A predator-prey system involving group defense: a connection matrix approach. Nonlin. anal. Th. math. Appl. 14, 955-969 (1990) · Zbl 0724.34015
[30] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. functional anal. 7, 487-513 (1971) · Zbl 0212.16504
[31] Riebesell, J. F.: Paradox of enrichment in competitive systems. Ecology 55, 183-187 (1974)
[32] Rosenzweig, M. L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385-387 (1971)
[33] Rosenzweig, M. L.: Reply to mccallister et al.. Science 175, 564-565 (1972)
[34] Rosenzweig, M. L.: Reply to gilpin. Science 177, 904 (1972)
[35] Ruan, S.; Freedman, H. I.: Persistence in three-species food chain models with group defense. Math. biosci. 107, 111-125 (1991) · Zbl 0752.92027
[36] Schaffer, W. H.; Rosenzweig, M. L.: Homage to the red queen. Part I. Coevolution of predators and their victims. Theoret. pop. Biol. 14, 135-157 (1978) · Zbl 0383.92019
[37] Tener, J. S.: Muskoxen. (1965)
[38] Thingstad, T. F.; Langeland, T. I.: Dynamics of a chemostat culture: the effect of a delay in a cell response. J. theor. Biol. 48, 149-159 (1974)
[39] Wolkowicz, G.: Bifurcation analysis of a predator-prey system involving group defense. SIAM J. Appl. math. 48, 1-15 (1988) · Zbl 0657.92015
[40] Yang, R. D.; Humphrey, A. E.: Dynamics and steady state studies of phenal biodegradation in pure and mixed cultures. Biotechnol. bioeng. 17, 1121-1235 (1975)