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Three-species food-chain models with mutual interference and time delays. (English) Zbl 0592.92024
Summary: Models of three-species food chains incorporating mutual interference among predators and time delays due to gestation are proposed. We consider the case that for such models an interior equilibrium exists. In the case that the equilibrium is stable with no delays, the length of delay is estimated for which stability continues to hold. Further, conditions are derived under which there can be no change of stability.

92D40 Ecology
34D99 Stability theory for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
[1] Beddington, J.R., Mutual interference between parasites and predators and its effect on searching efficiency, J. animal ecology, 44, 331-340, (1975)
[2] G.J. Butler, H.I. Freedman, and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., to appear.
[3] Chuma, J.; van den Driessche, P., A general second-order transcendental equation, Appl. math. notes, 5, 85-96, (1980) · Zbl 0443.34063
[4] Cooke, K.L.; Grossman, Z., Discrete delay distributed delay and stability switches, J. math. anal. appl., 86, 592-627, (1982) · Zbl 0492.34064
[5] Cushing, J.M., Integrodifferential equations and delay models in population dynamics, () · Zbl 0363.92014
[6] Driver, R.D., Ordinary and delay differential equations, (1977), Springer Heidelberg · Zbl 0374.34001
[7] Erbe, L.H.; Freedman, H.I., Modeling persistence and mutual interference among subpopulations of ecological communities, Bull. math. biol., 47, 295-304, (1985) · Zbl 0571.92026
[8] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[9] 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
[10] H.I. Freedman and V.S.H. Rao, Stability criteria for a system involving two time delays, SIAM J. Appl. Math., to appear. · Zbl 0624.34066
[11] Freedman, H.I.; So, J.W.-H., Global stability and persistence of simple food chains, Math. biosci., 76, 69-86, (1985) · Zbl 0572.92025
[12] Freedman, H.I.; Waltman, P., Persistence in models of three interacting predator-prey populations, Math. biosci, 68, 213-231, (1984) · Zbl 0534.92026
[13] Gopalsamy, K., Harmless delays in model ecosystems, Bull. math. biol., 45, 295-309, (1983) · Zbl 0514.34060
[14] Hale, J., Theory of functional differential equations, (1977), Springer Heidelberg
[15] Hassell, M.P.; May, R.M., Stability in insect host-parasite models, J. animal ecology, 42, 693-726, (1973)
[16] Hastings, A., Age-dependent predation is not a simple process I, Theoret. population biol., 23, 247-362, (1983) · Zbl 0507.92016
[17] A. Hastings, Age-dependent predation is not a simple process II, Theoret. Population Biol., to appear. · Zbl 0541.92018
[18] MacDonald, N., Time lags in biological models, () · Zbl 0403.92020
[19] May, R.M., Time-delay versus stability in population models with two and three trophic levels, Ecology, 54, 315-325, (1973)
[20] Nunney, L., The effect of long time delays in predator-prey systems, Theoret. population biol., 27, 202-221, (1985) · Zbl 0566.92020
[21] R.D. Nussbaum and A.J.B. Potter, Cyclic differential equations and period three solutions of differential-delay equations, J. Differential Equations 46:379-408. · Zbl 0533.34050
[22] Rao, V.S.H.; Freedman, H.I., A model predator-prey system with mutual interference and time delay, (), 317-319, Lecture notes in Biomathematics · Zbl 0524.92021
[23] Rogers, D.J.; Hassell, M.P., General models for insect parasite and searching behaviour: interference, J. animal ecology, 43, 239-253, (1974)
[24] Sánchez, D.A., Ordinary differential equations and stability theory: an introduction, (1968), Freeman San Francisco · Zbl 0179.12502
[25] Thingstad, T.F.; Langeland, T.I., Dynamics of a chemostat culture: the effect of a delay in a cell response, J. theoret. biol., 48, 149-159, (1974)
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