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Oscillation and global attractivity in a periodic Nicholson’s blowflies model. (English) Zbl 1012.34067

Here, the authors consider the following nonlinear delay differential equation \[ N'(t)=-\delta N(t)+P(t)N(t-m\omega)e^{-\alpha N(t-n\omega)},\tag{*} \] where \(m\) is a positive integer, \(\delta(t)\) and \(P(t)\) are positive \(\omega\)-periodic functions. In the non-delay case, they show that equation (*) has a unique positive periodic solution and provide sufficient conditions for their global attractivity. In the delay case, they provide sufficient conditions for the oscillation of all positive solutions.

MSC:

34K11 Oscillation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
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[1] Gurney, W. S.; Blythe, S. P.; Nisbet, R. M., Nicholson’s blowflies (revisited), Nature, 287, 17-21 (1980)
[2] Győri, I.; Ladas, G., Linearized oscillations for equations with piecewise constant arguments, Differential and Integral Equations, 2, 123-131 (1989) · Zbl 0723.34058
[3] Győri, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0780.34048
[4] Kocic, V. L.; Ladas, G., Oscillation and global attractivity in the discrete model of Nicholson’s blowflies, Applicable Analysis, 38, 21-31 (1990) · Zbl 0715.39003
[5] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993), Kluwer: Kluwer Dordrecht · Zbl 0787.39001
[6] Kulenovic, M. R.S; Ladas, G.; Sficas, Y. S., Global attractivity in Nicholson’s blowflies, Applicable Analysis, 43, 109-124 (1992) · Zbl 0754.34078
[7] Nicholson, A. J., An outline of the dynamics of animal populations, Austral. J. Zool., 2, 9-25 (1954)
[8] Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer: Kluwer Dordrecht · Zbl 0752.34039
[9] Gopalsamy, K.; Kulenovic, M. R.S; Ladas, G., Oscillation and global attractivity in respiratory dynamics, Dyn. Stab. Systems, 4, 131-139 (1989) · Zbl 0683.92009
[10] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[11] I. Kubiaczyk and S.H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Mathl. Comput. Modelling; I. Kubiaczyk and S.H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Mathl. Comput. Modelling · Zbl 1069.34107
[12] Mackey, M. C.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 287-289 (1977) · Zbl 1383.92036
[13] Kulenovic, M. R.S; Ladas, G., Linearized oscillation in population dynamics, Bull. Math. Biol., 44, 615-627 (1987) · Zbl 0634.92013
[14] Nicholson, A. J., The balance of animal population, J. Animal Ecology, 2, 132-178 (1993)
[15] Gopalsamy, K.; He, X.-Z, Oscillation and convergence in almost periodic competition system, Acta Appl. Math., 46, 247-266 (1997) · Zbl 0872.34050
[16] Gopalsamy, K.; Kulenovic, M. R.S; Ladas, G., Environmental periodicity and time delays in a ‘Food Limited’ population model, J. Math. Anal. Appl., 147, 545-555 (1990) · Zbl 0701.92021
[17] Gopalsamy, K.; Trofimchuk, S. I., Almost periodic solutions of Lasota-Wazewska type delay differential equations, J. Math. Anal. Appl., 237, 106-127 (1999) · Zbl 0936.34058
[18] Graef, J. R.; Qian, C.; Spikes, P. W., Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull., 39, 275-283 (1996) · Zbl 0870.34073
[19] Hui, F.; Li, J., On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl., 256, 8-17 (2001) · Zbl 0995.34073
[20] Kulenovic, M. R.S; Ladas, G.; Sficas, Y. S., Global attractivity in population dynamics, Computers Math. Applic., 18, 10/11, 925-928 (1989) · Zbl 0686.92019
[21] Lalli, B. S.; Zhang, B. G., On a periodic delay population model, Quart. Appl. Math., 52, 35-42 (1994) · Zbl 0788.92022
[22] S.H. Saker and S. Agarwal, Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics, Computers Math. Applic.; S.H. Saker and S. Agarwal, Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics, Computers Math. Applic. · Zbl 1041.34073
[23] Yan, J.; Feng, Q., Global attractivity and oscillation in a nonlinear delay equation, Nonlinear Analysis, 43, 101-108 (2001) · Zbl 0987.34065
[24] Barbalat, L., Systemes d’ equations differentielles d’oscillations nonlineaires, Rev. Roumaine Math. Pures. Appl., 4, 267-270 (1959) · Zbl 0090.06601
[25] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. Z., (Oscillation Theory of Differential Equations with Deviating Arguments (1987), Marcel Dekker: Marcel Dekker New York) · Zbl 0622.34071
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