×

Dynamics of a class of non-autonomous systems of two non-interacting preys with common predator. (English) Zbl 1069.34071

The system \[ y_i'= y_i(a_i- c_i y_i- b_i y_3),\quad i= 1,2, \]
\[ y_3'= y_3(- a_3+ b_3 y_1+ b_4 y_2), \] with positive, variable coefficients is investigated with respect to permanence, extinction and global stability. For periodic coefficients also periodic solutions are studied.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Azar, C.; Holmberg, J.; Lindgren, K., Stability analysis of harvesting in a predator-preymodel, J. Theort. Biol., 174, 13-19 (1995) · doi:10.1006/jtbi.1995.0076
[2] Barbalat, L., Systemes d’ equations differentielles d’oscillations nonlineaires, Rev. Roumaine Math. Pures. Appl., 4, 267-270 (1959) · Zbl 0090.06601
[3] Beretta, E.; Kuang, Y., Convergence results in a well-known predator prey system, J. Math. Anal. Appl., 204, 840-853 (1996) · Zbl 0876.92021 · doi:10.1006/jmaa.1996.0471
[4] Bhattachaya, D. K.; Karan, S., Pest managment of two non-interacting pests in presence of common predator, J. Appl. Math. Computing, 13, 301-322 (2003) · Zbl 1047.34033
[5] Cavani, M.; Lizana, M., Stable perriodic for a predator-prey model with delay, J. Math. Anal. Appl., 249, 324-339 (2000) · Zbl 0997.92035 · doi:10.1006/jmaa.2000.6802
[6] Feng, M.; Wang, K., Global existence of positive periodic solutions of periodic-predator-prey system with infinite delays, J. Math. Anal. Appl., 262, 1-11 (2001) · Zbl 0995.34063 · doi:10.1006/jmaa.2000.7181
[7] Gaines, R. E.; Mawhin, J. L., Coincidence degree and nonlinear differential equations (1977), Berlin: Springer, Berlin · Zbl 0339.47031
[8] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics (1992), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0752.34039
[9] He, X.-Z., Stability and delays in a predator prey system, J. Math. Anal. Appl., 198, 355-370 (1996) · Zbl 0873.34062 · doi:10.1006/jmaa.1996.0087
[10] Krawcewicz, W.; Wu, J., Theory of degrees with applications to bifurcations and differential equations (1997), New York: John Wily, New York · Zbl 0882.58001
[11] Kuang, Y., Delay differential equations with applications in population dynamics (1993), New York: Academic press, New York · Zbl 0777.34002
[12] Kumar, S.; Sirvastava, S. K.; Chingakham, P., Hopf bifurcation and stability analysis in a harvasted one-predator-two-prey model, Appl. Math. Comp., 126, 107-118 (2002) · Zbl 1017.92041 · doi:10.1016/S0096-3003(01)00033-9
[13] Li, Y., Positive periodic solution for neutral delay model, Acta Math. Scinica, 39, 790-795 (1996)
[14] Li, Y., Periodic solutions of a periodic neutral delay equations, J. Math. Anal. Appl., 214, 11-21 (1997) · Zbl 0894.34075 · doi:10.1006/jmaa.1997.5576
[15] Li, Y., Periodic solution of a periodic delay predator-prey system, Proc. Amer. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057 · doi:10.1090/S0002-9939-99-05210-7
[16] Saker, S. H., Oscillation and global attractivity in a periodic delay hematopoiesis model, J. Appl. Math. and Computing, 13, 1-2, 287-300 (2003) · Zbl 1055.34127 · doi:10.1007/BF02936093
[17] Saker, S. H., Oscillation and global attractivity of hematopoiesis model with delay time, Appl. Math. Comp., 136, 2-3, 27-36 (2003) · Zbl 1026.34082
[18] Saker, S. H.; Agarwal, Sheba, Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics, Comp. Math. Appl., 44, 5-6, 623-632 (2002) · Zbl 1041.34073 · doi:10.1016/S0898-1221(02)00177-3
[19] Saker, S. H.; Agarwal, S., Oscillation and global attractivity in a periodic Nicholson’s Blowflies model, Mathl. Comp. Modelling, 35, 719-731 (2002) · Zbl 1012.34067 · doi:10.1016/S0895-7177(02)00043-2
[20] S. H. Saker and S. Agarwal,Oscillation and global attractivity of a periodic survival red blood cells model, Journal Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, in press. · Zbl 1078.34062
[21] Saker, S. H.; Agarwal, S., Oscillation and global attractivity in nonlinear delay periodic model of population dynamics, Applicable Analysis, 81, 787-799 (2002) · Zbl 1041.34061 · doi:10.1080/0003681021000004429
[22] Takeuchi, Y.; Adachi, N., Existence and bifurcation of stable equilibrium in two-prey, one predator communities, Bull. Math. Biol., 45, 877-900 (1983) · Zbl 0524.92025
[23] Tang, S.; Chen, L., Global qualitative analysis for a ratio-dependent predator-prey model with delay, J. Math. Anal. Appl., 266, 401-419 (2002) · Zbl 1069.34122 · doi:10.1006/jmaa.2001.7751
[24] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence, and stability of stage-structured predator-prey model, J. Math. Anal. Appl., 262, 499-528 (2001) · Zbl 0997.34069 · doi:10.1006/jmaa.2001.7543
[25] Xu, R.; Chaplain, M. A. J., Persistence and global stability in a delayed Gause-type predator-prey system with dominating instantaneous negative feedbacks, J. Math. Anal. Appl., 256, 148-162 (2002) · Zbl 1013.34074 · doi:10.1006/jmaa.2001.7701
[26] Upadhyay, R. K.; Iyengar, S. R.; Rai, V., Stability and complexity in ecological systems, Chaos, Solitons and Fractals, 11, 533-542 (2000) · Zbl 0943.92033 · doi:10.1016/S0960-0779(98)00112-X
[27] Zhang, Z.; Wang, Z., Periodic solutions for a two-species nonautonomous competition Lotka-Volterra pathch system with time delay, J. Math. Anal. Appl., 265, 38-48 (2002) · Zbl 1003.34060 · doi:10.1006/jmaa.2001.7682
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.