Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator. (English) Zbl 1151.34067

Summary: A delayed predator-prey system with stage structure for the predator is studied. It is found that the time delay is harmless for permanence of the stage-structured system. If \(\alpha \beta < 1\), sufficient conditions which guarantee the global stability of positive equilibrium are given. If \(\alpha \beta > 1\), we show that the unique positive equilibrium is locally asymptotically stable when the time delay \(\tau ^{*}\) is sufficiently small, while loss of stability by a Hopf bifurcation can occur as the delay increases.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
34K13 Periodic solutions to functional-differential equations
Full Text: DOI


[1] Freedman, H. I.; Gopalsamy, K., Nonoccurance of stability switching in systems with discrete delays, Can. Math. Bull., 31, 52-58 (1988) · Zbl 0607.34062
[2] Huo, H. F.; Li, W. T., Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput., 148, 1, 35-46 (2004) · Zbl 1057.34093
[3] Kuang, Y., Delay Differential Equation with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[4] Liu, S.; Chen, L.; Liu, Z., Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274, 67-684 (2002) · Zbl 1039.34068
[5] May, R. M., Time delay versus stability in population models with two or trophic levels, Ecology, 54, 315-325 (1973)
[6] Wang, L.; Li, W., Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146, 167-185 (2003) · Zbl 1029.92025
[7] Huo, H.; Li, W., Periodic solutions of a periodic Lotka-Volterra system with delays, Appl. Math. Comput., 156, 787-803 (2004) · Zbl 1069.34099
[8] Blythe, S. P.; Nisbet, R. M.; Gurney, W. S.C., Stability switches in distributed delay models, J. Math. Anal. Appl., 109, 388-396 (1985) · Zbl 0589.92018
[9] Ma, Z., Stability of predation models with time delays, Appl. Anal., 22, 169-192 (1986) · Zbl 0592.92020
[10] Chen, L.; Song, X.; Lu, Z., Mathematical Models and Methods in Ecology (2003), Science Press: Science Press Beijing, (in Chinese)
[11] Aiello, W. G.; Freedman, H. I., A time-delay model of single-species growth with stage structure, Math. Biosci., 101, 139-153 (1990) · Zbl 0719.92017
[12] Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 3, 855-869 (1992) · Zbl 0760.92018
[13] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Global stability of a stage-structured predator-prey model with prey dispersal, Appl. Math. Comput., 171, 293-314 (2005) · Zbl 1080.92069
[14] Freedman, H. I.; Wu, J., persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. Appl. Math., 2, 351-371 (1991) · Zbl 0732.92021
[15] Liu, S.; Chen, L.; Luo, G., Extinction and permanence in competitive stage structured system with time-delays, Nonlinear Anal. Th. Mech. Appl., 51, 1347-1361 (2002) · Zbl 1021.34065
[16] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay, Appl. Math. Comput., 159, 863-880 (2004) · Zbl 1056.92063
[17] Wang, W.; Ma, Z.; Freedman, H. I., Global stability of Volterra models with time delay, J. Math. Anal. Appl., 160, 51-59 (1991) · Zbl 0760.34058
[18] Xiao, Y.; Chen, L.; van den Bosch, F., Dynamical behavior for a stage-structured SIR infection disease model, Nonlinear Anal.: RWA, 3, 175-190 (2002) · Zbl 1007.92032
[19] Hinggins, K.; Hastings, A.; Botsford, L., Density dependence and age structure: nonlinear dynamics and population behavior, Am. Nat., 149, 247-269 (1997)
[20] Wang, W.; Chen, L., A predator-prey system with stage stricture for predator, Comput. Math. Appl., 33, 83-92 (1997)
[21] Wang, W., Global dynamics of a population model with stage structure a predator-prey system with stage structure for predator, (Chen, L.; Ruan, S.; Zhu, J., Advanced Topics in Biomathematics (1998), World Scientific), 253-257 · Zbl 0986.92026
[22] Zhang, X.; Chen, L.; Neumann, A. U., The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., 168, 201-210 (1974) · Zbl 0961.92037
[23] Gurney, W. S.C.; Nisbet, R. M.; Lawton, J. H., The systematic formulation of tractable single species population models incorporating age structure, J. Animal Ecol., 52, 479-485 (1983)
[24] Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52, 855-869 (1992) · Zbl 0760.92018
[25] Freedman, H. I.; Sree Hari Rao, V., The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol., 45, 991 (1983) · Zbl 0535.92024
[26] Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer New York · Zbl 0425.34048
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.