Hopf bifurcation in a delayed food-limited model with feedback control. (English) Zbl 1306.93033
Summary: In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results.
MSC:
| 93B52 | Feedback control |
| 92D40 | Ecology |
| 70K50 | Bifurcations and instability for nonlinear problems in mechanics |
| 70K42 | Equilibria and periodic trajectories for nonlinear problems in mechanics |
References:
| [1] | Smith, F.E.: Population dynamics in Daphnia magna. Ecology 44, 651-663 (1963) · doi:10.2307/1933011 |
| [2] | Gopalsamy, K., Kulenovic, M.R.S., Ladas, G.: Time lags in a food-limited population model. Appl. Anal. 31, 225-237 (1988) · Zbl 0639.34070 · doi:10.1080/00036818808839826 |
| [3] | Gopalsamy, K., Kulenovic, M.R.S., Ladas, G.: Environmental periodicity and time delay in a food-limited populationmodel. J. Math. Anal. Appl. 147, 545-555 (1990) · Zbl 0701.92021 · doi:10.1016/0022-247X(90)90369-Q |
| [4] | So, J.W.H., Yu, J.S.: On the uniform stability for a food-limited population model with time delay. Proc. R. Soc. Edinburgh Sect. A 125, 991-1002 (1995) · Zbl 0844.34079 · doi:10.1017/S0308210500022605 |
| [5] | Wan, A.Y., Wei, J.J.: Hopf bifurcation analysis of a food-limited population model with delay. Nonlinear Anal. RWA 11, 1087-1095 (2010) · Zbl 1183.37155 · doi:10.1016/j.nonrwa.2009.01.052 |
| [6] | Su, Y., Wan, A.Y., Wei, J.J.: Bifurcation analysis in a diffusive food-limited model with time delay. Appl. Anal. 89, 1161-1181 (2010) · Zbl 1201.35037 · doi:10.1080/00036810903116010 |
| [7] | Gourley, S.A., So, J.W.H.: Dynamics of a food-limited populationmodel incorporating nonlocal delays on a finite domain. J. Math. Biol. 44, 49-78 (2002) · Zbl 0993.92027 · doi:10.1007/s002850100109 |
| [8] | Tang, S.Y., Chen, L.S.: Global attractivity in a food-limited population model with impulsive effects. J. Math. Anal. Appl. 292, 211-221 (2004) · Zbl 1062.34055 · doi:10.1016/j.jmaa.2003.11.061 |
| [9] | Chen, F.D., Sun, D.X., Shi, J.L.: Periodicity in a food-limited population model with toxicants and state dependent delays. J. Math. Anal. Appl. 288, 136-146 (2003) · Zbl 1087.34045 · doi:10.1016/S0022-247X(03)00586-9 |
| [10] | Feng, W., Lu, X.: On diffusive population models with toxicants and time delays. J. Math. Anal. Appl. 233, 373-386 (1999) · Zbl 0927.35049 · doi:10.1006/jmaa.1999.6332 |
| [11] | Wang, Z.C., Li, W.T.: Monotone travelling fronts of a food-limited population model with nonlocal delay. Nonlinear Anal. RWA 8, 699-712 (2007) · Zbl 1152.35408 · doi:10.1016/j.nonrwa.2006.03.001 |
| [12] | Gourley, S.A.: Wave front solution of a diffusive delay model for populations of Daphnia magna. Comput. Math. Appl. 42, 1421-1430 (2001) · Zbl 0998.92029 · doi:10.1016/S0898-1221(01)00251-6 |
| [13] | Davidson, F.A., Gourley, S.A.: The effects of temporal delays in amodel for a food-limited diffusing population. J. Math. Anal. Appl. 261, 633-648 (2001) · Zbl 0992.35047 · doi:10.1006/jmaa.2001.7563 |
| [14] | Gopalsamy, K., Weng, P.X.: Feedback regulation of logistic growth. Int. J. Math. Sci. 16, 177-192 (1993) · Zbl 0765.34058 · doi:10.1155/S0161171293000213 |
| [15] | Aizerman, M.A., Gantmacher, F.R.: Absolute stability of regulator systems. Holden Day, San Francisco (1964) · Zbl 0123.28401 |
| [16] | Lefschetz, S.: Stability of nonlinear control systems. Academic Press, New York (1965) · Zbl 0136.08801 |
| [17] | Song, Y.L., Yuan, S.L.: Bifurcation analysis for a regulated logistic growth model. Appl. Math. Model. 31, 1729-1738 (2007) · Zbl 1167.34377 · doi:10.1016/j.apm.2006.06.006 |
| [18] | Fang, S.L., Jiang, M.H.: Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays. Commun. Nonlinear Sci. Numer. Simulat. 14, 4292-4303 (2009) · Zbl 1221.37183 |
| [19] | Gopalsamy, K., Weng, P.X.: Global attractivity in a competition system with feedback controls. Comput. Math. Appl. 45, 665-676 (2003) · Zbl 1059.93111 · doi:10.1016/S0898-1221(03)00026-9 |
| [20] | Hu, H.X., Teng, Z.T., Gao, S.J.: Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls. Nonlinear Anal. RWA 10, 2508-2520 (2009) · Zbl 1163.45301 · doi:10.1016/j.nonrwa.2008.05.011 |
| [21] | Li, Z., Han, M.A., Chen, F.D.: Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays. Nonlinear Anal. RWA 14, 402-413 (2013) · Zbl 1268.34147 · doi:10.1016/j.nonrwa.2012.07.004 |
| [22] | Xu, R.: Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response. Nonlinear Dyn. 67, 1683-1693 (2012) · Zbl 1242.92063 · doi:10.1007/s11071-011-0096-1 |
| [23] | Fan, Y.H., Wang, L.L.: Global asymptotical stability of a logistic model with feedback control. Nonlinear Anal. RWA 11, 2686-2697 (2010) · Zbl 1200.34057 · doi:10.1016/j.nonrwa.2009.09.016 |
| [24] | Chakraborty, K., Haldar, S., Kar, T.K.: Global stability and bifurcation analysis of a delay induced prey-predator system with stage structure. Nonlinear Dyn. 73, 1307-1325 (2013) · Zbl 1281.92066 · doi:10.1007/s11071-013-0864-1 |
| [25] | Meng, X.Y., Huo, H.F., Zhang, X.B., Xiang, H.: Stability and Hopf bifurcation in a three-species system with feedback delays. Nonlinear Dyn. 64, 349-364 (2011) · doi:10.1007/s11071-010-9866-4 |
| [26] | Chen, F.D., Yang, J.H., Chen, L.J.: Note on the persistent property of a feedback control system with delays. Nonlinear Anal. RWA 11, 1061-1066 (2010) · Zbl 1187.34106 · doi:10.1016/j.nonrwa.2009.01.045 |
| [27] | Zhang, G.D., Shen, Y., Chen, B.S.: Hopf bifurcation of a predator-prey system with predator harvesting and two delays. Nonlinear Dyn. 73, 2119-2131 (2013) · Zbl 1281.92076 · doi:10.1007/s11071-013-0928-2 |
| [28] | Xia, Y.H.: Global analysis of an impulsive delayed Lotka-Volterra competition system. Commun. Nonlinear Sci. Numer. Simulat. 16, 1597-1616 (2011) · Zbl 1221.34206 · doi:10.1016/j.cnsns.2010.07.014 |
| [29] | Wang, X.H., Liu, H.H., Xu, C.L.: Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay. Nonlinear Dyn. 69, 2155-2167 (2012) · Zbl 1263.34070 · doi:10.1007/s11071-012-0416-0 |
| [30] | Wang, Y., Jiang, W.H., Wang, H.B.: Stability and global Hopf bifurcation in toxic phytoplankton-zooplankton model with delay and selective harvesting. Nonlinear Dyn. 73, 881-896 (2013) · Zbl 1281.92074 · doi:10.1007/s11071-013-0839-2 |
| [31] | Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and applications of Hopf bifurcation. Cambridge University Press, Cambridge (1981) · Zbl 0474.34002 |
| [32] | Kuang, Y.: Delay differential equations with applications in population dynamics. Academic Press, New York (1993) · Zbl 0777.34002 |
| [33] | Barbalat, I.: Systems d’equations differential d’oscillations nonlinearities. Rev. Roumaine Math. Pure Appl. 4, 267-270 (1959) · Zbl 0090.06601 |
| [34] | Hale, J.K.: Theory of functional differential equations. Spring-Verlag, New York (1977) · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2 |
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.
