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Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters. (English) Zbl 1190.34106
The authors consider a delayed population model with delay-dependent parameters. They prove analytically that the positive equilibrium switches from being stable to unstable and then back to stable as the delay $\tau$ increases, and Hopf bifurcations occur between the two critical values of stability changes. The critical values for the stability switches and Hopf bifurcations can be analytically determined. Using the perturbation approach and Floquet technique, they also obtain an approximation to the bifurcating periodic solution and derive formulas for determining the direction and stability of the Hopf bifurcations. Finally, they illustrate their results by some numerical examples.
##### MSC:
 34K60 Qualitative investigation and simulation of models 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
##### References:
 [1] Hale, J.: Theory of functional differential equations, (1977) [2] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039 [3] Ding, X.; Li, W.: Local Hopf bifurcation and global existence of periodic solutions in a kind of physiological system, Nonlinear anal. RWA 8, 1459-1471 (2007) · Zbl 1163.34053 · doi:10.1016/j.nonrwa.2006.07.013 [4] Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981) [5] 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 · doi:10.1016/0025-5564(90)90019-U [6] 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 · doi:10.1137/0152048 [7] Cooke, K. L.; Den Driesche, P. Van: Analysis of an SEIRS epidemic model with two delays, J. math. Biol. 35, 240-260 (1996) · Zbl 0865.92019 · doi:10.1007/s002850050051 [8] Cooke, K. L.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models, J. math. Biol. 39, 332-352 (1999) · Zbl 0945.92016 · doi:10.1007/s002850050194 [9] Bence, J. R.; Nisbet, R. M.: Space-limited recruitment in open systems: the importance of time delays, Ecology 70, 1434-1441 (1989) [10] Nisbet, R. M.; Gurney, W. S. C.: Modelling fluctuating populations, (1982) · Zbl 0593.92013 [11] Nisbet, R. M.; Gurney, W. S. C.; Metz, J. A. J.: Stage structure models applied in evolutionary ecology, Biomathematics 18, 428-449 (1989) [12] Mackey, M. C.; Glass, L.: Oscillations and chaos in physiological control systems, Science 197, 287-289 (1977) [13] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002 [14] Mackey, M. C.: Periodic auto-immune hemolytic anemia: an induced dynamical disease, Bull. math. Biol. 41, 829-834 (1979) · Zbl 0414.92012 [15] Mackey, M. C.: Some models in hemopoiesis: predictions and problems in biomathematics and cell kinetics, (1981) [16] Hale, J.; Sternberg, N.: Onset of chaos in differential delay equations, J. comput. Phys. 77, 221-239 (1988) · Zbl 0644.65050 · doi:10.1016/0021-9991(88)90164-7 [17] Barclay, H. J.; Den Driessche, P. Van: A model for a species with two life history stages and added mortality, Ecol. model. 11, 157-166 (1980) [18] Fisher, M. E.; Goh, B. S.: Stability results for delayed recruitment models in population dynamics, J. math. Biol. 19, 147-156 (1984) · Zbl 0533.92017 · doi:10.1007/BF00275937 [19] Tognetti, K.: The two stage stochastic model, Math. biosci. 25, 195-204 (1975) [20] Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. anal. 33, 1144-1165 (2002) · Zbl 1013.92034 · doi:10.1137/S0036141000376086 [21] Stokes, A.: On the stability of a limit cycle of autonomous functional differential equations, Contr. diff. Eqns. 3, 121-140 (1964) · Zbl 0135.30903 [22] Der Heiden, U. An; Walther, H. O.: Existence of chaos in control systems with delayed feedback, J. differential equations 47, 273-295 (1983) · Zbl 0477.93040 · doi:10.1016/0022-0396(83)90037-2