Hopf bifurcation in an age-structured population model with two delays. (English) Zbl 1314.35195
Summary: This paper is devoted to the study of an age-structured population system with Riker type birth function. Two time lag factors is considered for the model. One lag lies in the birth process and the another is in the birth function. We investigate some dynamical properties of the equation by using integrated semigroup theory, through which we obtain some conditions of asymptotical stability and Hopf bifurcation occurring at positive steady state for the system. The obtained results show how the two delays affect these dynamical properties.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 35B32 | Bifurcations in context of PDEs |
| 35K55 | Nonlinear parabolic equations |
| 92D25 | Population dynamics (general) |
Keywords:
population system; age-structured; integrated semigroup; asymptotical stability; Hopf bifurcationReferences:
| [1] | A. S. Ackleh, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle,, SIAM J. Appl. Math., 69, 1644 (2009) · Zbl 1181.92076 · doi:10.1137/080723673 |
| [2] | L. J. S. Allen, The effects of vaccination in an age-dependent model for varicella and herpes zoster,, IEEE Trans. Autom. Contr., 43, 779 (1998) |
| [3] | S. Anita, <em>Analysis and Control of Age-dependent Population Dynamics,</em>, Kluwer Academic Publishers (2000) · Zbl 0960.92026 · doi:10.1007/978-94-015-9436-3 |
| [4] | W. Arendt, <em>Vector valued Laplace transforms and Cauchy problems,</em>, Israel J. Math., 59, 327 (1987) · Zbl 0637.44001 · doi:10.1007/BF02774144 |
| [5] | W. Arendt, <em>Vector-Valued Laplace Transforms and Cauchy Problems,</em>, Birkh \(\ddot{\mbox a}\) user (2001) · Zbl 0978.34001 · doi:10.1007/978-3-0348-5075-9 |
| [6] | O. Arino, A survey of structured cell population dynamics,, Acta Bio., 43, 3 (1995) |
| [7] | O. Arino, A survey of cellpopulation dynamics,, J. Theor. Med., 1, 35 (1997) · Zbl 0964.92506 |
| [8] | D. M. Auslander, Dynamics of interacting populatons,, J. Frank. Inst., 297, 345 (1974) |
| [9] | A. Calsina, Global dynamics and optimal life history of a structured population model,, SIAM J. Appl. Math., 59, 1667 (1999) · Zbl 0944.35011 · doi:10.1137/S0036139997331239 |
| [10] | A. Calsina, Stability and instability of equilibria of an equation of size structured population dynamics,, J. Math. Anal. Appl., 286, 435 (2003) · Zbl 1032.35035 · doi:10.1016/S0022-247X(03)00464-5 |
| [11] | J. Chu, Hopf bifurcation in a size-structured population dynamic model with random growth,, J. Diff. Eq., 247, 956 (2009) · Zbl 1172.35322 · doi:10.1016/j.jde.2009.04.003 |
| [12] | J. M. Cushing, The dynamics of hierarchical age-structured populations,, J. Math. Biol., 32, 705 (1994) · Zbl 0823.92018 · doi:10.1007/BF00163023 |
| [13] | J. M. Cushing, <em>An Introduction to Structured Population Dynamics,</em>, SIAM (1998) · Zbl 0939.92026 · doi:10.1137/1.9781611970005 |
| [14] | A. Ducrot, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems,, J. Math. Anal. Appl., 341, 501 (2008) · Zbl 1142.34036 · doi:10.1016/j.jmaa.2007.09.074 |
| [15] | A. Ducrotc, Projectors on the generalized eigenspaces for partial diffrential equations with time delay,, Fields Inst. Comm., 64, 353 (2013) · Zbl 1272.35118 · doi:10.1007/978-1-4614-4523-4_14 |
| [16] | K-J. Engel, <em>One-Parameter Semigroups for Linear Evolution Equations,</em>, Springer (2000) · Zbl 0952.47036 |
| [17] | J. K. Hale, <em>Theory of Functional Differential Equations,</em>, Springer-Verlag (1977) · Zbl 0352.34001 |
| [18] | B. D. Hassard, <em>Theory and Applications of Hopf Bifurcaton,</em>, Cambridge Univ. Press (1981) · Zbl 0474.34002 |
| [19] | H. Kellermann, Integrated semigroups,, J. Funct. Anal., 84, 160 (1989) · Zbl 0689.47014 · doi:10.1016/0022-1236(89)90116-X |
| [20] | Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states,, Israel. J. Math., 70, 1 (1990) · Zbl 0732.58037 · doi:10.1007/BF02807217 |
| [21] | Z. Liu, Hopf bifurcation for non-densely defined Cauchy problems,, Zeitschrift fur angewandte Math. Phys., 62, 191 (2011) · Zbl 1242.34112 · doi:10.1007/s00033-010-0088-x |
| [22] | P. Magal, Perturbation of a globally stable steady state and uniform persistence,, J. Dynam. Diff. Eq., 21, 1 (2009) · Zbl 1173.37025 · doi:10.1007/s10884-008-9127-0 |
| [23] | P. Magal, Compact attractors for time-periodic age structured population models,, Electron. J. Diff. Equ., 2001, 1 (2001) · Zbl 0992.35019 |
| [24] | P. Magal, On integrated semigroups and age structured models in \(L^p\) spaces,, Diff. Int. Eq., 20, 197 (2007) · Zbl 1212.35238 |
| [25] | P. Magal, On semilinear Cauchy problems with non-dense domain,, Adv. in Diff. Equ., 14, 1041 (2009) · Zbl 1225.47042 |
| [26] | P. Magal, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009) · Zbl 1191.35045 · doi:10.1090/S0065-9266-09-00568-7 |
| [27] | P. Magal, Sustained oscillations in an evolutionary epidemiological model of influenza A drift,, Proceedings of Royal Society A: Mathematical, 466, 965 (2010) · Zbl 1195.35290 · doi:10.1098/rspa.2009.0435 |
| [28] | J. A. J. Metz, <em>The Dynamics of Physiologically Structured Populations,</em>, Springer (1986) · Zbl 0614.92014 · doi:10.1007/978-3-662-13159-6 |
| [29] | A. Pazy, <em>Semigroups of linear operators and applications to partial differential equations,</em>, Springer-Verlag (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [30] | W. E. Ricker, Stock and recruitment,, J. Fish. Res. Board Can., 11, 559 (1954) |
| [31] | W. E. Ricker, Computation and interpretation of biological studies of fish populations,, Bull. Fish. Res. Board Can, 191 (1975) |
| [32] | Y. Su, Periodicity and synchronization in blood-stage malaria infection,, J. Math. Biol., 63, 557 (2011) · Zbl 1311.92182 · doi:10.1007/s00285-010-0381-5 |
| [33] | H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Diff. Int. Equ., 3, 1035 (1990) · Zbl 0734.34059 |
| [34] | H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems,, J. Math. Anal. Appl., 152, 416 (1990) · Zbl 0738.47037 · doi:10.1016/0022-247X(90)90074-P |
| [35] | G. F. Webb, <em>Theory of Nonlinear Age-dependent Population Dynamics,</em>, Marcell Dekker (1985) · Zbl 0555.92014 |
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.
