# American Institute of Mathematical Sciences

2015, 20(6): 1735-1757. doi: 10.3934/dcdsb.2015.20.1735

## Hopf bifurcation for a spatially and age structured population dynamics model

 1 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 2 University of Bordeaux, IMB, UMR CNRS 5251, 33076 Bordeaux, France

Received  November 2013 Revised  February 2014 Published  June 2015

This paper is devoted to the study of a spatially and age structured population dynamics model. We study the stability and Hopf bifurcation of the positive equilibrium of the model by using a bifurcation theory in the context of integrated semigroups. This problem is a first example for Hopf bifurcation for a spatially and age/size structured population dynamics model. Bifurcation analysis indicates that Hopf bifurcation occurs at a positive age/size dependent steady state of the model. The results are confirmed by some numerical simulations.
Citation: Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
##### References:
 [1] S. Bertoni, Periodic solutions for non-linear equations of structured populations,, J. Math. Anal. Appl., 220 (1998), 250. doi: 10.1006/jmaa.1997.5878. [2] P. Bi and X. Fu, Hopf bifurcation in an age-dependent population model with delayed birth process,, I. J. Bifurcation and Chaos, 22 (2012). doi: 10.1142/S0218127412501465. [3] C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin and W. M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity,, J. Math. Biol., 27 (1989), 233. doi: 10.1007/BF00275810. [4] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth,, Journal of Differential Equations, 247 (2009), 956. doi: 10.1016/j.jde.2009.04.003. [5] J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase,, Discrete and Continuous Dynamical Systems, 33 (2013), 4891. doi: 10.3934/dcds.2013.33.4891. [6] J. Chu, P. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model,, J. Nonlinear Sci., 21 (2011), 521. doi: 10.1007/s00332-010-9091-9. [7] M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions,, Arch. Rational Mech. Anal., 67 (1977), 53. doi: 10.1007/BF00280827. [8] J. M. Cushing, Model stability and instability in age structured populations,, J. Theoret. Biol., 86 (1980), 709. doi: 10.1016/0022-5193(80)90307-0. [9] J. M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics,, Comput. Math. Appl., 9 (1983), 459. doi: 10.1016/0898-1221(83)90060-3. [10] A. Ducrot, Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections,, Journal of Differential Equations, 250 (2011), 410. doi: 10.1016/j.jde.2010.09.019. [11] A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems,, J. Math. Anal. Appl., 341 (2008), 501. doi: 10.1016/j.jmaa.2007.09.074. [12] A. Ducrot, P. Magal and O. Seydi, Nonlinear boundary conditions derived by singular perturbation in age structured population dynamics model,, J. Appl. Anal. Comput., 1 (2011), 373. [13] M. Doumic, A. Marciniak-Czochra, B. Perthame and J. P. Zubelli, A structured population model of cell differentiation,, SIAM J. Appl. Math., 71 (2011), 1918. doi: 10.1137/100816584. [14] H. Inaba, Mathematical analysis for an evolutionary epidemic model,, in Mathematical Models in Medical and Health Sciences (eds. M. A. Horn, (1998), 213. [15] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model,, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, (2002), 337. doi: 10.1007/978-1-4613-0065-6_19. [16] T. Kostova and J. Li, Oscillations and stability due to juvenile competitive effects on adult fertility,, Comput. Math. Appl., 32 (1996), 57. doi: 10.1016/S0898-1221(96)00197-6. [17] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, Zeitschrift fur Angewandte Mathematik und Physik, 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. [18] P. Magal, Compact attractors for time-periodic age structured population models,, Electron. J. Differential Equations, 2001 (2001), 1. [19] P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. [20] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, Advances in Differential Equations, 14 (2009), 1041. [21] P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift,, Proc. R. Soc. A, 466 (2010), 965. doi: 10.1098/rspa.2009.0435. [22] K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population,, Nonlinearity, 23 (2010), 55. doi: 10.1088/0951-7715/23/1/003. [23] J. Prüss, On the qualitative behaviour of populations with age-specific interactions,, Comput. Math. Appl., 9 (1983), 327. doi: 10.1016/0898-1221(83)90020-2. [24] W. E. Ricker, Stock and recruitment,, J. Fish. Res. Board Canada, 11 (1954), 559. doi: 10.1139/f54-039. [25] W. E. Ricker, Computation and interpretation of biological studies of fish populations,, Bull. Fish. Res. Bd. Canada, 191 (1975). [26] Y. Su, S. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection,, Journal of Mathematical Biology, 63 (2011), 557. doi: 10.1007/s00285-010-0381-5. [27] J. H. Swart, Hopf bifurcation and the stability of nonlinear age-dependent population models,, Comput. Math. Appl., 15 (1988), 555. doi: 10.1016/0898-1221(88)90280-5. [28] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. [29] H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems,, J. Math. Anal. Appl., 152 (1990), 416. doi: 10.1016/0022-247X(90)90074-P. [30] H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in Advances in Mathematical Population Dynamics-Molecules, (1997), 691. [31] Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model,, J. Math. Anal. Appl., 385 (2012), 1134. doi: 10.1016/j.jmaa.2011.07.038. [32] J. Wu, Theory and Applications of Partial Functional-Differential Equations,, Springer-Verlag, (1996). doi: 10.1007/978-1-4612-4050-1. [33] P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age structure in human hosts and its application to treatment strategies,, Math. Biosci., 205 (2007), 83. doi: 10.1016/j.mbs.2006.06.006.

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##### References:
 [1] S. Bertoni, Periodic solutions for non-linear equations of structured populations,, J. Math. Anal. Appl., 220 (1998), 250. doi: 10.1006/jmaa.1997.5878. [2] P. Bi and X. Fu, Hopf bifurcation in an age-dependent population model with delayed birth process,, I. J. Bifurcation and Chaos, 22 (2012). doi: 10.1142/S0218127412501465. [3] C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin and W. M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity,, J. Math. Biol., 27 (1989), 233. doi: 10.1007/BF00275810. [4] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth,, Journal of Differential Equations, 247 (2009), 956. doi: 10.1016/j.jde.2009.04.003. [5] J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase,, Discrete and Continuous Dynamical Systems, 33 (2013), 4891. doi: 10.3934/dcds.2013.33.4891. [6] J. Chu, P. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model,, J. Nonlinear Sci., 21 (2011), 521. doi: 10.1007/s00332-010-9091-9. [7] M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions,, Arch. Rational Mech. Anal., 67 (1977), 53. doi: 10.1007/BF00280827. [8] J. M. Cushing, Model stability and instability in age structured populations,, J. Theoret. Biol., 86 (1980), 709. doi: 10.1016/0022-5193(80)90307-0. [9] J. M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics,, Comput. Math. Appl., 9 (1983), 459. doi: 10.1016/0898-1221(83)90060-3. [10] A. Ducrot, Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections,, Journal of Differential Equations, 250 (2011), 410. doi: 10.1016/j.jde.2010.09.019. [11] A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems,, J. Math. Anal. Appl., 341 (2008), 501. doi: 10.1016/j.jmaa.2007.09.074. [12] A. Ducrot, P. Magal and O. Seydi, Nonlinear boundary conditions derived by singular perturbation in age structured population dynamics model,, J. Appl. Anal. Comput., 1 (2011), 373. [13] M. Doumic, A. Marciniak-Czochra, B. Perthame and J. P. Zubelli, A structured population model of cell differentiation,, SIAM J. Appl. Math., 71 (2011), 1918. doi: 10.1137/100816584. [14] H. Inaba, Mathematical analysis for an evolutionary epidemic model,, in Mathematical Models in Medical and Health Sciences (eds. M. A. Horn, (1998), 213. [15] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model,, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, (2002), 337. doi: 10.1007/978-1-4613-0065-6_19. [16] T. Kostova and J. Li, Oscillations and stability due to juvenile competitive effects on adult fertility,, Comput. Math. Appl., 32 (1996), 57. doi: 10.1016/S0898-1221(96)00197-6. [17] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, Zeitschrift fur Angewandte Mathematik und Physik, 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. [18] P. Magal, Compact attractors for time-periodic age structured population models,, Electron. J. Differential Equations, 2001 (2001), 1. [19] P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. [20] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, Advances in Differential Equations, 14 (2009), 1041. [21] P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift,, Proc. R. Soc. A, 466 (2010), 965. doi: 10.1098/rspa.2009.0435. [22] K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population,, Nonlinearity, 23 (2010), 55. doi: 10.1088/0951-7715/23/1/003. [23] J. Prüss, On the qualitative behaviour of populations with age-specific interactions,, Comput. Math. Appl., 9 (1983), 327. doi: 10.1016/0898-1221(83)90020-2. [24] W. E. Ricker, Stock and recruitment,, J. Fish. Res. Board Canada, 11 (1954), 559. doi: 10.1139/f54-039. [25] W. E. Ricker, Computation and interpretation of biological studies of fish populations,, Bull. Fish. Res. Bd. Canada, 191 (1975). [26] Y. Su, S. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection,, Journal of Mathematical Biology, 63 (2011), 557. doi: 10.1007/s00285-010-0381-5. [27] J. H. Swart, Hopf bifurcation and the stability of nonlinear age-dependent population models,, Comput. Math. Appl., 15 (1988), 555. doi: 10.1016/0898-1221(88)90280-5. [28] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. [29] H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems,, J. Math. Anal. Appl., 152 (1990), 416. doi: 10.1016/0022-247X(90)90074-P. [30] H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in Advances in Mathematical Population Dynamics-Molecules, (1997), 691. [31] Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model,, J. Math. Anal. Appl., 385 (2012), 1134. doi: 10.1016/j.jmaa.2011.07.038. [32] J. Wu, Theory and Applications of Partial Functional-Differential Equations,, Springer-Verlag, (1996). doi: 10.1007/978-1-4612-4050-1. [33] P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age structure in human hosts and its application to treatment strategies,, Math. Biosci., 205 (2007), 83. doi: 10.1016/j.mbs.2006.06.006.
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