# American Institute of Mathematical Sciences

August  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 and 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), 250-267. 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), 1250146, 16pp. 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), 233-258. 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), 956-1000. 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), 4891-4921. 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), 521-562. 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), 53-72. doi: 10.1007/BF00280827. [8] J. M. Cushing, Model stability and instability in age structured populations, J. Theoret. Biol., 86 (1980), 709-730. 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), 459-478. 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), 410-449. 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), 501-518. 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), 373-395. [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), 1918-1940. 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, G. Simonett and G. F. Webb), Vanderbilt Univ. Press, Nashville, TN, 1998, 213-236. [15] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory (eds. C. Castillo-Chavez, et al.), Springer-Verlag, New York, 2002, 337-359. 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), 57-70. 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), 191-222. 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), 1-35. [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), vi+71 pp. 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), 1041-1084. [21] P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. A, 466 (2010), 965-992. doi: 10.1098/rspa.2009.0435. [22] K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75. 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), 327-339. doi: 10.1016/0898-1221(83)90020-2. [24] W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623. 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), 557-574. 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), 555-564. 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), 1035-1066. [29] H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447. 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, Cells and Man (eds. O. Arino, D. Axelrod and M. Kimmel), Ser. Math. Biol. Med., 6, World Sci. Publ., River Edge, NJ, 1997, 691-711. [31] Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150. doi: 10.1016/j.jmaa.2011.07.038. [32] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer-Verlag, New York, 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), 83-107. doi: 10.1016/j.mbs.2006.06.006.

show all references

##### References:
 [1] S. Bertoni, Periodic solutions for non-linear equations of structured populations, J. Math. Anal. Appl., 220 (1998), 250-267. 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), 1250146, 16pp. 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), 233-258. 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), 956-1000. 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), 4891-4921. 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), 521-562. 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), 53-72. doi: 10.1007/BF00280827. [8] J. M. Cushing, Model stability and instability in age structured populations, J. Theoret. Biol., 86 (1980), 709-730. 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), 459-478. 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), 410-449. 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), 501-518. 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), 373-395. [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), 1918-1940. 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, G. Simonett and G. F. Webb), Vanderbilt Univ. Press, Nashville, TN, 1998, 213-236. [15] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory (eds. C. Castillo-Chavez, et al.), Springer-Verlag, New York, 2002, 337-359. 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), 57-70. 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), 191-222. 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), 1-35. [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), vi+71 pp. 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), 1041-1084. [21] P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. A, 466 (2010), 965-992. doi: 10.1098/rspa.2009.0435. [22] K. Pakdaman, B. Perthame and D. Salort, Dynamics of a structured neuron population, Nonlinearity, 23 (2010), 55-75. 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), 327-339. doi: 10.1016/0898-1221(83)90020-2. [24] W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623. 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), 557-574. 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), 555-564. 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), 1035-1066. [29] H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447. 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, Cells and Man (eds. O. Arino, D. Axelrod and M. Kimmel), Ser. Math. Biol. Med., 6, World Sci. Publ., River Edge, NJ, 1997, 691-711. [31] Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150. doi: 10.1016/j.jmaa.2011.07.038. [32] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer-Verlag, New York, 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), 83-107. doi: 10.1016/j.mbs.2006.06.006.
 [1] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [2] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [3] Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082 [4] Sebastian Aniţa, Ana-Maria Moşsneagu. Optimal harvesting for age-structured population dynamics with size-dependent control. Mathematical Control and Related Fields, 2019, 9 (4) : 607-621. doi: 10.3934/mcrf.2019043 [5] Abed Boulouz. A spatially and size-structured population model with unbounded birth process. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022038 [6] Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891 [7] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [8] Min He. On continuity in parameters of integrated semigroups. Conference Publications, 2003, 2003 (Special) : 403-412. doi: 10.3934/proc.2003.2003.403 [9] Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112 [10] Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233 [11] Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095 [12] Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure and Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637 [13] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [14] L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203 [15] Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 [16] Miriam Kiessling, Sascha Kurz, Jörg Rambau. The integrated size and price optimization problem. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 669-693. doi: 10.3934/naco.2012.2.669 [17] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [18] Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107 [19] Z.-R. He, M.-S. Wang, Z.-E. Ma. Optimal birth control problems for nonlinear age-structured population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 589-594. doi: 10.3934/dcdsb.2004.4.589 [20] Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265

2020 Impact Factor: 1.327