June  2019, 24(6): 2535-2549. doi: 10.3934/dcdsb.2018264

Stability and bifurcation in an age-structured model with stocking rate and time delays

1. 

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China

2. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding authors

Received  November 2017 Revised  May 2018 Published  October 2018

In this paper, a predator-prey model with age structure, stocking rate and two delays is investigated. We show that Hopf bifurcation occurs when one of the time delay $τ$ crosses a sequence of critical values, by applyingHopf bifurcation theory for abstract Cauchy problems with non-dense domain. Numerical simulations are included to verify our results and a summary is also given.

Citation: Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
References:
[1]

M. AdimyH. Bouzahir and K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Non. Anal., 46 (2001), 91-112.  doi: 10.1016/S0362-546X(99)00447-2.  Google Scholar

[2]

M. Adimy and K. Ezzinbi, A class of linear partial neutral functional differential equations with nondense domain, J. Diff. Equ., 147 (1998), 285-332.  doi: 10.1006/jdeq.1998.3446.  Google Scholar

[3]

J. Cao and R. Yuan, Bifurcation analysis in a modified Lesile-Gower model with Holling type Ⅱ functional response and delay, Nonl. Dyn., 84 (2016), 1341-1352.  doi: 10.1007/s11071-015-2572-5.  Google Scholar

[4]

S. ChenJ. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Non. Sci., 23 (2013), 1-38.  doi: 10.1007/s00332-012-9138-1.  Google Scholar

[5]

J. ChuZ. LiuP. Magal and S. Ruan, Normal form for an age structured model, J. Dyn. Diff. Equ., 28 (2016), 733-761.  doi: 10.1007/s10884-015-9500-8.  Google Scholar

[6]

J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250.  doi: 10.1007/BF01832847.  Google Scholar

[7]

Y. Du and Y. Lou, S-Shaped Global Bifurcation Curve and Hopf Bifurcation of Positive Solutions to a Predator-Prey Model, J. Diff. Equ., 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394.  Google Scholar

[8]

A. DucrotZ. 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.  Google Scholar

[9]

A. DucrotP. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, Inf. Dimens. Dyn. Syst., 64 (2013), 353-390.  doi: 10.1007/978-1-4614-4523-4_14.  Google Scholar

[10]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monographs CNR, 7, Giadini Editori e Stampatori, Pisa, 1994. Google Scholar

[11]

S. JitsuroK. Rie and M. Rinko, On a predator prey system of Holling type, Proc. Ameri. Math. Soc., 125 (1997), 2041-2050.  doi: 10.1090/S0002-9939-97-03901-4.  Google Scholar

[12]

H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[13]

Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Non. Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x.  Google Scholar

[14]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.  Google Scholar

[15]

Z. LiuP. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Diff. Equ., 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.  Google Scholar

[16]

Z. LiuP. Magal and S. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Dis. Cont. Dyn. Syst., 21 (2016), 537-555.  doi: 10.3934/dcdsb.2016.21.537.  Google Scholar

[17]

Z. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Science China, 60 (2017), 1371-1398.  doi: 10.1007/s11425-016-0371-8.  Google Scholar

[18]

P. Magal and S. Ruan, On integrated semigroups and age structured models in Lp spaces, Diff. Integ. Equ., 20 (2007), 197-239.   Google Scholar

[19]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differ. Equ., 14 (2009), 1041-1084.   Google Scholar

[20]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and apllications on Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), ⅵ+71 pp. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[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.  Google Scholar

[22]

P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by nonlinear an age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695.  Google Scholar

[23]

J. Murray, Mathematical Biology I: An Introduction, Springer, 2002.  Google Scholar

[24]

Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21.  doi: 10.1016/j.jmaa.2004.06.056.  Google Scholar

[25]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574.  doi: 10.1007/s00285-010-0381-5.  Google Scholar

[26]

H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.  Google Scholar

[27]

H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Ana. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[28]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of none-densely defined operators, Diff. Integ. Equ., 3 (1990), 1035-1066.   Google Scholar

[29]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in Mathematical Population Dynamics: Molecules, Cells and Man, World Scientific Publishing, River Edge, 6 (1997), 691-711.  Google Scholar

[30]

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.  Google Scholar

[31]

W. WangG. MuloneF. Salemi and V. Salone, Permanence and Stability of a Stage-Structured Predator-Prey Model, J. Math. Anal. Appl., 262 (2001), 499-528.  doi: 10.1006/jmaa.2001.7543.  Google Scholar

[32]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Diff. Equ., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[33]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, New York N., 1985  Google Scholar

[34]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.  doi: 10.1007/s002850100097.  Google Scholar

[35]

D. YanY. Cao and X. Fu, Asymptotic analysis of a size-structured cannibalism population model with delayed birth process, Disc. Cont. Dyn. Syst. B, 21 (2017), 1975-1998.  doi: 10.3934/dcdsb.2016032.  Google Scholar

[36]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equ., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[37]

T. ZhaoY. Kuang and H. Smith, Global exisnece of periodic solutions in a class of delayed Gause-type predator-prey systems, Non. Anal., 28 (1997), 1373-1394.  doi: 10.1016/0362-546X(95)00230-S.  Google Scholar

[38]

G. Zhu and J. Wei, Global stability and bifurcation analysis of a delayed predator-prey system with prey immigration, Elect. J. Diff. Equ., 13 (2016), Paper No. 13, 20 pp.  Google Scholar

show all references

References:
[1]

M. AdimyH. Bouzahir and K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Non. Anal., 46 (2001), 91-112.  doi: 10.1016/S0362-546X(99)00447-2.  Google Scholar

[2]

M. Adimy and K. Ezzinbi, A class of linear partial neutral functional differential equations with nondense domain, J. Diff. Equ., 147 (1998), 285-332.  doi: 10.1006/jdeq.1998.3446.  Google Scholar

[3]

J. Cao and R. Yuan, Bifurcation analysis in a modified Lesile-Gower model with Holling type Ⅱ functional response and delay, Nonl. Dyn., 84 (2016), 1341-1352.  doi: 10.1007/s11071-015-2572-5.  Google Scholar

[4]

S. ChenJ. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Non. Sci., 23 (2013), 1-38.  doi: 10.1007/s00332-012-9138-1.  Google Scholar

[5]

J. ChuZ. LiuP. Magal and S. Ruan, Normal form for an age structured model, J. Dyn. Diff. Equ., 28 (2016), 733-761.  doi: 10.1007/s10884-015-9500-8.  Google Scholar

[6]

J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250.  doi: 10.1007/BF01832847.  Google Scholar

[7]

Y. Du and Y. Lou, S-Shaped Global Bifurcation Curve and Hopf Bifurcation of Positive Solutions to a Predator-Prey Model, J. Diff. Equ., 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394.  Google Scholar

[8]

A. DucrotZ. 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.  Google Scholar

[9]

A. DucrotP. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, Inf. Dimens. Dyn. Syst., 64 (2013), 353-390.  doi: 10.1007/978-1-4614-4523-4_14.  Google Scholar

[10]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monographs CNR, 7, Giadini Editori e Stampatori, Pisa, 1994. Google Scholar

[11]

S. JitsuroK. Rie and M. Rinko, On a predator prey system of Holling type, Proc. Ameri. Math. Soc., 125 (1997), 2041-2050.  doi: 10.1090/S0002-9939-97-03901-4.  Google Scholar

[12]

H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[13]

Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Non. Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x.  Google Scholar

[14]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.  Google Scholar

[15]

Z. LiuP. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Diff. Equ., 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.  Google Scholar

[16]

Z. LiuP. Magal and S. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Dis. Cont. Dyn. Syst., 21 (2016), 537-555.  doi: 10.3934/dcdsb.2016.21.537.  Google Scholar

[17]

Z. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Science China, 60 (2017), 1371-1398.  doi: 10.1007/s11425-016-0371-8.  Google Scholar

[18]

P. Magal and S. Ruan, On integrated semigroups and age structured models in Lp spaces, Diff. Integ. Equ., 20 (2007), 197-239.   Google Scholar

[19]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differ. Equ., 14 (2009), 1041-1084.   Google Scholar

[20]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and apllications on Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), ⅵ+71 pp. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[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.  Google Scholar

[22]

P. Magal and H. R. Thieme, Eventual compactness for a semiflow generated by nonlinear an age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695.  Google Scholar

[23]

J. Murray, Mathematical Biology I: An Introduction, Springer, 2002.  Google Scholar

[24]

Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21.  doi: 10.1016/j.jmaa.2004.06.056.  Google Scholar

[25]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574.  doi: 10.1007/s00285-010-0381-5.  Google Scholar

[26]

H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.  Google Scholar

[27]

H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Ana. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[28]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of none-densely defined operators, Diff. Integ. Equ., 3 (1990), 1035-1066.   Google Scholar

[29]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in Mathematical Population Dynamics: Molecules, Cells and Man, World Scientific Publishing, River Edge, 6 (1997), 691-711.  Google Scholar

[30]

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.  Google Scholar

[31]

W. WangG. MuloneF. Salemi and V. Salone, Permanence and Stability of a Stage-Structured Predator-Prey Model, J. Math. Anal. Appl., 262 (2001), 499-528.  doi: 10.1006/jmaa.2001.7543.  Google Scholar

[32]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Diff. Equ., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[33]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, New York N., 1985  Google Scholar

[34]

D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43 (2001), 268-290.  doi: 10.1007/s002850100097.  Google Scholar

[35]

D. YanY. Cao and X. Fu, Asymptotic analysis of a size-structured cannibalism population model with delayed birth process, Disc. Cont. Dyn. Syst. B, 21 (2017), 1975-1998.  doi: 10.3934/dcdsb.2016032.  Google Scholar

[36]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equ., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[37]

T. ZhaoY. Kuang and H. Smith, Global exisnece of periodic solutions in a class of delayed Gause-type predator-prey systems, Non. Anal., 28 (1997), 1373-1394.  doi: 10.1016/0362-546X(95)00230-S.  Google Scholar

[38]

G. Zhu and J. Wei, Global stability and bifurcation analysis of a delayed predator-prey system with prey immigration, Elect. J. Diff. Equ., 13 (2016), Paper No. 13, 20 pp.  Google Scholar

Figure 1.  A solution of (1) that tends to the positive steady states when $\tau = 3.5 < \tau_0$
Figure 2.  For $\tau = 6>\tau_0$, the solution will oscillate periodically
[1]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657

[2]

Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020137

[3]

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

[4]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046

[5]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[6]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

[7]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[8]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045

[9]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[10]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183

[11]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

[12]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098

[13]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024

[14]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

[15]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147

[16]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197

[17]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

[18]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499

[19]

Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667

[20]

Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (162)
  • HTML views (532)
  • Cited by (0)

Other articles
by authors

[Back to Top]