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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

Shengqin Xu 1, , Chuncheng Wang 2,, and  Dejun Fan 2,,

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  June 2019 Early access  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.

Keywords: Hopf bifurcation, age structure, time delays, stocking rate, non-densely defined, integrated semigroup.
Mathematics Subject Classification: Primary: 35B32; Secondary: 92D30.
Citation: Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
References:
[1]

M. Adimy, H. 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.

[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.

[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.

[4]

S. Chen, J. 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.

[5]

J. Chu, Z. Liu, P. 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.

[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.

[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.

[8]

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.

[9]

A. Ducrot, P. 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.

[10]

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

[11]

S. Jitsuro, K. 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.

[12]

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

[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.

[14]

Z. Liu, P. 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.

[15]

Z. Liu, P. 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.

[16]

Z. Liu, P. 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.

[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.

[18]

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

[19]

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

[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.

[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]

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.

[23]

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

[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.

[25]

Y. Su, S. 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.

[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.

[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.

[28]

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

[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.

[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.

[31]

W. Wang, G. Mulone, F. 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.

[32]

J. Wang, J. 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.

[33]

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

[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.

[35]

D. Yan, Y. 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.

[36]

F. Yi, J. 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.

[37]

T. Zhao, Y. 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.

[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.

show all references

References:
[1]

M. Adimy, H. 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.

[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.

[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.

[4]

S. Chen, J. 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.

[5]

J. Chu, Z. Liu, P. 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.

[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.

[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.

[8]

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.

[9]

A. Ducrot, P. 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.

[10]

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

[11]

S. Jitsuro, K. 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.

[12]

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

[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.

[14]

Z. Liu, P. 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.

[15]

Z. Liu, P. 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.

[16]

Z. Liu, P. 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.

[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.

[18]

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

[19]

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

[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.

[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]

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.

[23]

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

[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.

[25]

Y. Su, S. 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.

[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.

[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.

[28]

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

[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.

[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.

[31]

W. Wang, G. Mulone, F. 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.

[32]

J. Wang, J. 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.

[33]

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

[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.

[35]

D. Yan, Y. 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.

[36]

F. Yi, J. 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.

[37]

T. Zhao, Y. 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.

[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.

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