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

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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

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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

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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

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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

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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
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