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  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.

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. 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.

[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. 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.

[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.

[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. 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.

[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.

[10]

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

[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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[10]

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

[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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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 $\tau = 3.5 < \tau_0$
Figure 2.  For $\tau = 6>\tau_0$, the solution will oscillate periodically
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