Figure 1.
A solution of (1) that tends to the positive steady states when $\tau = 3.5 < \tau_0$
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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
| 1. | Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China |
| 2. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
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:
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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. |
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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 2.
For $\tau = 6>\tau_0$ , the solution will oscillate periodically
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