American Institute of Mathematical Sciences

Advanced
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  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 & 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[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

Tools

Metrics

  • PDF downloads (189)
  • HTML views (554)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences