December 2018, 23(10): 4063-4085. doi: 10.3934/dcdsb.2018126

Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity

1. 

School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710126, China

2. 

School of Mathematics, Northwest University, Xi'an, Shaanxi 710127, China

Received  June 2017 Revised  October 2017 Published  April 2018

Fund Project: The first author was supported by the NSF of China (11401453,11671315). The second author was supported by the NSF of China (11501446), Natural Science Research Fund of Northwest University (14NW17), and Scientific Research Plan Projects of Education Department of Shaanxi Provincial Government (15JK1765)

A non-local delayed reaction-diffusion model with a quiescent stage is investigated. It is shown that the spreading speed of this model without quasi-monotonicity is linearly determinate and coincides with the minimal wave speed of traveling waves.

Citation: Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J. A. Goldstein (Ed. ), Partial Differential Equations and Related Topics, in: Lecture Notes in Math., vol. 446, Springer-Verlag, 1975, pp. 5–49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[3]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4.

[4]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-deferential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[5]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15. doi: 10.1016/j.matpur.2012.10.009.

[6]

J. FangJ. J. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A, 466 (2010), 1919-1934. doi: 10.1098/rspa.2009.0577.

[7]

J. FangK. LanG. Seo and J. Wu, Spatial dynamics of an age-structured population model of Asian clams, SIAM J. Appl. Math., 74 (2014), 959-979. doi: 10.1137/130930273.

[8]

K. P. Hadeler, T. Hillen and M. A. Lewis, Biological modeling with quiescent phases, in: C. Cosner, S. Cantrell, S. Ruan (Eds. ), Spatial Ecology, Taylor and Francis, 2009. (Chapter 5)

[9]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Can. Appl. Math. Q., 10 (2002), 473-499.

[10]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

[11]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.

[12]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.

[13]

W. T. LiS. Ruan and Z. C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delay, J. Nonlin. Sci., 17 (2007), 505-525. doi: 10.1007/s00332-007-9003-9.

[14]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.

[15]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.

[16]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.

[17]

R. Lui, Biological growth and spread modeled by systems of recursions, I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.

[18]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.

[19]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.

[20]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[21]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of mathematical monographs, Province, RI: American Mathematical Society, 1994.

[22]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.

[23]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2243-2266. doi: 10.3934/dcdsb.2012.17.2243.

[24]

Z. C. WangW. T. Li and S. Ruan, Travelling wave-fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[25]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.

[26]

S. L. Wu and C. H. Hsu, Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1085-1112. doi: 10.1017/S0308210512001412.

[27]

S. L. WuC. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009.

[28]

S. L. Wu and H. Q. Zhao, Traveling fronts for a delayed reaction-diffusion system with a quiescent stage, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3610-3621. doi: 10.1016/j.cnsns.2011.01.012.

[29]

Z. Xu and D. Xiao, Spreading speeds and uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delays, J. Differential Equations, 260 (2016), 268-303. doi: 10.1016/j.jde.2015.08.049.

[30]

X. Yu and X.-Q. Zhao, A nonlocal spatial model for Lyme disease, J. Differential Equations, 261 (2016), 340-372. doi: 10.1016/j.jde.2016.03.014.

[31]

P. Zhang and W. T. Li, Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal., 72 (2010), 2178-2189. doi: 10.1016/j.na.2009.10.016.

[32]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A, 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.

[33]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional-differential equa-tions, Can. Appl. Math. Q., 4 (1996), 421-444.

[34]

H. Q. Zhao and S. Liu, Spatial dynamics for a non-quasi-monotone reaction-diffusion system with delay and quiescent stage, Appl. Math. Model., 40 (2016), 4291-4301. doi: 10.1016/j.apm.2015.11.036.

[35]

X.-Q. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dyn. Differ. Equ., 18 (2006), 1001-1019. doi: 10.1007/s10884-006-9044-z.

[36]

X. -Q. Zhao, Dynamical Systems in Population Biology, second edition, Springer, New York, 2017.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J. A. Goldstein (Ed. ), Partial Differential Equations and Related Topics, in: Lecture Notes in Math., vol. 446, Springer-Verlag, 1975, pp. 5–49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[3]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4.

[4]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-deferential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[5]

A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl., 100 (2013), 1-15. doi: 10.1016/j.matpur.2012.10.009.

[6]

J. FangJ. J. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A, 466 (2010), 1919-1934. doi: 10.1098/rspa.2009.0577.

[7]

J. FangK. LanG. Seo and J. Wu, Spatial dynamics of an age-structured population model of Asian clams, SIAM J. Appl. Math., 74 (2014), 959-979. doi: 10.1137/130930273.

[8]

K. P. Hadeler, T. Hillen and M. A. Lewis, Biological modeling with quiescent phases, in: C. Cosner, S. Cantrell, S. Ruan (Eds. ), Spatial Ecology, Taylor and Francis, 2009. (Chapter 5)

[9]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Can. Appl. Math. Q., 10 (2002), 473-499.

[10]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

[11]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.

[12]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.

[13]

W. T. LiS. Ruan and Z. C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delay, J. Nonlin. Sci., 17 (2007), 505-525. doi: 10.1007/s00332-007-9003-9.

[14]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.

[15]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.

[16]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.

[17]

R. Lui, Biological growth and spread modeled by systems of recursions, I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.

[18]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.

[19]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.

[20]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[21]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of mathematical monographs, Province, RI: American Mathematical Society, 1994.

[22]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.

[23]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2243-2266. doi: 10.3934/dcdsb.2012.17.2243.

[24]

Z. C. WangW. T. Li and S. Ruan, Travelling wave-fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[25]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.

[26]

S. L. Wu and C. H. Hsu, Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1085-1112. doi: 10.1017/S0308210512001412.

[27]

S. L. WuC. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009.

[28]

S. L. Wu and H. Q. Zhao, Traveling fronts for a delayed reaction-diffusion system with a quiescent stage, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3610-3621. doi: 10.1016/j.cnsns.2011.01.012.

[29]

Z. Xu and D. Xiao, Spreading speeds and uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delays, J. Differential Equations, 260 (2016), 268-303. doi: 10.1016/j.jde.2015.08.049.

[30]

X. Yu and X.-Q. Zhao, A nonlocal spatial model for Lyme disease, J. Differential Equations, 261 (2016), 340-372. doi: 10.1016/j.jde.2016.03.014.

[31]

P. Zhang and W. T. Li, Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage, Nonlinear Anal., 72 (2010), 2178-2189. doi: 10.1016/j.na.2009.10.016.

[32]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A, 463 (2007), 1029-1043. doi: 10.1098/rspa.2006.1806.

[33]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional-differential equa-tions, Can. Appl. Math. Q., 4 (1996), 421-444.

[34]

H. Q. Zhao and S. Liu, Spatial dynamics for a non-quasi-monotone reaction-diffusion system with delay and quiescent stage, Appl. Math. Model., 40 (2016), 4291-4301. doi: 10.1016/j.apm.2015.11.036.

[35]

X.-Q. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dyn. Differ. Equ., 18 (2006), 1001-1019. doi: 10.1007/s10884-006-9044-z.

[36]

X. -Q. Zhao, Dynamical Systems in Population Biology, second edition, Springer, New York, 2017.

[1]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591

[2]

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707

[3]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[4]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067

[5]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

[6]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769

[7]

Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843

[8]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043

[9]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[10]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[11]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

[12]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347

[13]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[14]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785

[15]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[16]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029

[17]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681

[18]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41

[19]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19

[20]

Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (217)
  • HTML views (590)
  • Cited by (0)

Other articles
by authors

[Back to Top]