American Institute of Mathematical Sciences

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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

Zhenguo Bai 1, and  Tingting Zhao 2,

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.

Keywords: Non-local reaction-diffusion model, traveling waves, spreading speed, minimal wave speed.
Mathematics Subject Classification: Primary: 35K57, 92D25; Secondary: 35R10.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Fang, J. 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.  Google Scholar

[7]

J. Fang, K. Lan, G. 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.  Google Scholar

[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) Google Scholar

[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.   Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

W. T. Li, S. 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.  Google Scholar

[14]

B. Li, H. 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.  Google Scholar

[15]

X. Liang, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Z. C. Wang, W. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. L. Wu, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Fang, J. 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.  Google Scholar

[7]

J. Fang, K. Lan, G. 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.  Google Scholar

[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) Google Scholar

[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.   Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

W. T. Li, S. 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.  Google Scholar

[14]

B. Li, H. 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.  Google Scholar

[15]

X. Liang, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

Z. C. Wang, W. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

S. L. Wu, C. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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