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: |
[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. 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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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. 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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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. 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.![]() ![]() ![]() |
[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. 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.![]() ![]() ![]() |
[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.
![]() ![]() |