Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T11:43:14.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition waves for lattice Fisher-KPP equations with time and space dependence

Part of: Representations of solutions Infinite-dimensional dissipative dynamical systems Partial differential equations Functional-differential and differential-difference equations

Published online by Cambridge University Press:  28 April 2020

Ning Wang ,
Zhi-Cheng Wang  and
Xiongxiong Bao
Ning Wang
Affiliation:
School Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu730000, People's Republic of China ([email protected]; [email protected])
Zhi-Cheng Wang
Affiliation:
School Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu730000, People's Republic of China ([email protected]; [email protected])
Xiongxiong Bao
Affiliation:
School of Science, Chang'an University, Xi'an, Shaanxi710064, People's Republic of China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the existence results for generalized transition waves of space periodic and time heterogeneous lattice Fisher-KPP equations. By constructing appropriate subsolutions and supersolutions, we show that there is a critical wave speed such that a transition wave solution exists as soon as the least mean of wave speed is above this critical speed. Moreover, the critical speed we construct is proved to be minimal in some particular cases, such as space-time periodic or space independent.

Keywords

Lattice differential equationsgeneralized transition wavesspace periodictime heterogeneous

MSC classification

Primary: 34K31: Lattice functional-differential equations
Secondary: 37L60: Lattice dynamics 35B51: Comparison principles 35C07: Traveling wave solutions 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 573 - 600
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aronson, D. G. and Weinberger, H. F.. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 (1978), 33–76.CrossRefGoogle Scholar
2Bao, X. and Li, W.-T.. Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats. Nonlinear Anal. Real World Appl. 51 (2020), 102975.CrossRefGoogle Scholar
3Bao, X., Li, W.-T. and Wang, Z.-C.. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Commun. Pure Appl. Anal. 19 (2020), 253–277.CrossRefGoogle Scholar
4Berestycki, H. and Hamel, F.. Generalized travelling waves for reaction-diffusion equations. In Perspectives in nonlinear partial differential equations. Contemp. Math., vol. 446, pp. 101–123 (Providence: Amer. Math. Soc., 2007).CrossRefGoogle Scholar
5Berestycki, H. and Hamel, F.. Generalized transition waves and their properties. Comm. Pure Appl. Math. 65 (2012), 592–648.CrossRefGoogle Scholar
6Berestycki, H., Hamel, F. and Nadirashvili, N.. The speed of propagation for KPP type problems, I - periodic framework. J. Eur. Math. Soc. 7 (2005), 172–213.Google Scholar
7Cao, F. and Shen, W.. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete Cont. Dyn. Syst. 37 (2017), 4697–4727.CrossRefGoogle Scholar
8Cao, F. and Shen, W.. Stability and uniqueness of generalized traveling waves of lattice Fisher-KPP equations in heterogeneous media. Sci. Sin. Math. 47 (2017), 1787– 1808.Google Scholar
9Chen, X., Fu, S.-C. and Guo, J.-S.. Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices. SIAM J. Math. Anal. 38 (2006), 233–258.CrossRefGoogle Scholar
10Chen, X. and Guo, J.-S.. Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J. Differ. Eqn. 184 (2002), 549–569.CrossRefGoogle Scholar
11Chen, X. and Guo, J.-S.. Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann. 326 (2003), 123–146.CrossRefGoogle Scholar
12Chen, Y.-Y., Guo, J.-S. and Hamel, F.. Traveling waves for a lattice dynamical system arising in a diffusive endemic model. Nonlinearity 30 (2017), 2334–2359.CrossRefGoogle Scholar
13Chen, X., Guo, J.-S. and Wu, C.-C.. Traveling waves in discrete periodic media for bistable dynamics. Arch. Ration. Mech. Anal. 189 (2008), 189–236.CrossRefGoogle Scholar
14Fang, J., Wei, J. and Zhao, X.-Q.. Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. Proc. R. Soc. A 466 (2010), 1919–1934.CrossRefGoogle Scholar
15Fang, J., Yu, X. and Zhao, X.-Q.. Traveling waves and spreading speeds for time-space periodic monotone systems. J. Funct. Anal. 272 (2017), 4222–4262.CrossRefGoogle Scholar
16Guo, J.-S. and Hamel, F.. Front propagation for discrete periodic monostable equations. Math. Ann. 335 (2006), 489–525.CrossRefGoogle Scholar
17Guo, J.-S. and Wu, C.-H.. Recent developments on wave propagation in 2-species competition systems. Discrete Cont. Dyn. Syst. Ser. B 17 (2012), 2713–2724.CrossRefGoogle Scholar
18Hamel, F. and Rossi, L.. Transition fronts for the Fisher-KPP equation. Trans. Am. Math. Soc. 368 (2016), 8675–8713.CrossRefGoogle Scholar
19Henry, D.. Geometric theory of semilinear parabolic equations. Lecture Notes in Math, vol. 840 (Berlin: Springer-Verlag, 1981).CrossRefGoogle Scholar
20Hoffman, A., Hupkes, H. J. and Van Vleck, E. S.. Entire solutions for bistable lattice differential equations with obstacles. Mem. Am. Math. Soc. 250 (2017), no. 1188, 119 pp.Google Scholar
21Huang, J. and Shen, W.. Speeds of spread and propagation for KPP models in time almost and space periodic media. SIAM J. Appl. Dyn. Syst. 8 (2009), 790–821.CrossRefGoogle Scholar
22Kyrychko, Y., Gourley, S. A. and Bartuccelli, M. V.. Dynamics of a stage-structured population model on an isolated finite lattice. SIAM J. Math. Anal. 37 (2006), 1688–1708.CrossRefGoogle Scholar
23Liang, X. and Zhao, X.-Q.. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60 (2007), 1–40.CrossRefGoogle Scholar
24Liang, X. and Zhao, X.-Q.. Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259 (2010), 857–903.CrossRefGoogle Scholar
25Ma, S. and Zou, X.. Propagation and its failure in a lattice delayed differential equation with global interaction. J. Differ. Eqn. 212 (2005), 129–190.CrossRefGoogle Scholar
26Miller, R. K. and Michel, A. N.. Ordinary differential equations (New York: Academic Press, 1982).Google Scholar
27Nadin, G.. Traveling fronts in space-time periodic media. J. Math. Pures Appl. 92 (2009), 232–262.CrossRefGoogle Scholar
28Nadin, G., Roquejoffre, J. M., Ryzhik, L. and Zlatoš, A.. Existence and non-existence of Fisher-KPP transition fronts. Arch. Ration. Mech. Anal. 203 (2012), 217–246.Google Scholar
29Nadin, G. and Rossi, L.. Propagation phenomena for time heterogeneous KPP reaction-diffusion equations. J. Math. Pures Appl. 98 (2012), 633–653.CrossRefGoogle Scholar
30Nadin, G. and Rossi, L.. Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients. Anal. PDE 8 (2015), 1351–1377.CrossRefGoogle Scholar
31Nadin, G. and Rossi, L.. Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations. Arch. Ration. Mech. Anal. 223 (2017), 1239–1267.CrossRefGoogle Scholar
32Nolen, J., Rudd, M. and Xin, J.. Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. Dyn. PDE 2 (2005), 1–24.Google Scholar
33Nolen, J. and Xin, J.. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete Cont. Dyn. Syst. 13 (2005), 1217–1234.CrossRefGoogle Scholar
34Salako, R. B. and Shen, W.. Long time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts. Stochastics Dyn. 19 (2019), 1950046.CrossRefGoogle Scholar
35Salako, R. B. and Shen, W.. Long time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds. arXiv:1806.01354, 2018.Google Scholar
36Shen, W.. Traveling waves in diffusive random media. J. Dyn. Differ. Eqn. 16 (2004), 1011–1060.CrossRefGoogle Scholar
37Shen, W.. Existence of generalized traveling waves in time recurrent and space periodic monostable equations. J. Appl. Anal. Comput. 1 (2011), 69–93.Google Scholar
38Shen, W.. Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations. J. Dyn. Differ. Eqn. 23 (2011), 1–44.CrossRefGoogle Scholar
39Shigesada, N. and Kawasaki, K., Biological invasions: theory and practice. Oxford Series in Ecology and Evolution (Oxford: Oxford University Press, 1997).Google Scholar
40Shorrocks, B. and Swingland, I. R.. Living in a patch environment (New York: Oxford University Press, 1990).Google Scholar
41Wang, Z.-C., Li, W.-T. and Wu, J.. Entire solutions in delayed lattice differential equations with monostable nonlinearity. SIAM J. Math. Anal. 40 (2009), 2392–2420.CrossRefGoogle Scholar
42Weinberger, H. F.. On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45 (2002), 511–548.CrossRefGoogle Scholar
43Zinner, B., Harris, G and Hudson, W.. Traveling wavefronts for the discrete Fisher's equation. J. Differ. Eqn. 105 (1993), 46–62.CrossRefGoogle Scholar