Spreading and generalized propagating speeds of discrete KPP models in time varying environments

Wenxian SHEN

PDF(358 KB)
PDF(358 KB)
Front. Math. China ›› 2009, Vol. 4 ›› Issue (3) : 523-562. DOI: 10.1007/s11464-009-0032-6
RESEARCH ARTICLE
RESEARCH ARTICLE

Spreading and generalized propagating speeds of discrete KPP models in time varying environments

Author information +
History +

Abstract

The current paper deals with spatial spreading and front propagating dynamics for spatially discrete KPP (Kolmogorov, Petrovsky and Paskunov) models in time recurrent environments, which include time periodic and almost periodic environments as special cases. The notions of spreading speed interval, generalized propagating speed interval, and traveling wave solutions are first introduced, which are proper modifications of those introduced for spatially continuous KPP models in time almost periodic environments. Among others, it is then shown that the spreading speed interval in a given direction is the minimal generalized propagating speed interval in that direction. Some important upper and lower bounds for the spreading and generalized propagating speed intervals are provided. When the environment is unique ergodic and the so called linear determinacy condition is satisfied, it is shown that the spreading speed interval in any direction is a singleton (called the spreading speed), which equals the classical spreading speed if the environment is actually periodic. Moreover, in such a case, a variational principle for the spreading speed is established and it is shown that there is a front of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction.

Keywords

Discrete KPP (Kolmogorov / Petrovsky and Paskunov) model / time varying environment / front / front solution / spreading speed interval / generalized propagating speed interval / traveling wave solution / linear determinacy / variational principle / compact flow / recurrent function

Cite this article

Download citation ▾
Wenxian SHEN. Spreading and generalized propagating speeds of discrete KPP models in time varying environments. Front Math Chin, 2009, 4(3): 523‒562 https://doi.org/10.1007/s11464-009-0032-6

References

[1]
Alikakos N D, Bates P W, Chen X. Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans Amer Math Soc, 1999, 351: 2777-2805
CrossRef Google scholar
[2]
Aronson D G, Weinberger H F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Math, Vol 466. New York: Springer-Verlag, 1975, 5-49
CrossRef Google scholar
[3]
Aronson D G, Weinberger H F. Multidimensional nonlinear diffusions arising in population genetics. Adv Math, 1978, 30: 33-76
CrossRef Google scholar
[4]
Berestycki H, Hamel F, Nadirashili N. The speed of propagation for KPP type problems, I—Periodic framework. J Eur Math Soc, 2005, 7: 172-213
[5]
Berestycki H, Hamel F, Roques L. Analysis of periodically fragmented environment model: II—Biological invasions and pulsating traveling fronts. J Math Pures Appl, 2005, 84: 1101-1146
CrossRef Google scholar
[6]
Chen X, Guo J-S. Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J Diff Eq, 2002, 184(2): 549-569
CrossRef Google scholar
[7]
Chen X, Guo J-S. Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math Ann, 2003, 326(1): 123-146
CrossRef Google scholar
[8]
Conlon J G, Doering C R. On travelling waves for the stochastic Fisher-Kolmogorov- Petrovsky-Piscunov equation. J Stat Phys, 2005, 120(3-4): 421-477
CrossRef Google scholar
[9]
Dumortier F, Popovic N, Kaper T J. The critical wave speed for the Fisher- Kolmogorov-Petrowskii-Piscounov equation with cut-off. Nonlinearity, 2007, 20(4): 855-877
CrossRef Google scholar
[10]
Fife P C, Mcleod J B. The approach of solutions of nonlinear diffusion equations to traveling front solutions. Arch Ration Mech Anal, 1977, 65: 335-361
CrossRef Google scholar
[11]
Fink A M. Almost Periodic Differential Equations. Lecture Notes in Mathematics, Vol 377. Berlin/Heidelberg/New York: Springer-Verlag, 1974
[12]
Fisher R. The wave of advance of advantageous genes. Ann of Eugenics, 1937, 7: 335-369
[13]
Freidlin M, Gärtner J. On the propagation of concentration waves in periodic and random media. Soviet Math Dokl, 1979, 20: 1282-1286
[14]
Guo J-S, Hamel F. Front propagation for discrete periodic monostable equations. Math Ann, 2006, 335(3): 489-525
CrossRef Google scholar
[15]
Heinze S, Papanicolaou G, Stevens A. A variational principle for propagation speeds in inhomogeneous media. SIAM J Appl Math, 2001, 62: 129-148
CrossRef Google scholar
[16]
Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math, Vol 840. Berlin: Springer-Verlag, 1981
[17]
Hudson W, Zinner B. Existence of traveling waves for a generalized discrete Fisher’s equation. Comm Appl Nonlinear Anal, 1994, 1(3): 23-46
[18]
Johnson R A, Palmer K J, Sell G R. Ergodic properties of linear dynamical systems. SIAM J Math Anal, 1987, 18(1): 1-33
CrossRef Google scholar
[19]
Kametaka Y. On the nonlinear diffusion equation of Kolmogorov-Petrovskii- Piskunov type. Osaka J Math, 1976, 13: 11-66
[20]
Kolmogorov A, Petrowsky I, Piscunov N. A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul Moskovskogo Gos Univ, 1937, 1: 1-26
[21]
Lewis L A, Li B, Weinberger H F. Spreading speed and linear determinacy for twospecies competition models. J Math Biol, 2002, 45(3): 219-233
CrossRef Google scholar
[22]
Liang X, Yi Y, Zhao X-Q. Spreading speeds and traveling waves for periodic evolution systems. J Diff Eq, 2006, 231(1): 57-77
CrossRef Google scholar
[23]
Liang X, Zhao X-Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math, 2007, 60(1): 1-40
CrossRef Google scholar
[24]
Lui R. Biological growth and spread modeled by systems of recursions. Math Biosciences, 1989, 93: 269-312
CrossRef Google scholar
[25]
Ma S, Zou X. Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay. J Diff Eq, 2005, 217(1): 54-87
CrossRef Google scholar
[26]
Majda A J, Souganidis P E. Large scale front dynamics for turbulent reactiondiffusion equations with separated velocity scales. Nonlinearity, 1994, 7: 1-30
CrossRef Google scholar
[27]
Majda A J, Souganidis P E. Flame fronts in a turbulent combustion model with fractal velocity fields. Comm Pure and Appl Math, 1998, LI: 1337-1348
CrossRef Google scholar
[28]
Mueller C, Sowers R B. Random travelling waves for the KPP equation with noise. J Funct Anal, 1995, 128(2): 439-498
CrossRef Google scholar
[29]
Nolen J, Rudd M, Xin J. Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. Dynamics of PDE, 2005, 2: 1-24
[30]
Nolen J, Xin J. Reaction-diffusion front speeds in spatially-temporally periodic shear flows. Multiscale Model Simul, 2003, 1: 554-570
CrossRef Google scholar
[31]
Nolen J, Xin J. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete and Continuous Dynamical Systems, 2005, 13: 1217-1234
CrossRef Google scholar
[32]
Nolen J, Xin J. A variational principle based study of KPP minimal front speeds in random shears. Nonlinearity, 2005, 18: 1655-1675
CrossRef Google scholar
[33]
Popovic N, Kaper T J. Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations. J Dynam Diff Eq, 2006, 18(1): 103-139
CrossRef Google scholar
[34]
Ryzhik L, A Zlatoš A. KPP pulsating front speed-up by flows. Commun Math Sci, 2007, 5(3): 575-593
[35]
Sattinger D H. On the stability of waves of nonlinear parabolic systems. Advances in Math, 1976, 22: 312-355
CrossRef Google scholar
[36]
Sell G R. Topological Dynamics and Ordinary Differential Equations. New York: Van Norstand Reinhold Company, 1971
[37]
Shen W. Traveling waves in time almost periodic structures governed by bistable nonlinearities, I. Stability and uniqueness. J Diff Eq, 1999, 159: 1-54
CrossRef Google scholar
[38]
Shen W. Traveling waves in time almost periodic structures governed by bistable nonlinearities, II. Existence. J Diff Eq, 1999, 159: 55-101
CrossRef Google scholar
[39]
Shen W. Dynamical systems and traveling waves in almost periodic structures. J Diff Eq, 2001, 169: 493-548
CrossRef Google scholar
[40]
Shen W. Traveling waves in diffusive random media. J Dynam Diff Eq, 2004, 16: 1011-1060
CrossRef Google scholar
[41]
Shen W. Traveling waves in time dependent bistable equations. Diff and Integral Eq, 2006, 19: 241-278
[42]
Shigesada N, Kawasaki K. Biological Invasions: Theory and Practice. Oxford Series in Ecology and Evolution. Oxford: Oxford University Press, 1997
[43]
Shorrocks B, Swingland I R. Living in a Patch Environment. New York: Oxford University Press, 1990
[44]
Thieme H, Zhao X-Q. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J Diff Eq, 2003, 195: 430-470
CrossRef Google scholar
[45]
Uchiyama K. The behavior of solutions of some nonlinear diffusion equations for large time. J Math Kyoto Univ, 1978, 18: 453-508
[46]
Weinberger H F. Long-time behavior of a class of biology models. SIAM J Math Anal, 1982, 13: 353-396
CrossRef Google scholar
[47]
Weinberger H F. On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J Math Biol, 2002, 45: 511-548
CrossRef Google scholar
[48]
Weinberger H F, Lewis M A, Li B. Analysis of linear determinacy for spread in cooperative models. J Math Biol, 2002, 45(3): 183-218
CrossRef Google scholar
[49]
Wu J, Zou X. Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J Diff Eq, 1997, 135(2): 315-357
CrossRef Google scholar
[50]
Wu J, Zou X. Traveling wave fronts of reaction-diffusion systems with delays. J Dynam Diff Eq, 2001, 13: 651-687
CrossRef Google scholar
[51]
Xin J. Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity. J Dynam Diff Eq, 1991, 3(4): 541-573
CrossRef Google scholar
[52]
Xin J. Front propagation in heterogeneous media. SIAM Review, 2000, 42: 161-230
CrossRef Google scholar
[53]
Xin J. KPP front speeds in random shears and the parabolic Anderson problem. Methods and Applications of Analysis, 2003, 10: 191-198
[54]
Yi Y. Almost Automorphic and Almost Periodic Dynamics in Skew-product Semiflows, Part I. Almost Automorphy and Almost Periodicity. Memoirs of Amer Math Soc, 647. Providence: Amer Math Soc, 1998
[55]
Zinner B, Harris G, Hudson W. Traveling wavefronts for the discrete Fisher’s equation. J Diff Eq, 1993, 105(1): 46-62
CrossRef Google scholar
[56]
Zlatoš A. Pulsating front speed-up and quenching of reaction by fast advection. Nonlinearity, 2007, 20: 2907-2921
CrossRef Google scholar

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
AI Summary AI Mindmap
PDF(358 KB)

Accesses

Citations

Detail

Sections
Recommended

/