Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models

Shen, Wenxian

doi:10.1090/S0002-9947-10-04950-0

Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models
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Trans. Amer. Math. Soc. 362 (2010), 5125-5168 Request permission

Abstract:

Spatial spread and front propagation dynamics is one of the most important dynamical issues in KPP models. Such dynamics of KPP models in time independent or periodic media has been widely studied. Recently, the author of the current paper with Huang established some theoretical foundation for the study of spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. A notion of spreading speed intervals for such models was introduced in the above-mentioned paper and was shown to be the natural extension of the classical concept of the spreading speeds for time independent or periodic KPP models and that it could be used for more general time dependent KPP models. A notion of generalized propagating speed intervals of front solutions and a notion of traveling wave solutions to time almost periodic and space periodic KPP models were also introduced, which are the generalizations of wave speeds and traveling wave solutions in time independent or periodic KPP models.

The aim of the current paper is to gain some further qualitative and quantitative understanding of the spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. By applying the principal Lyapunov exponent and the principal Floquet bundle theory for time almost periodic parabolic equations, we provide various useful estimates for spreading and generalized propagating speeds for such KPP models. Under the so-called linear determinacy condition, we show that the spreading speed interval in any given direction is a singleton (called the spreading speed). Moreover, in such a case we establish a variational principle for the spreading speed and prove that there is a front solution of speed $c$ in a given direction if and only if $c$ is greater than or equal to the spreading speed in that direction. Both the estimates and variational principle provide important and efficient tools for the spreading speeds analysis as well as the spreading speeds computation. Based on the variational principle, the influence of time and space variation of the media on the spreading speeds is also discussed in this paper. It is shown that the time and space variation cannot slow down the spatial spread and that it indeed speeds up the spatial spread except in certain degenerate cases, which provides deep insights into the understanding of the influence of the inhomogeneity of the underline media on the spatial spread in KPP models.

References

Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2777–2805. MR 1467460, DOI 10.1090/S0002-9947-99-02134-0
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
Henri Berestycki and François Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 101–123. MR 2373726, DOI 10.1090/conm/446/08627
Henri Berestycki, François Hamel, and Nikolai Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 2, 173–213. MR 2127993, DOI 10.4171/JEMS/26
H. Berestycki, F. Hamel, and N. Nadirashili, The speed of propagation for KPP type problems. II. General domains, preprint.
Henri Berestycki, François Hamel, and Lionel Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. (9) 84 (2005), no. 8, 1101–1146 (English, with English and French summaries). MR 2155900, DOI 10.1016/j.matpur.2004.10.006
Xinfu Chen and Jong-Shenq Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), no. 2, 549–569. MR 1929888, DOI 10.1006/jdeq.2001.4153
Xinfu Chen and Jong-Sheng Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003), no. 1, 123–146. MR 1981615, DOI 10.1007/s00208-003-0414-0
Joseph G. Conlon and Charles R. Doering, On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation, J. Stat. Phys. 120 (2005), no. 3-4, 421–477. MR 2182316, DOI 10.1007/s10955-005-5960-2
Freddy Dumortier, Nikola Popović, and Tasso J. Kaper, The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off, Nonlinearity 20 (2007), no. 4, 855–877. MR 2307884, DOI 10.1088/0951-7715/20/4/004
Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361. MR 442480, DOI 10.1007/BF00250432
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799
R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7(1937), 335-369.
Ju. Gertner and M. I. Freĭdlin, The propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk SSSR 249 (1979), no. 3, 521–525 (Russian). MR 553200
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Jong-Shenq Guo and François Hamel, Front propagation for discrete periodic monostable equations, Math. Ann. 335 (2006), no. 3, 489–525. MR 2221123, DOI 10.1007/s00208-005-0729-0
F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decay and monotonicity, prepint.
S. Heinze, G. Papanicolaou, and A. Stevens, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math. 62 (2001), no. 1, 129–148. MR 1857539, DOI 10.1137/S0036139999361148
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
J. H. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, preprint.
W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary value problems for functional-differential equations, World Sci. Publ., River Edge, NJ, 1995, pp. 187–199. MR 1375475
William Hudson and Bertram Zinner, Existence of traveling waves for a generalized discrete Fisher’s equation, Comm. Appl. Nonlinear Anal. 1 (1994), no. 3, 23–46. MR 1295491
Juraj Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations 226 (2006), no. 2, 541–557. MR 2237690, DOI 10.1016/j.jde.2006.02.008
Juraj Húska, Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: the divergence case, Indiana Univ. Math. J. 55 (2006), no. 3, 1015–1043. MR 2244596, DOI 10.1512/iumj.2006.55.2868
Juraj Húska and Peter Poláčik, The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations 16 (2004), no. 2, 347–375. MR 2105779, DOI 10.1007/s10884-004-2784-8
Juraj Húska, Peter Poláčik, and Mikhail V. Safonov, Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 5, 711–739 (English, with English and French summaries). MR 2348049, DOI 10.1016/j.anihpc.2006.04.006
V. Hutson, W. Shen, and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1669–1679. MR 1814096, DOI 10.1090/S0002-9939-00-05808-1
Russell A. Johnson, Kenneth J. Palmer, and George R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal. 18 (1987), no. 1, 1–33. MR 871817, DOI 10.1137/0518001
Yoshinori Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka Math. J. 13 (1976), no. 1, 11–66. MR 422875
A. Kolmogorov, I. Petrowsky, and N. Piscunov, 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., 1 (1937), 1-26.
Mark A. Lewis, Bingtuan Li, and Hans F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol. 45 (2002), no. 3, 219–233. MR 1930975, DOI 10.1007/s002850200144
Bingtuan Li, Hans F. Weinberger, and Mark A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci. 196 (2005), no. 1, 82–98. MR 2156610, DOI 10.1016/j.mbs.2005.03.008
Xing Liang and Xiao-Qiang Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007), no. 1, 1–40. MR 2270161, DOI 10.1002/cpa.20154
Xing Liang, Yingfei Yi, and Xiao-Qiang Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations 231 (2006), no. 1, 57–77. MR 2287877, DOI 10.1016/j.jde.2006.04.010
Roger Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci. 93 (1989), no. 2, 269–295. MR 984281, DOI 10.1016/0025-5564(89)90026-6
Shiwang Ma and Xingfu Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations 217 (2005), no. 1, 54–87. MR 2170528, DOI 10.1016/j.jde.2005.05.004
Andrew J. Majda and Panagiotis E. Souganidis, Large-scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity 7 (1994), no. 1, 1–30. MR 1260130
Andrew J. Majda and Panagiotis Souganidis, Flame fronts in a turbulent combustion model with fractal velocity fields, Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1337–1348. MR 1639220, DOI 10.1002/(SICI)1097-0312(199811/12)51:11/12<1337::AID-CPA4>3.3.CO;2-2
H. Matano, Talks presented at various conferences.
Janusz Mierczyński and Wenxian Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Differential Equations 191 (2003), no. 1, 175–205. MR 1973287, DOI 10.1016/S0022-0396(03)00016-0
J. Mierczyński and W. Shen, “Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,” Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.
Carl Mueller and Richard B. Sowers, Random travelling waves for the KPP equation with noise, J. Funct. Anal. 128 (1995), no. 2, 439–498. MR 1319963, DOI 10.1006/jfan.1995.1038
James Nolen, Matthew Rudd, and Jack Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ. 2 (2005), no. 1, 1–24. MR 2142338, DOI 10.4310/DPDE.2005.v2.n1.a1
James Nolen and Jack Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete Contin. Dyn. Syst. 13 (2005), no. 5, 1217–1234. MR 2166666, DOI 10.3934/dcds.2005.13.1217
James Nolen and Jack Xin, A variational principle based study of KPP minimal front speeds in random shears, Nonlinearity 18 (2005), no. 4, 1655–1675. MR 2150348, DOI 10.1088/0951-7715/18/4/013
James Nolen and Jack Xin, A variational principle for KPP front speeds in temporally random shear flows, Comm. Math. Phys. 269 (2007), no. 2, 493–532. MR 2274555, DOI 10.1007/s00220-006-0144-8
Jim Nolen and Jack Xin, Reaction-diffusion front speeds in spatially-temporally periodic shear flows, Multiscale Model. Simul. 1 (2003), no. 4, 554–570. MR 2029591, DOI 10.1137/S1540345902420234
P. Poláčik, On uniqueness of positive entire solutions and other properties of linear parabolic equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 1, 13–26. MR 2121246, DOI 10.3934/dcds.2005.12.13
P. Poláčik and Ignác Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Differential Equations 5 (1993), no. 2, 279–303. MR 1223450, DOI 10.1007/BF01053163
Nikola Popović and Tasso J. Kaper, Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations, J. Dynam. Differential Equations 18 (2006), no. 1, 103–139. MR 2220193, DOI 10.1007/s10884-005-9002-1
Lenya Ryzhik and Andrej Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci. 5 (2007), no. 3, 575–593. MR 2352332
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
Wenxian Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations 159 (1999), no. 1, 1–54. MR 1726918, DOI 10.1006/jdeq.1999.3651
Wenxian Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. II. Existence, J. Differential Equations 159 (1999), no. 1, 55–101. MR 1726919, DOI 10.1006/jdeq.1999.3652
Wenxian Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations 169 (2001), no. 2, 493–548. Special issue in celebration of Jack K. Hale’s 70th birthday, Part 4 (Atlanta, GA/Lisbon, 1998). MR 1808473, DOI 10.1006/jdeq.2000.3906
Wenxian Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations 16 (2004), no. 4, 1011–1060. MR 2110054, DOI 10.1007/s10884-004-7832-x
Wenxian Shen, Traveling waves in time dependent bistable equations, Differential Integral Equations 19 (2006), no. 3, 241–278. MR 2215558
Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
Wenxian Shen and Yingfei Yi, Convergence in almost periodic Fisher and Kolmogorov models, J. Math. Biol. 37 (1998), no. 1, 84–102. MR 1636648, DOI 10.1007/s002850050121
Horst R. Thieme and Xiao-Qiang Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations 195 (2003), no. 2, 430–470. MR 2016819, DOI 10.1016/S0022-0396(03)00175-X
K\B{o}hei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494, DOI 10.1215/kjm/1250522506
W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719–751. MR 187014, DOI 10.2307/2373071
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), no. 3, 353–396. MR 653463, DOI 10.1137/0513028
Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224, DOI 10.1007/s00285-002-0169-3
Hans F. Weinberger, Mark A. Lewis, and Bingtuan Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol. 45 (2002), no. 3, 183–218. MR 1930974, DOI 10.1007/s002850200145
Jianhong Wu and Xingfu Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential Equations 135 (1997), no. 2, 315–357. MR 1441274, DOI 10.1006/jdeq.1996.3232
Jianhong Wu and Xingfu Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001), no. 3, 651–687. MR 1845097, DOI 10.1023/A:1016690424892
Xue Xin, Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity, J. Dynam. Differential Equations 3 (1991), no. 4, 541–573. MR 1129560, DOI 10.1007/BF01049099
Jack Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000), no. 2, 161–230. MR 1778352, DOI 10.1137/S0036144599364296
Jack Xin, KPP front speeds in random shears and the parabolic Anderson problem, Methods Appl. Anal. 10 (2003), no. 2, 191–197. MR 2074747
Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
B. Zinner, G. Harris, and W. Hudson, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), no. 1, 46–62. MR 1237977, DOI 10.1006/jdeq.1993.1082
Andrej Zlatoš, Pulsating front speed-up and quenching of reaction by fast advection, Nonlinearity 20 (2007), no. 12, 2907–2921. MR 2368331, DOI 10.1088/0951-7715/20/12/009