Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy

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June 2016, 36(6): 3317-3338. doi: 10.3934/dcds.2016.36.3317

Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy

Fang Li ^1, , Xing Liang ^1, and Wenxian Shen ^2,

1.	School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China, China
2.	Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310

Received May 2015 Revised October 2015 Published December 2015

Abstract
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In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In this first part, we show that a spreading-vanishing dichotomy occurs for such free boundary problems, that is, the species either successfully spreads to all the new environment and stabilizes at a time almost periodic positive solution, or it fails to establish and dies out eventually. The results of this part extend the existing results on spreading-vanishing dichotomy for time and space independent, or time periodic and space independent, or time independent and space periodic diffusive KPP equations with free boundaries. The extension is nontrivial and is ever done for the first time.

Keywords: principal Lyapunov exponent, time almost periodic environment, free boundary, spreading-vanishing dichotomy, time almost periodic positive solution., part metric, Diffusive KPP equation.

Mathematics Subject Classification: 35K20, 35R35, 35K57, 35B15, 92B0.

Citation: Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317

References:

[1]	H. Berestycki and F. Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., 65 (2012), 592. doi: 10.1002/cpa.21389.
[2]	H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146. doi: 10.1016/j.jfa.2008.06.030.
[3]	H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4764932.
[4]	Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II,, J. Differential Equations, 250 (2011), 4336. doi: 10.1016/j.jde.2011.02.011.
[5]	Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, J. Functional Analysis, 265 (2013), 2089. doi: 10.1016/j.jfa.2013.07.016.
[6]	Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089.
[7]	Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model,, Ann. I. H. Poincaré-AN, 32 (2015), 279. doi: 10.1016/j.anihpc.2013.11.004.
[8]	Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., 17 (2015), 2673. doi: 10.4171/JEMS/568.
[9]	A. M. Fink, Almost Periodic Differential Equations,, Lectures Notes in Mathematics, (1974).
[10]	R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[11]	J. Huang and W. Shen, Speeds of spread and propagation of KPP models in time almost and space periodic media,, SIAM J. Appl. Dyn. Syst., 8 (2009), 790. doi: 10.1137/080723259.
[12]	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.
[13]	L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity,, J. Dynam. Differential Equations, 26 (2014), 181. doi: 10.1007/s10884-014-9351-8.
[14]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018.
[15]	J. L. Lockwood, M. F. Hoppes and M. P. Marchetti, Invasion Ecology,, Blackwell Publishing, (2007).
[16]	J. Mierczynski and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations,, J. Differential Equations, 191 (2003), 175. doi: 10.1016/S0022-0396(03)00016-0.
[17]	J. Mierczynski and W. Shen, Lyapunov exponents and asymptotic dynamics in random kolmogorov models,, J. Evolution Equations, 4 (2004), 371. doi: 10.1007/s00028-004-0160-0.
[18]	J. Mierczynski and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., (2008). doi: 10.1201/9781584888963.
[19]	G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002.
[20]	G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 633. doi: 10.1016/j.matpur.2012.05.005.
[21]	J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts,, Arch. Ration. Mech. Anal., 203* (2012), 217. doi: 10.1007/s00205-011-0449-4.*
[22]	J. Nolen and J. 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), 1217. doi: 10.3934/dcds.2005.13.1217.
[23]	J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears,, Nonlinearity, 18 (2005), 1655. doi: 10.1088/0951-7715/18/4/013.
[24]	A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1.
[25]	R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete and Continuous Dynamical Systems, 33 (2013), 2007. doi: 10.3934/dcds.2013.33.2007.
[26]	W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models,, Trans. Amer. Math. Soc., 362 (2010), 5125. doi: 10.1090/S0002-9947-10-04950-0.
[27]	W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations,, J. Dynam. Differential Equations, 23 (2011), 1. doi: 10.1007/s10884-010-9200-3.
[28]	W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations,, J. Appl. Anal. Comput., 1 (2011), 69.
[29]	W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflow,, Memoirs of the American Mathmatical Society, (1998). doi: 10.1090/memo/0647.
[30]	N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997).
[31]	P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones,, Nonlinear Anal., 26 (1996), 1753. doi: 10.1016/0362-546X(95)00015-N.
[32]	T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero,, Nonlinearity, 27 (2014), 2409. doi: 10.1088/0951-7715/27/9/2409.
[33]	A. C. Thompson, On certain contraction mappings in a partially ordered vector space,, in Proceedings of the American Mathematical Society, (1963), 438.
[34]	H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3.
[35]	A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 89. doi: 10.1016/j.matpur.2011.11.007.
[36]	X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differential Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2.

show all references

References:

[1]	H. Berestycki and F. Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., 65 (2012), 592. doi: 10.1002/cpa.21389.
[2]	H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146. doi: 10.1016/j.jfa.2008.06.030.
[3]	H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4764932.
[4]	Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II,, J. Differential Equations, 250 (2011), 4336. doi: 10.1016/j.jde.2011.02.011.
[5]	Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, J. Functional Analysis, 265 (2013), 2089. doi: 10.1016/j.jfa.2013.07.016.
[6]	Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089.
[7]	Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model,, Ann. I. H. Poincaré-AN, 32 (2015), 279. doi: 10.1016/j.anihpc.2013.11.004.
[8]	Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., 17 (2015), 2673. doi: 10.4171/JEMS/568.
[9]	A. M. Fink, Almost Periodic Differential Equations,, Lectures Notes in Mathematics, (1974).
[10]	R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[11]	J. Huang and W. Shen, Speeds of spread and propagation of KPP models in time almost and space periodic media,, SIAM J. Appl. Dyn. Syst., 8 (2009), 790. doi: 10.1137/080723259.
[12]	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.
[13]	L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity,, J. Dynam. Differential Equations, 26 (2014), 181. doi: 10.1007/s10884-014-9351-8.
[14]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018.
[15]	J. L. Lockwood, M. F. Hoppes and M. P. Marchetti, Invasion Ecology,, Blackwell Publishing, (2007).
[16]	J. Mierczynski and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations,, J. Differential Equations, 191 (2003), 175. doi: 10.1016/S0022-0396(03)00016-0.
[17]	J. Mierczynski and W. Shen, Lyapunov exponents and asymptotic dynamics in random kolmogorov models,, J. Evolution Equations, 4 (2004), 371. doi: 10.1007/s00028-004-0160-0.
[18]	J. Mierczynski and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., (2008). doi: 10.1201/9781584888963.
[19]	G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002.
[20]	G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 633. doi: 10.1016/j.matpur.2012.05.005.
[21]	J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts,, Arch. Ration. Mech. Anal., 203* (2012), 217. doi: 10.1007/s00205-011-0449-4.*
[22]	J. Nolen and J. 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), 1217. doi: 10.3934/dcds.2005.13.1217.
[23]	J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears,, Nonlinearity, 18 (2005), 1655. doi: 10.1088/0951-7715/18/4/013.
[24]	A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1.
[25]	R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete and Continuous Dynamical Systems, 33 (2013), 2007. doi: 10.3934/dcds.2013.33.2007.
[26]	W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models,, Trans. Amer. Math. Soc., 362 (2010), 5125. doi: 10.1090/S0002-9947-10-04950-0.
[27]	W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations,, J. Dynam. Differential Equations, 23 (2011), 1. doi: 10.1007/s10884-010-9200-3.
[28]	W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations,, J. Appl. Anal. Comput., 1 (2011), 69.
[29]	W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflow,, Memoirs of the American Mathmatical Society, (1998). doi: 10.1090/memo/0647.
[30]	N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997).
[31]	P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones,, Nonlinear Anal., 26 (1996), 1753. doi: 10.1016/0362-546X(95)00015-N.
[32]	T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero,, Nonlinearity, 27 (2014), 2409. doi: 10.1088/0951-7715/27/9/2409.
[33]	A. C. Thompson, On certain contraction mappings in a partially ordered vector space,, in Proceedings of the American Mathematical Society, (1963), 438.
[34]	H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3.
[35]	A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 89. doi: 10.1016/j.matpur.2011.11.007.
[36]	X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differential Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2.

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