American Institute of Mathematical Sciences

Advanced
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

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 and Continuous Dynamical Systems, 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), 592-648. 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), 2146-2189. 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), 115619, 23pp. 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), 4336-4366. 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), 2089-2142. 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), 377-405. 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), 279-305. 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), 2673-2724. doi: 10.4171/JEMS/568.

[9]

A. M. Fink, Almost Periodic Differential Equations, Lectures Notes in Mathematics, Springer-Verlag, Berlin-New York, 1974.

[10]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. 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), 790-821. 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), 1-26.

[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), 181-215. 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), 857-903. 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), 175-205. 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), 371-390. 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., Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781584888963.

[19]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262. 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), 633-653. 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), 217-246. 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), 1217-1234. 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), 1655-1675. doi: 10.1088/0951-7715/18/4/013.

[24]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 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, Ser. A, 33 (2013), 2007-2031. 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), 5125-5168. 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), 1-44. 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), 69-93.

[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, Oxford, Oxford University Press, 1997.

[31]

P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones, Nonlinear Anal., 26 (1996), 1753-1777. 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), 2409-2416. 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, 14, 1963, 438-443.

[34]

H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. 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), 89-102. 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), 494-509. 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), 592-648. 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), 2146-2189. 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), 115619, 23pp. 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), 4336-4366. 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), 2089-2142. 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), 377-405. 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), 279-305. 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), 2673-2724. doi: 10.4171/JEMS/568.

[9]

A. M. Fink, Almost Periodic Differential Equations, Lectures Notes in Mathematics, Springer-Verlag, Berlin-New York, 1974.

[10]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. 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), 790-821. 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), 1-26.

[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), 181-215. 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), 857-903. 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), 175-205. 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), 371-390. 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., Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781584888963.

[19]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262. 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), 633-653. 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), 217-246. 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), 1217-1234. 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), 1655-1675. doi: 10.1088/0951-7715/18/4/013.

[24]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 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, Ser. A, 33 (2013), 2007-2031. 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), 5125-5168. 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), 1-44. 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), 69-93.

[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, Oxford, Oxford University Press, 1997.

[31]

P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones, Nonlinear Anal., 26 (1996), 1753-1777. 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), 2409-2416. 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, 14, 1963, 438-443.

[34]

H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. 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), 89-102. 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), 494-509. doi: 10.1016/S0022-0396(02)00054-2.

[1]

Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121
[2]

Qiaoling Chen, Fengquan Li, Feng Wang. A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13
[3]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256
[4]

Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022
[5]

Yan Zhang. Asymptotic behavior of a nonlocal KPP equation with an almost periodic nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5183-5199. doi: 10.3934/dcds.2016025
[6]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[7]

Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220
[8]

Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240
[9]

Ming-Zhen Xin, Bin-Guo Wang. Spatial dynamics of an epidemic model in time almost periodic and space periodic media. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022116
[10]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[11]

Bin-Guo Wang, Wan-Tong Li, Lizhong Qiang. An almost periodic epidemic model in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 271-289. doi: 10.3934/dcdsb.2016.21.271
[12]

Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138
[13]

Lizhong Qiang, Bin-Guo Wang. An almost periodic malaria transmission model with time-delayed input of vector. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1525-1546. doi: 10.3934/dcdsb.2017073
[14]

P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure and Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161
[15]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[16]

Shuang-Ming Wang, Zhaosheng Feng, Zhi-Cheng Wang, Liang Zhang. Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2005-2034. doi: 10.3934/cpaa.2021145
[17]

Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218
[18]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
[19]

Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080
[20]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (171)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy