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November  2018, 17(6): 2517-2545. doi: 10.3934/cpaa.2018120

Entire solutions in a two-dimensional nonlocal lattice dynamical system

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

2. 

School of Science, Chang'an University, Xi'an, Shaanxi 710064, China

* Corresponding author

Received  December 2017 Revised  March 2018 Published  June 2018

This paper is concerned with entire solutions for a two-dimensional periodic lattice dynamical system with nonlocal dispersal. In the bistable case, by applying comparison principle and constructing appropriate upper- and lowersolutions, two different types of entire solutions are constructed. The first type behaves like a monostable front merges with a bistable front and one chases another from the same side; while the other type can be represented by two monostable fronts merge and converge to a single bistable front. In the monostable case, we first establish the existence and properties of spatially periodic solutions which connect two steady states. Then new types of entire solutions are constructed by mixing a heteroclinic orbit of the spatially averaged ordinary differential equations with traveling wave fronts with different speeds. Further, for a class of special heterogeneous reaction function, we establish the uniqueness and continuous dependence of the entire solution on parameters, such as wave speeds and shifted variables.

Citation: Fang-Di Dong, Wan-Tong Li, Li Zhang. Entire solutions in a two-dimensional nonlocal lattice dynamical system. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2517-2545. doi: 10.3934/cpaa.2018120
References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[2]

C. P. ChengW. T. Li and G. Lin, Travelling wave solutions in periodic monostable equations on a two-dimensional spatial lattice, IMA J. Appl. Math., 80 (2015), 1254-1272.  doi: 10.1093/imamat/hxu038.  Google Scholar

[3]

C. P. ChengY. H. Su and Z. Feng, Wave propagation for monostable 2-D lattice differential equations with delay, Internat. J.Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350077, 11 pp.  doi: 10.1142/S0218127413500776.  Google Scholar

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C. P. ChengW. T. Li and Z. C. Wang, Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice, Nonlinear Anal. Real World Appl., 13 (2012), 1873-1890.  doi: 10.1016/j.nonrwa.2011.12.016.  Google Scholar

[5]

C. P. ChengW. T. Li and Z. C. Wang, Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice, Discrete Contin. Dyn. Syst. Ser, 13 (2010), 559-575.  doi: 10.3934/dcdsb.2010.13.559.  Google Scholar

[6]

C. P. ChengW. T. Li and Z. C. Wang, Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice, IMA J. Appl. Math., 73 (2008), 592-618.  doi: 10.1093/imamat/hxn003.  Google Scholar

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J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

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X. ChenS. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.  Google Scholar

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X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

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X. ChenJ. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959.  Google Scholar

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X. ChenJ. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.  Google Scholar

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Y. FukaoY. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.   Google Scholar

[14]

J. S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[15]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[16]

J. S. Guo and C. H. Wu, Front propagation for a two-dimensional periodic monostable lattice dynamical system, Discrete Contin. Dyn.Syst., 26 (2010), 197-223.  doi: 10.3934/dcds.2010.26.197.  Google Scholar

[17]

J. S. Guo and C. H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[18]

J. S. Guo and C. H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math., 45 (2008), 327-346.   Google Scholar

[19]

J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.   Google Scholar

[20]

J. S. GuoY. WangC. H. Wu and C. C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[21]

F. Hamel and N. Nadirashvili, Entire solution of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[22]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[23]

W. T. LiZ. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.  Google Scholar

[24]

W. T. LiJ. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[25]

W. T. Li, J. B. Wang and X. Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., https://doi.org/10.1007/s00332-018-9445-2. doi: 10.1007/s00332-018-9445-2.  Google Scholar

[26]

W. T. LiL. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[27]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[28]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[29]

S. MaP. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890.  doi: 10.1016/j.na.2005.10.042.  Google Scholar

[30]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.  Google Scholar

[31]

C. C. Wu, Uniqueness of traveling waves for a two-dimensional bistable periodic lattice dynamical system, Abstr. Appl. Anal., Volume 2012, Article ID 289168, 10 pages. doi: 10.1155/2012/289168.  Google Scholar

[32]

C. H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8.  Google Scholar

[33]

S. L. Wu and C. H. Hsu, Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.  doi: 10.3934/dcds.2016.36.2329.  Google Scholar

[34]

Z. C. WangW. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[35]

Z. C. WangW. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[36]

S. L. WuY. J. Sun and S. Y. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 921-946.  doi: 10.3934/dcds.2013.33.921.  Google Scholar

[37]

S. L. WuZ. X. Shi and F. Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049.  Google Scholar

[38]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

[39]

L. ZhangW. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.  Google Scholar

show all references

References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  Google Scholar

[2]

C. P. ChengW. T. Li and G. Lin, Travelling wave solutions in periodic monostable equations on a two-dimensional spatial lattice, IMA J. Appl. Math., 80 (2015), 1254-1272.  doi: 10.1093/imamat/hxu038.  Google Scholar

[3]

C. P. ChengY. H. Su and Z. Feng, Wave propagation for monostable 2-D lattice differential equations with delay, Internat. J.Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350077, 11 pp.  doi: 10.1142/S0218127413500776.  Google Scholar

[4]

C. P. ChengW. T. Li and Z. C. Wang, Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice, Nonlinear Anal. Real World Appl., 13 (2012), 1873-1890.  doi: 10.1016/j.nonrwa.2011.12.016.  Google Scholar

[5]

C. P. ChengW. T. Li and Z. C. Wang, Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice, Discrete Contin. Dyn. Syst. Ser, 13 (2010), 559-575.  doi: 10.3934/dcdsb.2010.13.559.  Google Scholar

[6]

C. P. ChengW. T. Li and Z. C. Wang, Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice, IMA J. Appl. Math., 73 (2008), 592-618.  doi: 10.1093/imamat/hxn003.  Google Scholar

[7]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[8]

X. ChenS. C. Fu and J. S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.  doi: 10.1137/050627824.  Google Scholar

[9]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[10]

X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[11]

X. ChenJ. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959.  Google Scholar

[12]

X. ChenJ. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.  Google Scholar

[13]

Y. FukaoY. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.   Google Scholar

[14]

J. S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[15]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[16]

J. S. Guo and C. H. Wu, Front propagation for a two-dimensional periodic monostable lattice dynamical system, Discrete Contin. Dyn.Syst., 26 (2010), 197-223.  doi: 10.3934/dcds.2010.26.197.  Google Scholar

[17]

J. S. Guo and C. H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009.  Google Scholar

[18]

J. S. Guo and C. H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math., 45 (2008), 327-346.   Google Scholar

[19]

J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010), 17-28.   Google Scholar

[20]

J. S. GuoY. WangC. H. Wu and C. C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373.  Google Scholar

[21]

F. Hamel and N. Nadirashvili, Entire solution of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[22]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[23]

W. T. LiZ. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.  Google Scholar

[24]

W. T. LiJ. B. Wang and L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016), 2472-2501.  doi: 10.1016/j.jde.2016.05.006.  Google Scholar

[25]

W. T. Li, J. B. Wang and X. Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., https://doi.org/10.1007/s00332-018-9445-2. doi: 10.1007/s00332-018-9445-2.  Google Scholar

[26]

W. T. LiL. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[27]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[28]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[29]

S. MaP. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal., 65 (2006), 1858-1890.  doi: 10.1016/j.na.2005.10.042.  Google Scholar

[30]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.  Google Scholar

[31]

C. C. Wu, Uniqueness of traveling waves for a two-dimensional bistable periodic lattice dynamical system, Abstr. Appl. Anal., Volume 2012, Article ID 289168, 10 pages. doi: 10.1155/2012/289168.  Google Scholar

[32]

C. H. Wu, A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems, J. Dynam. Differential Equations, 28 (2016), 317-338.  doi: 10.1007/s10884-016-9524-8.  Google Scholar

[33]

S. L. Wu and C. H. Hsu, Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.  doi: 10.3934/dcds.2016.36.2329.  Google Scholar

[34]

Z. C. WangW. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[35]

Z. C. WangW. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[36]

S. L. WuY. J. Sun and S. Y. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 921-946.  doi: 10.3934/dcds.2013.33.921.  Google Scholar

[37]

S. L. WuZ. X. Shi and F. Y. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049.  Google Scholar

[38]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

[39]

L. ZhangW. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.  Google Scholar

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