• Previous Article
    Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian
  • CPAA Home
  • This Issue
  • Next Article
    Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329

[2]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[3]

Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure and Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001

[4]

Léo Girardin. Competition in periodic media:Ⅰ-Existence of pulsating fronts. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1341-1360. doi: 10.3934/dcdsb.2017065

[5]

Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531

[6]

Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815

[7]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006

[8]

Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure and Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038

[9]

Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29 (5) : 3535-3550. doi: 10.3934/era.2021051

[10]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197

[11]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure and Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019

[12]

Fang-Di Dong, Wan-Tong Li, Shi-Liang Wu, Li Zhang. Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1031-1060. doi: 10.3934/dcdsb.2020152

[13]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385

[14]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107

[15]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[16]

Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128

[17]

Wenxian Shen, Xiaoxia Xie. Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1023-1047. doi: 10.3934/dcdsb.2017051

[18]

Ningning Ye, Zengyun Hu, Zhidong Teng. Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1361-1384. doi: 10.3934/cpaa.2022022

[19]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[20]

Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (281)
  • HTML views (177)
  • Cited by (1)

Other articles
by authors

[Back to Top]