January  2017, 16(1): 131-150. doi: 10.3934/cpaa.2017006

Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system

1. 

College of Science, National University of Defense Technology, Changsha, 410073, People's Republic of China

2. 

School of Mathematics and Computational Science, Hunan First Normal University, Changsha, 410205, People's Republic of China

3. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People's Republic of China

Jianhua Huang, E-mail address: jhhuang32@nudt.edu.cn

Received  June 2016 Revised  August 2016 Published  November 2016

This paper is concerned with the asymptotic behavior and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed nonlocal dispersal competitive system. We first prove the existence results by applying abstract theories. And then, we show that the traveling wave fronts decay exponentially at both infinities. At last, the strict monotonicity and uniqueness of traveling wave fronts are obtained by using the sliding method in the absent of intraspecific competitive delays. Based on the uniqueness, the exact decay rate of the stronger competitor is established under certain conditions.

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

P. BatesP. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037. Google Scholar

[2]

K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Philos. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494. Google Scholar

[3]

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

[4]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar

[5]

X. ChenS. Fu and J. 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

[6]

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

[7]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. Google Scholar

[8]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. Google Scholar

[9]

O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. TMA, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. Google Scholar

[10]

J. Guo and C. 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

[11]

X. Hou and A. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. RWA, 9 (2008), 2196-2213. doi: 10.1016/j.nonrwa.2007.07.007. Google Scholar

[12]

Y. Kan-on, Parameter dependence of propagation speed of traveling waves for competitiondiffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. Google Scholar

[13]

A. LeungX. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl., 338 (2008), 902-924. doi: 10.1016/j.jmaa.2007.05.066. Google Scholar

[14]

K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal. TMA, 75 (2012), 3705-3722. doi: 10.1016/j.na.2012.01.024. Google Scholar

[15]

K. Li and X. Li, Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497. doi: 10.1016/j.jmaa.2011.11.055. Google Scholar

[16]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation, Nonlinear Analysis TMA, 72 (2010), 3659-3668. doi: 10.1016/j.na.2009.12.047. Google Scholar

[17]

G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reactiondiffusion equations, J. Math. Anal. Appl., 385 (2012), 1094-1106. doi: 10.1016/j.jmaa.2011.07.033. Google Scholar

[18]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reactiondiffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. doi: 10.1016/j.jde.2005.05.004. Google Scholar

[19]

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

[20]

J. Medlock and M. Kot, Spreading disease: integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222. doi: 10.1016/S0025-5564(03)00041-5. Google Scholar

[21]

S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424. doi: 10.1016/j.jmaa.2008.05.057. Google Scholar

[22]

S. PanW. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. Google Scholar

[23]

S. PanW. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Analysis TMA, 72 (2010), 3150-3158. doi: 10.1016/j.na.2009.12.008. Google Scholar

[24]

K. Schumacher, Travelling-front solutions for integro-differential equations, I, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54. Google Scholar

[25]

Y. SunW. Li and Z. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Analysis TMA, 74 (2011), 814-826. doi: 10.1016/j.na.2010.09.032. Google Scholar

[26]

H. Thieme and X. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Dynam. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar

[27]

S. WuH. Zhao and S. Liu, Asymptotic stability of traveling waves for delayed reactiondiffusion equations with crossing-monostability, Z. Angew. Math. Phys., 62 (2011), 377-397. doi: 10.1007/s00033-010-0112-1. Google Scholar

[28]

Z. Xu and P. Weng, Traveling waves in a convolution model with infinite distributed delay and non-monotonicity, Nonlinear Analysis RWA, 12 (2011), 633-647. doi: 10.1016/j.nonrwa.2010.07.006. Google Scholar

[29]

H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: an abstract approach, arXiv: 0807.3612v4.Google Scholar

[30]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648. Google Scholar

[31]

Z. Yu and R. Yuan, Travelling wave solutions in non-local convolution diffusive competitivecooperative systems, IMA J. Appl. Math., 76 (2011), 493-513. doi: 10.1093/imamat/hxq048. Google Scholar

[32]

Z. Yu, Uniqueness of critical traveling wave for delayed lattice equation, Proc. Amer. Math. Soc., 140 (2012), 3853-3859. doi: 10.1090/S0002-9939-2012-11225-0. Google Scholar

[33]

Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems, Chaos, Solitons & Fractals, 45 (2012), 1361-1367. doi: 10.1016/j.chaos.2012.07.002. Google Scholar

[34]

Z. Yu and M. Mei, Asymptotics and uniqueness of travelling waves for non-monotone delayed systems on 2D lattices, Canad. Math. Bull., 56 (2013), 659-672. doi: 10.4153/CMB-2011-180-4. Google Scholar

[35]

G. Zhang, Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity, Nonlinear Anal. TMA, 74 (2011), 6518-6529. doi: 10.1016/j.na.2011.06.035. Google Scholar

[36]

G. Zhang, Traveling waves in a nonlocal dispersal population model with age-structure, Nonlinear Analysis TMA, 74 (2011), 5030-5047. doi: 10.1016/j.na.2011.04.069. Google Scholar

[37]

G. ZhangW. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

[38]

G. ZhangW. Li and Z. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124. doi: 10.1016/j.jde.2012.01.014. Google Scholar

[39]

G. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844. doi: 10.1007/s00033-013-0353-x. Google Scholar

[40]

X. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117. Google Scholar

[41]

X. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations, 18 (2006), 1001-1019. doi: 10.1007/s10884-006-9044-z. Google Scholar

show all references

References:
[1]

P. BatesP. FifeX. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136. doi: 10.1007/s002050050037. Google Scholar

[2]

K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Philos. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494. Google Scholar

[3]

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

[4]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations, 2 (1997), 125-160. Google Scholar

[5]

X. ChenS. Fu and J. 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

[6]

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

[7]

J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. Google Scholar

[8]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002. Google Scholar

[9]

O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. TMA, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. Google Scholar

[10]

J. Guo and C. 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

[11]

X. Hou and A. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. RWA, 9 (2008), 2196-2213. doi: 10.1016/j.nonrwa.2007.07.007. Google Scholar

[12]

Y. Kan-on, Parameter dependence of propagation speed of traveling waves for competitiondiffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. Google Scholar

[13]

A. LeungX. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl., 338 (2008), 902-924. doi: 10.1016/j.jmaa.2007.05.066. Google Scholar

[14]

K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal. TMA, 75 (2012), 3705-3722. doi: 10.1016/j.na.2012.01.024. Google Scholar

[15]

K. Li and X. Li, Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497. doi: 10.1016/j.jmaa.2011.11.055. Google Scholar

[16]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation, Nonlinear Analysis TMA, 72 (2010), 3659-3668. doi: 10.1016/j.na.2009.12.047. Google Scholar

[17]

G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reactiondiffusion equations, J. Math. Anal. Appl., 385 (2012), 1094-1106. doi: 10.1016/j.jmaa.2011.07.033. Google Scholar

[18]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reactiondiffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. doi: 10.1016/j.jde.2005.05.004. Google Scholar

[19]

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

[20]

J. Medlock and M. Kot, Spreading disease: integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222. doi: 10.1016/S0025-5564(03)00041-5. Google Scholar

[21]

S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424. doi: 10.1016/j.jmaa.2008.05.057. Google Scholar

[22]

S. PanW. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. Google Scholar

[23]

S. PanW. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Analysis TMA, 72 (2010), 3150-3158. doi: 10.1016/j.na.2009.12.008. Google Scholar

[24]

K. Schumacher, Travelling-front solutions for integro-differential equations, I, J. Reine Angew. Math., 316 (1980), 54-70. doi: 10.1515/crll.1980.316.54. Google Scholar

[25]

Y. SunW. Li and Z. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Analysis TMA, 74 (2011), 814-826. doi: 10.1016/j.na.2010.09.032. Google Scholar

[26]

H. Thieme and X. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Dynam. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar

[27]

S. WuH. Zhao and S. Liu, Asymptotic stability of traveling waves for delayed reactiondiffusion equations with crossing-monostability, Z. Angew. Math. Phys., 62 (2011), 377-397. doi: 10.1007/s00033-010-0112-1. Google Scholar

[28]

Z. Xu and P. Weng, Traveling waves in a convolution model with infinite distributed delay and non-monotonicity, Nonlinear Analysis RWA, 12 (2011), 633-647. doi: 10.1016/j.nonrwa.2010.07.006. Google Scholar

[29]

H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: an abstract approach, arXiv: 0807.3612v4.Google Scholar

[30]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953. doi: 10.2977/prims/1260476648. Google Scholar

[31]

Z. Yu and R. Yuan, Travelling wave solutions in non-local convolution diffusive competitivecooperative systems, IMA J. Appl. Math., 76 (2011), 493-513. doi: 10.1093/imamat/hxq048. Google Scholar

[32]

Z. Yu, Uniqueness of critical traveling wave for delayed lattice equation, Proc. Amer. Math. Soc., 140 (2012), 3853-3859. doi: 10.1090/S0002-9939-2012-11225-0. Google Scholar

[33]

Z. Yu and R. Yuan, Existence and asymptotics of traveling waves for nonlocal diffusion systems, Chaos, Solitons & Fractals, 45 (2012), 1361-1367. doi: 10.1016/j.chaos.2012.07.002. Google Scholar

[34]

Z. Yu and M. Mei, Asymptotics and uniqueness of travelling waves for non-monotone delayed systems on 2D lattices, Canad. Math. Bull., 56 (2013), 659-672. doi: 10.4153/CMB-2011-180-4. Google Scholar

[35]

G. Zhang, Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity, Nonlinear Anal. TMA, 74 (2011), 6518-6529. doi: 10.1016/j.na.2011.06.035. Google Scholar

[36]

G. Zhang, Traveling waves in a nonlocal dispersal population model with age-structure, Nonlinear Analysis TMA, 74 (2011), 5030-5047. doi: 10.1016/j.na.2011.04.069. Google Scholar

[37]

G. ZhangW. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

[38]

G. ZhangW. Li and Z. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124. doi: 10.1016/j.jde.2012.01.014. Google Scholar

[39]

G. Zhang and R. Ma, Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844. doi: 10.1007/s00033-013-0353-x. Google Scholar

[40]

X. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117. Google Scholar

[41]

X. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations, 18 (2006), 1001-1019. doi: 10.1007/s10884-006-9044-z. Google Scholar

[1]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093

[2]

Fang-Di Dong, Wan-Tong Li, Jia-Bing Wang. Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6291-6318. doi: 10.3934/dcds.2017272

[3]

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

[4]

Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467

[5]

Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218

[6]

Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567

[7]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

[8]

Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3067-3075. doi: 10.3934/dcdsb.2018300

[9]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111

[10]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227

[11]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[12]

Chin-Chin Wu. Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2813-2827. doi: 10.3934/dcds.2017121

[13]

Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253

[14]

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

[15]

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

[16]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[17]

Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-13. doi: 10.3934/dcdsb.2019102

[18]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659

[19]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1483-1508. doi: 10.3934/cpaa.2019071

[20]

Jianhua Huang, Xingfu Zou. Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 925-936. doi: 10.3934/dcds.2003.9.925

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (13)
  • HTML views (95)
  • Cited by (0)

Other articles
by authors

[Back to Top]