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 and Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006
References:
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P. Bates, P. Fife, X. 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.

[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.

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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.

[4]

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

[5]

X. Chen, S. 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.

[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.

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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.

[8]

J. Coville, J. 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.

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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.

[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.

[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.

[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.

[13]

A. Leung, X. 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.

[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.

[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.

[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.

[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.

[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.

[19]

S. Ma, P. 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.

[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.

[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.

[22]

S. Pan, W. 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.

[23]

S. Pan, W. 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.

[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.

[25]

Y. Sun, W. 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.

[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.

[27]

S. Wu, H. 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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[34]

Z. Yu, 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.

[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.

[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.

[37]

G. Zhang, W. 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.

[38]

G. Zhang, W. 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.

[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.

[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.

[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.

show all references

References:
[1]

P. Bates, P. Fife, X. 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.

[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.

[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.

[4]

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

[5]

X. Chen, S. 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.

[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.

[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.

[8]

J. Coville, J. 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.

[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.

[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.

[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.

[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.

[13]

A. Leung, X. 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.

[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.

[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.

[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.

[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.

[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.

[19]

S. Ma, P. 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.

[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.

[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.

[22]

S. Pan, W. 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.

[23]

S. Pan, W. 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.

[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.

[25]

Y. Sun, W. 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.

[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.

[27]

S. Wu, H. 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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[34]

Z. Yu, 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.

[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.

[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.

[37]

G. Zhang, W. 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.

[38]

G. Zhang, W. 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.

[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.

[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.

[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.

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