October  2017, 10(5): 1107-1131. doi: 10.3934/dcdss.2017060

Invasion traveling wave solutions in temporally discrete random-diffusion systems with delays

1. 

College of Science, National University of Defense Technology, Changsha 410073, China

2. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

* Corresponding author: Jianhua Huang

Received  December 2016 Revised  January 2017 Published  June 2017

Fund Project: * The first two authors are supported by the NSF of China(No.11371367,11571126), and the third author is supported by Innovation Program of Shanghai Municipal Education Commission (No.14YZ096) and by the Hujiang Foundation of China (B14005)

This paper is devoted to the invasion traveling wave solutions for a temporally discrete delayed reaction-diffusion competitive system. The existence of invasion traveling wave solutions is established by using Schauder's fixed point Theorem. Ikeharaś theorem is applied to show the asymptotic behaviors. We further investigate the monotonicity and uniqueness invasion traveling waves with the help of sliding method and strong maximum principle.

Citation: Hui Xue, Jianhua Huang, Zhixian Yu. Invasion traveling wave solutions in temporally discrete random-diffusion systems with delays. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1107-1131. doi: 10.3934/dcdss.2017060
References:
[1]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (1998), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

J. Guo and X. Liang, The Minimal Speed of Traveling Fronts for the Lotka-Volterra Competition System, J. Dyn. Differ. Eqns., 23 (2011), 353-363. doi: 10.1007/s10884-011-9214-5. Google Scholar

[6]

C. HauptmannH. Touchette and M. Mackey, Influence of spatiotemporally correlated noise on structure formation in excitable media, Phys. Rev. E., 67 (2003). Google Scholar

[7]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 60 (1998), 435-448. doi: 10.1006/bulm.1997.0008. Google Scholar

[8]

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

[9]

C. H. Hsu and T. S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of traveling waves for a epidemic model, Nonlinearity, 26 (2013), 121-139; Corrigendum, 26 (2013), 2925-2928. doi: 10.1088/0951-7715/26/10/2925. Google Scholar

[10]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dyn. Differ. Eqns, 22 (2010), 285-297. doi: 10.1007/s10884-010-9159-0. Google Scholar

[11]

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differ. Eqns., 251 (2011), 1549-1561. doi: 10.1016/j.jde.2011.05.012. Google Scholar

[12]

J. H. Huang and X. F. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Disc. Contin. Dyn. Syst., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. Google Scholar

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J. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal. TMA., 27 (1996), 579-587. doi: 10.1016/0362-546X(95)00221-G. Google Scholar

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M. Kot, Discrete-time traveling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295. Google Scholar

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M. Kot and W. Schaffer, Discrete-time growth-dispersal models, J. Math. Biol., 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6. Google Scholar

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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

[17]

W. T. LiG. Lin and S. G. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003. Google Scholar

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G. Lin and W. T. Li, Traveling wavefronts in temporally discrete reaction-diffusion equations with delay, Nonlinear Anal., Real World Appl., 9 (2008), 197-205. doi: 10.1016/j.nonrwa.2006.11.003. Google Scholar

[19]

G. Y. Lv and M. X. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. RWA., 11 (2010), 1323-1329. doi: 10.1016/j.nonrwa.2009.02.020. Google Scholar

[20]

S. W. Ma, Traveling wavefronts for delayed rection-diffusion systems via a fixed point theorem, J. Differ. Equ., 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. Google Scholar

[21]

S. Mohamad and K. Gopalsamy, Dynamics of a class of discrete-time neural networks and their continuous-time counterparts, Math. Comput. Simulat., 53 (2000), 1-39. doi: 10.1016/S0378-4754(00)00168-3. Google Scholar

[22]

L. I. W. Roeger, Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes, Discret. Contin. Dyn. Syst. B., 9 (2008), 415-429. doi: 10.3934/dcdsb.2008.9.415. Google Scholar

[23]

T. Shibata and K. Kaneko, Coupled map gas: Structure formation and dynamics of interacting motile elements with internal dynamics, Physica D., 181 (2003), 197-214. doi: 10.1016/S0167-2789(03)00101-5. Google Scholar

[24]

M. Tang and P. Fife, Propagation fronts in competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. Google Scholar

[25]

J. van Vuuren, The existence of traveling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148. doi: 10.1093/imamat/55.2.135. Google Scholar

[26] D. V. Widder, The Laplace Tranform, Princeton University Press, Princeton, 1941. Google Scholar
[27]

J. Wu and X. Zou, Traveling wave fronts of reaction diffusion systems with delay, J Dynam. Diff. Eqns., 13 (2001), 651-687. doi: 10.1023/A:1016690424892. Google Scholar

[28]

J. Xia and Z. X. Yu, Traveling wave solutions in temporally discrete reaction-diffusion systems with delays, Z. Angew. Math. Mech., 91 (2011), 809-823. doi: 10.1002/zamm.201000157. Google Scholar

[29]

X. Yang and Y. Wang, Travelling wave and global attractivity in a competition-diffusion system with nonlocal delays, Comput. Math. Appl., 59 (2010), 3338-3350. doi: 10.1016/j.camwa.2010.03.020. Google Scholar

[30]

Z. X. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Eqns., 260 (2016), 241-267. doi: 10.1016/j.jde.2015.08.037. Google Scholar

[31]

Z. X. Yu and R. Yuan, Traveling waves of delayed reaction diffusion systems with applications, Nonlinear Anal. RWA., 12 (2011), 2475-2488. doi: 10.1016/j.nonrwa.2011.02.005. Google Scholar

[32]

Z. X. Yu and R. Yuan, Traveling waves for a Lotka-Volterra competition system with diffusion, Math. Comput. Model, 53 (2011), 1035-1043. doi: 10.1016/j.mcm.2010.11.061. Google Scholar

[33]

Z. Yu and R. Yuan, Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwanese J. Math., 17 (2013), 2163-2190. doi: 10.11650/tjm.17.2013.3794. Google Scholar

[34]

Z. X. Yu and H. K. Zhao, Traveling waves for competitive Lotka-Volterra systems with spatial diffusions and spatio-temporal delays, Appl. Math. Comput., 242 (2014), 669-678. doi: 10.1016/j.amc.2014.06.058. Google Scholar

[35]

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

show all references

References:
[1]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (1998), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

J. Guo and X. Liang, The Minimal Speed of Traveling Fronts for the Lotka-Volterra Competition System, J. Dyn. Differ. Eqns., 23 (2011), 353-363. doi: 10.1007/s10884-011-9214-5. Google Scholar

[6]

C. HauptmannH. Touchette and M. Mackey, Influence of spatiotemporally correlated noise on structure formation in excitable media, Phys. Rev. E., 67 (2003). Google Scholar

[7]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 60 (1998), 435-448. doi: 10.1006/bulm.1997.0008. Google Scholar

[8]

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

[9]

C. H. Hsu and T. S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of traveling waves for a epidemic model, Nonlinearity, 26 (2013), 121-139; Corrigendum, 26 (2013), 2925-2928. doi: 10.1088/0951-7715/26/10/2925. Google Scholar

[10]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dyn. Differ. Eqns, 22 (2010), 285-297. doi: 10.1007/s10884-010-9159-0. Google Scholar

[11]

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differ. Eqns., 251 (2011), 1549-1561. doi: 10.1016/j.jde.2011.05.012. Google Scholar

[12]

J. H. Huang and X. F. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Disc. Contin. Dyn. Syst., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. Google Scholar

[13]

J. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Anal. TMA., 27 (1996), 579-587. doi: 10.1016/0362-546X(95)00221-G. Google Scholar

[14]

M. Kot, Discrete-time traveling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295. Google Scholar

[15]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models, J. Math. Biol., 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6. Google Scholar

[16]

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

[17]

W. T. LiG. Lin and S. G. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[18]

G. Lin and W. T. Li, Traveling wavefronts in temporally discrete reaction-diffusion equations with delay, Nonlinear Anal., Real World Appl., 9 (2008), 197-205. doi: 10.1016/j.nonrwa.2006.11.003. Google Scholar

[19]

G. Y. Lv and M. X. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. RWA., 11 (2010), 1323-1329. doi: 10.1016/j.nonrwa.2009.02.020. Google Scholar

[20]

S. W. Ma, Traveling wavefronts for delayed rection-diffusion systems via a fixed point theorem, J. Differ. Equ., 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. Google Scholar

[21]

S. Mohamad and K. Gopalsamy, Dynamics of a class of discrete-time neural networks and their continuous-time counterparts, Math. Comput. Simulat., 53 (2000), 1-39. doi: 10.1016/S0378-4754(00)00168-3. Google Scholar

[22]

L. I. W. Roeger, Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes, Discret. Contin. Dyn. Syst. B., 9 (2008), 415-429. doi: 10.3934/dcdsb.2008.9.415. Google Scholar

[23]

T. Shibata and K. Kaneko, Coupled map gas: Structure formation and dynamics of interacting motile elements with internal dynamics, Physica D., 181 (2003), 197-214. doi: 10.1016/S0167-2789(03)00101-5. Google Scholar

[24]

M. Tang and P. Fife, Propagation fronts in competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. Google Scholar

[25]

J. van Vuuren, The existence of traveling plane waves in a general class of competition-diffusion systems, IMA J. Appl. Math., 55 (1995), 135-148. doi: 10.1093/imamat/55.2.135. Google Scholar

[26] D. V. Widder, The Laplace Tranform, Princeton University Press, Princeton, 1941. Google Scholar
[27]

J. Wu and X. Zou, Traveling wave fronts of reaction diffusion systems with delay, J Dynam. Diff. Eqns., 13 (2001), 651-687. doi: 10.1023/A:1016690424892. Google Scholar

[28]

J. Xia and Z. X. Yu, Traveling wave solutions in temporally discrete reaction-diffusion systems with delays, Z. Angew. Math. Mech., 91 (2011), 809-823. doi: 10.1002/zamm.201000157. Google Scholar

[29]

X. Yang and Y. Wang, Travelling wave and global attractivity in a competition-diffusion system with nonlocal delays, Comput. Math. Appl., 59 (2010), 3338-3350. doi: 10.1016/j.camwa.2010.03.020. Google Scholar

[30]

Z. X. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Eqns., 260 (2016), 241-267. doi: 10.1016/j.jde.2015.08.037. Google Scholar

[31]

Z. X. Yu and R. Yuan, Traveling waves of delayed reaction diffusion systems with applications, Nonlinear Anal. RWA., 12 (2011), 2475-2488. doi: 10.1016/j.nonrwa.2011.02.005. Google Scholar

[32]

Z. X. Yu and R. Yuan, Traveling waves for a Lotka-Volterra competition system with diffusion, Math. Comput. Model, 53 (2011), 1035-1043. doi: 10.1016/j.mcm.2010.11.061. Google Scholar

[33]

Z. Yu and R. Yuan, Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwanese J. Math., 17 (2013), 2163-2190. doi: 10.11650/tjm.17.2013.3794. Google Scholar

[34]

Z. X. Yu and H. K. Zhao, Traveling waves for competitive Lotka-Volterra systems with spatial diffusions and spatio-temporal delays, Appl. Math. Comput., 242 (2014), 669-678. doi: 10.1016/j.amc.2014.06.058. Google Scholar

[35]

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

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