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 and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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