October  2017, 10(5): 1063-1078. doi: 10.3934/dcdss.2017057

Traveling wave solutions of a reaction-diffusion predator-prey model

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

* the Corresponding author

Received  November 2016 Revised  January 2017 Published  June 2017

Fund Project: This work is supported by the Natural Science Foundation of China (Grant No.11471146).

This paper is concerned with the dynamics of traveling wave solutions for a reaction-diffusion predator-prey model with a nonlocal delay. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions which are easy to construct in practice. We also investigate the asymptotic behavior of traveling wave solutions by employing the standard asymptotic theory.

Citation: Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
References:
[1]

N. F. Britton, Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population model, SIAM. J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[2]

X. Chen and Z. J. Du, Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theory Dyn. Syst. , (2017). doi: 10.1007/s12346-017-0223-6.  Google Scholar

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C. Conley and R. Gardner, An application of the generalized morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018.  Google Scholar

[4]

Q. T. GanR. Xu and X. Zhang, Traveling wave of a three-species Lotka-Volterra food-chain model with spatial diffusion and time delays, Nonlinear. Anal., 11 (2010), 2817-2832.  doi: 10.1016/j.nonrwa.2009.10.006.  Google Scholar

[5]

S. B. HsuT. W. Hwang and Y. Kuang, Golbal dynamics of a predator-prey model with Hassell-Varley type functional response, Discret. Contin. Dyn. Syst.B., 10 (2008), 857-871.  doi: 10.3934/dcdsb.2008.10.857.  Google Scholar

[6]

J. H. Huang and X. F. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X.  Google Scholar

[7]

J. Huang and X. Zou, Traveling wave solutions in delayer reaction diffusion systems with partial monotonicity, Acta. Math. Appl. Sinica., 22 (2006), 243-256.  doi: 10.1007/s10255-006-0300-0.  Google Scholar

[8]

X. Liang and X. Q. Zhao, Asymptotic speed of spread and traveling waves for monotone semiflows with applications, Comm. Pure. Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[9]

Y. LinQ. R. Wang and K. Zhou, Traveling wave solutions in n-dimensional delayed reaction-diffusion systems with mixed monotonicity, J. Comput. Appl. Math., 243 (2013), 16-27.  doi: 10.1016/j.cam.2012.11.007.  Google Scholar

[10]

G. Y. Lv and M. X. Wang, Existence, uniqueness and asymptotic behavior of taraveling wave fronts for a vector disease model, Nonlinear Anal., 11 (2010), 2035-2043.  doi: 10.1016/j.nonrwa.2009.05.006.  Google Scholar

[11]

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

[12]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differetial equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[13]

X. H. ShangZ. J. Du and X. J. Lin, Traveling wave solutions in $n$-dimensional delayed reaction-diffusion systems and application to four-dimensional predator-prey systems, Math. Methods Appl. Sci., 39 (2016), 1607-1620.  doi: 10.1002/mma.3595.  Google Scholar

[14]

R. Xu, A reaction-diffusion predator-prey model with stage structure and nonlocal delay, Appl. Math. Comput., 175 (2006), 984-1006.  doi: 10.1016/j.amc.2005.08.014.  Google Scholar

[15]

R. Xu and Z. E. Ma, Global stability of a reaction-diffusion predator-prey model with a nonlocal delay, Math. Comput. Model., 50 (2009), 194-206.  doi: 10.1016/j.mcm.2009.02.011.  Google Scholar

[16]

R. Xu and X. Zhang, Gobal stability and traveling waves of a predator-prey model with diffusion and nonlocal maturation delay, Comm. Nonlinear. Sci. Numer. Simulat., 15 (2010), 3390-3401.  doi: 10.1016/j.cnsns.2009.12.031.  Google Scholar

[17]

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

[18]

X. Zhang and R. Xu, Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure, J. Math. Anal. Appl., 373 (2011), 475-484.  doi: 10.1016/j.jmaa.2010.07.044.  Google Scholar

[19]

K. Zhou and Q. R. Wang, Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity, J. Math. Anal. Appl., 372 (2010), 598-610.  doi: 10.1016/j.jmaa.2010.07.032.  Google Scholar

show all references

References:
[1]

N. F. Britton, Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population model, SIAM. J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[2]

X. Chen and Z. J. Du, Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theory Dyn. Syst. , (2017). doi: 10.1007/s12346-017-0223-6.  Google Scholar

[3]

C. Conley and R. Gardner, An application of the generalized morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018.  Google Scholar

[4]

Q. T. GanR. Xu and X. Zhang, Traveling wave of a three-species Lotka-Volterra food-chain model with spatial diffusion and time delays, Nonlinear. Anal., 11 (2010), 2817-2832.  doi: 10.1016/j.nonrwa.2009.10.006.  Google Scholar

[5]

S. B. HsuT. W. Hwang and Y. Kuang, Golbal dynamics of a predator-prey model with Hassell-Varley type functional response, Discret. Contin. Dyn. Syst.B., 10 (2008), 857-871.  doi: 10.3934/dcdsb.2008.10.857.  Google Scholar

[6]

J. H. Huang and X. F. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466.  doi: 10.1016/S0022-247X(02)00135-X.  Google Scholar

[7]

J. Huang and X. Zou, Traveling wave solutions in delayer reaction diffusion systems with partial monotonicity, Acta. Math. Appl. Sinica., 22 (2006), 243-256.  doi: 10.1007/s10255-006-0300-0.  Google Scholar

[8]

X. Liang and X. Q. Zhao, Asymptotic speed of spread and traveling waves for monotone semiflows with applications, Comm. Pure. Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[9]

Y. LinQ. R. Wang and K. Zhou, Traveling wave solutions in n-dimensional delayed reaction-diffusion systems with mixed monotonicity, J. Comput. Appl. Math., 243 (2013), 16-27.  doi: 10.1016/j.cam.2012.11.007.  Google Scholar

[10]

G. Y. Lv and M. X. Wang, Existence, uniqueness and asymptotic behavior of taraveling wave fronts for a vector disease model, Nonlinear Anal., 11 (2010), 2035-2043.  doi: 10.1016/j.nonrwa.2009.05.006.  Google Scholar

[11]

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

[12]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differetial equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[13]

X. H. ShangZ. J. Du and X. J. Lin, Traveling wave solutions in $n$-dimensional delayed reaction-diffusion systems and application to four-dimensional predator-prey systems, Math. Methods Appl. Sci., 39 (2016), 1607-1620.  doi: 10.1002/mma.3595.  Google Scholar

[14]

R. Xu, A reaction-diffusion predator-prey model with stage structure and nonlocal delay, Appl. Math. Comput., 175 (2006), 984-1006.  doi: 10.1016/j.amc.2005.08.014.  Google Scholar

[15]

R. Xu and Z. E. Ma, Global stability of a reaction-diffusion predator-prey model with a nonlocal delay, Math. Comput. Model., 50 (2009), 194-206.  doi: 10.1016/j.mcm.2009.02.011.  Google Scholar

[16]

R. Xu and X. Zhang, Gobal stability and traveling waves of a predator-prey model with diffusion and nonlocal maturation delay, Comm. Nonlinear. Sci. Numer. Simulat., 15 (2010), 3390-3401.  doi: 10.1016/j.cnsns.2009.12.031.  Google Scholar

[17]

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

[18]

X. Zhang and R. Xu, Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure, J. Math. Anal. Appl., 373 (2011), 475-484.  doi: 10.1016/j.jmaa.2010.07.044.  Google Scholar

[19]

K. Zhou and Q. R. Wang, Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity, J. Math. Anal. Appl., 372 (2010), 598-610.  doi: 10.1016/j.jmaa.2010.07.032.  Google Scholar

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