May  2020, 25(5): 1859-1870. doi: 10.3934/dcdsb.2020006

Traveling waves for a reaction-diffusion model with a cyclic structure

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author: Tianran Zhang

Received  December 2018 Revised  March 2019 Published  May 2020 Early access  December 2019

Fund Project: The author is supported by the National Natural Science Foundation of China (11871403), the Natural Science Foundation of Chongqing (cstc2018jcyjAX0439), and the Fundamental Research Funds for the Central Universities (XDJK2018C072).

In this paper, a reaction-diffusion model with a cyclic structure is studied, which includes the SIS disease-transmission model and the nutrient-phytoplankton model. The minimal wave speed $ c^* $ of traveling wave solutions is given. The existence of traveling semi-fronts with $ c>c^* $ is proved by Schauder's fixed-point theorem. The traveling semi-fronts are shown to be bounded by rescaling method and comparison principle. The existence of traveling semi-front with $ c = c^* $ is obtained by limit arguments. Finally, the traveling semi-fronts are shown to connect to the positive equilibrium by a Lyapunov function.

Citation: Tianran Zhang. Traveling waves for a reaction-diffusion model with a cyclic structure. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1859-1870. doi: 10.3934/dcdsb.2020006
References:
[1]

A. ArapostathisM. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations, 24 (1999), 1555-1571.  doi: 10.1080/03605309908821475.  Google Scholar

[2]

Z. G. Bai and S.-L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221-232.  doi: 10.1016/j.amc.2015.04.048.  Google Scholar

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H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

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D. T. Dimitrov and H. V. Kojouharov, Analysis and numerical simulation of phytoplankton-nutrient systems with nutrient loss, Math. Comput. Simulation, 70 (2005), 33-43.  doi: 10.1016/j.matcom.2005.03.001.  Google Scholar

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W. DingW. Z. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.  doi: 10.3934/dcdsb.2013.18.1291.  Google Scholar

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A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

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A. DucrotM. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commun. Pure Appl. Anal., 11 (2012), 97-113.  doi: 10.3934/cpaa.2012.11.97.  Google Scholar

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S.-C. Fu and J.-C. Tsai, Wave propagation in predator-prey systems, Nonlinearity, 28 (2015), 4389-4423.  doi: 10.1088/0951-7715/28/12/4389.  Google Scholar

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L. Girardin, Non-cooperative Fisher-KPP systems: Traveling waves and long-time behavior, Nonlinearity, 31 (2018), 108-164.  doi: 10.1088/1361-6544/aa8ca7.  Google Scholar

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C. H. Hsu and T. S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.  doi: 10.1088/0951-7715/26/1/121.  Google Scholar

[14]

W. Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dynam. Differential Equations, 24 (2012), 633-644.  doi: 10.1007/s10884-012-9255-4.  Google Scholar

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W. Z. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations, 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060.  Google Scholar

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A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1.  Google Scholar

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K.-Y. LamX. Y. Wang and T. R. Zhang, Traveling waves for a class of diffusive disease-transmission models with network structures, SIAM J. Math. Anal., 50 (2018), 5719-5748.  doi: 10.1137/17M1144258.  Google Scholar

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H. LiR. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.  Google Scholar

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Y. Li, W.-T. Li and Y.-R. Yang, Stability of traveling waves of a diffusive susceptible-infective-removed (SIR) epidemic model, J. Math. Phys., 57 (2016), 041504, 28 pp. doi: 10.1063/1.4947106.  Google Scholar

[20]

Z. H. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, International J. Biomathematics, 10 (2017), 1750071, 23 pp. doi: 10.1142/S1793524517500711.  Google Scholar

[21]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

[22]

T. R. Zhang and Y. Jin, Traveling waves for a reaction-diffusion-advection predator-prey model, Nonlinear Anal. Real World Appl., 36 (2017), 203-232.  doi: 10.1016/j.nonrwa.2017.01.011.  Google Scholar

[23]

T. R. ZhangW. D. Wang and K. F. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations, 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

[24]

L. Zhao and Z.-C. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033.  Google Scholar

[25]

L. ZhaoZ.-C. Wang and S. G. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.  doi: 10.1088/1361-6544/aa59ae.  Google Scholar

show all references

References:
[1]

A. ArapostathisM. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations, 24 (1999), 1555-1571.  doi: 10.1080/03605309908821475.  Google Scholar

[2]

Z. G. Bai and S.-L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221-232.  doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[3]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[4]

N. F. Britton, Essential Mathematical Biology, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2003. doi: 10.1007/978-1-4471-0049-2.  Google Scholar

[5]

C. J. DaiM. Zhao and H. G. Yu, Dynamics induced by delay in a nutrient-phytoplankton model with diffusion, Ecological Complexity, 26 (2016), 29-36.  doi: 10.1016/j.ecocom.2016.03.001.  Google Scholar

[6]

D. T. Dimitrov and H. V. Kojouharov, Analysis and numerical simulation of phytoplankton-nutrient systems with nutrient loss, Math. Comput. Simulation, 70 (2005), 33-43.  doi: 10.1016/j.matcom.2005.03.001.  Google Scholar

[7]

W. DingW. Z. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.  doi: 10.3934/dcdsb.2013.18.1291.  Google Scholar

[8]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[9]

A. DucrotM. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commun. Pure Appl. Anal., 11 (2012), 97-113.  doi: 10.3934/cpaa.2012.11.97.  Google Scholar

[10]

S.-C. Fu and J.-C. Tsai, Wave propagation in predator-prey systems, Nonlinearity, 28 (2015), 4389-4423.  doi: 10.1088/0951-7715/28/12/4389.  Google Scholar

[11]

L. Girardin, Non-cooperative Fisher-KPP systems: Traveling waves and long-time behavior, Nonlinearity, 31 (2018), 108-164.  doi: 10.1088/1361-6544/aa8ca7.  Google Scholar

[12]

F. M. Hilker and M. A. Lewis, Predator-prey systems in streams and rivers, Theor. Ecol., 3 (2010), 175-193.  doi: 10.1007/s12080-009-0062-4.  Google Scholar

[13]

C. H. Hsu and T. S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.  doi: 10.1088/0951-7715/26/1/121.  Google Scholar

[14]

W. Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dynam. Differential Equations, 24 (2012), 633-644.  doi: 10.1007/s10884-012-9255-4.  Google Scholar

[15]

W. Z. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations, 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060.  Google Scholar

[16]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1.  Google Scholar

[17]

K.-Y. LamX. Y. Wang and T. R. Zhang, Traveling waves for a class of diffusive disease-transmission models with network structures, SIAM J. Math. Anal., 50 (2018), 5719-5748.  doi: 10.1137/17M1144258.  Google Scholar

[18]

H. LiR. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.  Google Scholar

[19]

Y. Li, W.-T. Li and Y.-R. Yang, Stability of traveling waves of a diffusive susceptible-infective-removed (SIR) epidemic model, J. Math. Phys., 57 (2016), 041504, 28 pp. doi: 10.1063/1.4947106.  Google Scholar

[20]

Z. H. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, International J. Biomathematics, 10 (2017), 1750071, 23 pp. doi: 10.1142/S1793524517500711.  Google Scholar

[21]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

[22]

T. R. Zhang and Y. Jin, Traveling waves for a reaction-diffusion-advection predator-prey model, Nonlinear Anal. Real World Appl., 36 (2017), 203-232.  doi: 10.1016/j.nonrwa.2017.01.011.  Google Scholar

[23]

T. R. ZhangW. D. Wang and K. F. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations, 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

[24]

L. Zhao and Z.-C. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033.  Google Scholar

[25]

L. ZhaoZ.-C. Wang and S. G. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.  doi: 10.1088/1361-6544/aa59ae.  Google Scholar

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