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September  2021, 14(9): 3113-3139. doi: 10.3934/dcdss.2020340

Traveling waves in a nonlocal dispersal predator-prey model

School of Mathematics and Statistics, Lanzhou University, Gansu, Lanzhou 730000, People's Republic of China

* Corresponding author: Wan-Tong Li

Received  May 2019 Revised  October 2019 Published  September 2021 Early access  April 2020

Fund Project: The second author is supported by NSF of China grant 11731005, 11671180 and the third author is supported by NSF of China grant 11601205

This paper is concerned with the traveling wave solutions for a class of predator-prey model with nonlocal dispersal. By adopting the truncation method, we use Schauder's fixed-point theorem to obtain the existence of traveling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations for $ c>c_{*} $, in which $ c_* $ is the minimal wave speed. Meanwhile, through the limiting approach and the delicate analysis, we establish the existence of traveling wave solutions with the critical speed. Finally, we show the nonexistence of traveling waves for $ 0<c<c_{*} $ by the characteristic equation corresponding to the linearization of original system at the semi-trivial equilibrium. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the non-preserving of system itself.

Citation: Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3113-3139. doi: 10.3934/dcdss.2020340
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, Amer. Math. Soc., Providence, Rhode Island, 2010. doi: 10.1090/surv/165.

[2]

S. B. AiY. H. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Equations, 263 (2017), 7782-7814.  doi: 10.1016/j.jde.2017.08.021.

[3]

M. AlfaroJ. Coville and G. Raoul, Bistable travelling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791.  doi: 10.3934/dcds.2014.34.1775.

[4]

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

[5]

P. W. BatesP. C. FifeX. F. Ren and X. F. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[6]

X. X. BaoW. T. Li and W. X. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.

[7]

X. X. BaoW.-T. Li and Z.-C. Wang, Time periodic traveling curved fronts in the periodic Lotka-Volterra competition-diffusion system, J. Dynam. Differential Equations, 29 (2017), 981-1016.  doi: 10.1007/s10884-015-9512-4.

[8]

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

[9]

Y.-Y. ChenJ. S. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359.  doi: 10.1088/1361-6544/aa6b0a.

[10]

Y.-Y. ChenJ. S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071.

[11]

J. CovilleJ. 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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S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.

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S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $\mathbb{R}^4$, Trans. Amer. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810.

[14]

S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.  doi: 10.1137/0146063.

[15]

W. Ding and W. Z. Huang, Traveling wave solutions for some classes of diffusive predator-prey models, J. Dynam. Differential Equations, 28 (2016), 1293-1308.  doi: 10.1007/s10884-015-9472-8.

[16]

F. D. DongW. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272.

[17]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenice., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[18]

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.

[19]

C.-H. HsuC.-R. YangT.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008.

[20]

Y. L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, J. Math. Anal. Appl., 418 (2014), 163-184.  doi: 10.1016/j.jmaa.2014.03.085.

[21]

J. H. HuangG. Lu and S. G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.  doi: 10.1007/s00285-002-0171-9.

[22]

K. Hong and P. X. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-Ⅱ functional response and harvesting, Nonlinear Anal. Real World Appl., 14 (2013), 83-103.  doi: 10.1016/j.nonrwa.2012.05.004.

[23]

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.

[24]

A. N. KolmogorovI. G. Petrovskii and N. S. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A: Mat. Mekh., 1 (1937), 1-26. 

[25]

W.-T. LiY.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.

[26]

W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-Ⅲ functional response, Chaos Solitons Fractals, 37 (2008), 476-486.  doi: 10.1016/j.chaos.2006.09.039.

[27]

W.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.

[28]

X.-S. Li and G. Lin, Traveling wavefronts in nonlocal dispersal and cooperative Lotka-Volterra system with delays, Appl. Math. Comput., 204 (2008), 738-744.  doi: 10.1016/j.amc.2008.07.016.

[29]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.

[30]

G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.  doi: 10.1016/j.na.2013.10.024.

[31]

X. B. LinP. X. Weng and C. F. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dynam. Differential Equations, 23 (2011), 903-921.  doi: 10.1007/s10884-011-9220-7.

[32]

A. J. Lotka, Elements of Physicals Biology, Williams and Wilkins Company, Baltimore, 1925.

[33]

J. D. Murray, Mathematical Biology. I: An Introduction, 3$^rd$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[34]

S. X. PanW.-T. 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.

[35]

J. A. Sherratt, Invasion generates periodic traveling waves (wavetrains) in predator-prey models with nonlocal dispersal, SIAM J. Appl. Math., 76 (2016), 293-313.  doi: 10.1137/15M1027991.

[36]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.

[37]

F.-Y. Yang and W.-T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.

[38]

F.-Y. YangW.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129-147.  doi: 10.1016/j.nonrwa.2014.12.001.

[39]

F.-Y. YangY. LiW.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.

[40]

G.-B. ZhangW.-T. 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.

[41]

G.-B. ZhangW.-T. Li and Z.-C. 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.

[42]

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.

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, Amer. Math. Soc., Providence, Rhode Island, 2010. doi: 10.1090/surv/165.

[2]

S. B. AiY. H. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Equations, 263 (2017), 7782-7814.  doi: 10.1016/j.jde.2017.08.021.

[3]

M. AlfaroJ. Coville and G. Raoul, Bistable travelling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791.  doi: 10.3934/dcds.2014.34.1775.

[4]

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

[5]

P. W. BatesP. C. FifeX. F. Ren and X. F. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.  doi: 10.1007/s002050050037.

[6]

X. X. BaoW. T. Li and W. X. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.

[7]

X. X. BaoW.-T. Li and Z.-C. Wang, Time periodic traveling curved fronts in the periodic Lotka-Volterra competition-diffusion system, J. Dynam. Differential Equations, 29 (2017), 981-1016.  doi: 10.1007/s10884-015-9512-4.

[8]

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

[9]

Y.-Y. ChenJ. S. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359.  doi: 10.1088/1361-6544/aa6b0a.

[10]

Y.-Y. ChenJ. S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predator-prey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071.

[11]

J. CovilleJ. 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.

[12]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.

[13]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $\mathbb{R}^4$, Trans. Amer. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810.

[14]

S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.  doi: 10.1137/0146063.

[15]

W. Ding and W. Z. Huang, Traveling wave solutions for some classes of diffusive predator-prey models, J. Dynam. Differential Equations, 28 (2016), 1293-1308.  doi: 10.1007/s10884-015-9472-8.

[16]

F. D. DongW. T. Li and J. B. Wang, Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application, Discrete Contin. Dyn. Syst., 37 (2017), 6291-6318.  doi: 10.3934/dcds.2017272.

[17]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenice., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[18]

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.

[19]

C.-H. HsuC.-R. YangT.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008.

[20]

Y. L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, J. Math. Anal. Appl., 418 (2014), 163-184.  doi: 10.1016/j.jmaa.2014.03.085.

[21]

J. H. HuangG. Lu and S. G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.  doi: 10.1007/s00285-002-0171-9.

[22]

K. Hong and P. X. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-Ⅱ functional response and harvesting, Nonlinear Anal. Real World Appl., 14 (2013), 83-103.  doi: 10.1016/j.nonrwa.2012.05.004.

[23]

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.

[24]

A. N. KolmogorovI. G. Petrovskii and N. S. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A: Mat. Mekh., 1 (1937), 1-26. 

[25]

W.-T. LiY.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.

[26]

W.-T. Li and S.-L. Wu, Traveling waves in a diffusive predator-prey model with Holling type-Ⅲ functional response, Chaos Solitons Fractals, 37 (2008), 476-486.  doi: 10.1016/j.chaos.2006.09.039.

[27]

W.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.

[28]

X.-S. Li and G. Lin, Traveling wavefronts in nonlocal dispersal and cooperative Lotka-Volterra system with delays, Appl. Math. Comput., 204 (2008), 738-744.  doi: 10.1016/j.amc.2008.07.016.

[29]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.

[30]

G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.  doi: 10.1016/j.na.2013.10.024.

[31]

X. B. LinP. X. Weng and C. F. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dynam. Differential Equations, 23 (2011), 903-921.  doi: 10.1007/s10884-011-9220-7.

[32]

A. J. Lotka, Elements of Physicals Biology, Williams and Wilkins Company, Baltimore, 1925.

[33]

J. D. Murray, Mathematical Biology. I: An Introduction, 3$^rd$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[34]

S. X. PanW.-T. 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.

[35]

J. A. Sherratt, Invasion generates periodic traveling waves (wavetrains) in predator-prey models with nonlocal dispersal, SIAM J. Appl. Math., 76 (2016), 293-313.  doi: 10.1137/15M1027991.

[36]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0.

[37]

F.-Y. Yang and W.-T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.

[38]

F.-Y. YangW.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129-147.  doi: 10.1016/j.nonrwa.2014.12.001.

[39]

F.-Y. YangY. LiW.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.

[40]

G.-B. ZhangW.-T. 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.

[41]

G.-B. ZhangW.-T. Li and Z.-C. 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.

[42]

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.

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