December  2021, 20(12): 4083-4105. doi: 10.3934/cpaa.2021146

Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author

Received  September 2020 Revised  June 2021 Published  December 2021 Early access  September 2021

Fund Project: F. Y. Yang was partially supported by NSF of China (11601205) and W. T. Li was partially supported by NSF of China (11731005, 11671180)

This paper is concerned with the propagation dynamics of a nonlocal dispersal predator-prey model with two predators and one prey. Precisely, our main concern is the invasion process of the two predators into the habitat of one prey, when the two predators are weak competitors in the absence of prey. This invasion process is characterized by the spreading speed of the predators as well as the minimal wave speed of traveling waves connecting the predator-free state to the co-existence state. Particularly, the right-hand tail limit of wave profile is derived by the idea of contracting rectangle.

Citation: Feiying Yang, Wantong Li, Renhu Wang. Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4083-4105. doi: 10.3934/cpaa.2021146
References:
[1]

F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165.

[2]

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

[3]

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

[4]

Z. Bi and S. Pan, Dynamics of a predator-prey system with three species, Bound. Value Probl., 162 (2018), 25 pp. doi: 10.1186/s13661-018-1084-x.

[5]

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

[6]

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

[7]

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.

[8]

A. Ducrot, J. S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), 25 pp. doi: 10.1007/s00033-019-1188-x.

[9]

Z. DuZ. Feng and X. Zhang, Traveling wave phenomena of n-dimensional diffusive predator-prey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.  doi: 10.1016/j.nonrwa.2017.10.012.

[10]

F. D. Dong, W. T. Li and G. B. Zhang, Invasion traveling wave solutions of a predator-prey model with nonlocal dispersal, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 17 pp. doi: 10.1016/j.cnsns.2019.104926.

[11]

P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin (2003), 153–191. doi: 10.1007/978-3-662-05281-5_3.

[12]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.

[13]

J. S. GuoK. I. NakamuraT. Ogiwara and C. C. Wu, Traveling wave solutions for a predator-prey system with two predators and one prey, Nonlinear Anal. Real World Appl., 54 (2020), 103-111.  doi: 10.1016/j.nonrwa.2020.103111.

[14]

C. H. Hsu and J. J. Lin, Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models, Commun. Pure Appl. Anal., 18 (2019), 1483-1508.  doi: 10.3934/cpaa.2019071.

[15]

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.

[16]

Y. Jin and X. Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity, 22 (2009), 1167-1189.  doi: 10.1088/0951-7715/22/5/011.

[17]

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.

[18]

W. T. LiW. B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512.  doi: 10.3934/dcds.2017107.

[19]

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.

[20]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differ. Equ., 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.

[21]

J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[22]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.

[23]

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.

[24]

Y. J. SunW. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differ. Equ., 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.

[25]

X. J. Wang and G. Lin, Asymptotic spreading for a time-periodic predator-prey system, Commun. Pure Appl. Anal., 18 (2019), 2983-2999.  doi: 10.3934/cpaa.2019133.

[26]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.

[27]

C.C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14.  doi: 10.1016/j.aml.2018.11.022.

[28]

W. B. XuW. T. Li and G. Lin, Nonlocal dispersal cooperative systems: acceleration propagation among species, J. Differ. Equ., 268 (2020), 1081-1105.  doi: 10.1016/j.jde.2019.08.039.

[29]

F. Y. YangW. T. Li and J. B. Wang, Wave propagation for a class of non-local dispersal non-cooperative systems, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1965-1997.  doi: 10.1017/prm.2019.4.

[30]

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.

[31]

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.

[32]

G. B. ZhangW. T. Li and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equ., 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.

[33]

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.

[34]

T. 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.

[35]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differ. Equ., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165.

[2]

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

[3]

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

[4]

Z. Bi and S. Pan, Dynamics of a predator-prey system with three species, Bound. Value Probl., 162 (2018), 25 pp. doi: 10.1186/s13661-018-1084-x.

[5]

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

[6]

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

[7]

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.

[8]

A. Ducrot, J. S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), 25 pp. doi: 10.1007/s00033-019-1188-x.

[9]

Z. DuZ. Feng and X. Zhang, Traveling wave phenomena of n-dimensional diffusive predator-prey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.  doi: 10.1016/j.nonrwa.2017.10.012.

[10]

F. D. Dong, W. T. Li and G. B. Zhang, Invasion traveling wave solutions of a predator-prey model with nonlocal dispersal, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), 17 pp. doi: 10.1016/j.cnsns.2019.104926.

[11]

P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin (2003), 153–191. doi: 10.1007/978-3-662-05281-5_3.

[12]

J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.

[13]

J. S. GuoK. I. NakamuraT. Ogiwara and C. C. Wu, Traveling wave solutions for a predator-prey system with two predators and one prey, Nonlinear Anal. Real World Appl., 54 (2020), 103-111.  doi: 10.1016/j.nonrwa.2020.103111.

[14]

C. H. Hsu and J. J. Lin, Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models, Commun. Pure Appl. Anal., 18 (2019), 1483-1508.  doi: 10.3934/cpaa.2019071.

[15]

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.

[16]

Y. Jin and X. Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity, 22 (2009), 1167-1189.  doi: 10.1088/0951-7715/22/5/011.

[17]

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.

[18]

W. T. LiW. B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512.  doi: 10.3934/dcds.2017107.

[19]

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.

[20]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differ. Equ., 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.

[21]

J. D. Murray, Mathematical Biology, II, Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[22]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.

[23]

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.

[24]

Y. J. SunW. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differ. Equ., 251 (2011), 551-581.  doi: 10.1016/j.jde.2011.04.020.

[25]

X. J. Wang and G. Lin, Asymptotic spreading for a time-periodic predator-prey system, Commun. Pure Appl. Anal., 18 (2019), 2983-2999.  doi: 10.3934/cpaa.2019133.

[26]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.

[27]

C.C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14.  doi: 10.1016/j.aml.2018.11.022.

[28]

W. B. XuW. T. Li and G. Lin, Nonlocal dispersal cooperative systems: acceleration propagation among species, J. Differ. Equ., 268 (2020), 1081-1105.  doi: 10.1016/j.jde.2019.08.039.

[29]

F. Y. YangW. T. Li and J. B. Wang, Wave propagation for a class of non-local dispersal non-cooperative systems, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1965-1997.  doi: 10.1017/prm.2019.4.

[30]

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.

[31]

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.

[32]

G. B. ZhangW. T. Li and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equ., 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.

[33]

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.

[34]

T. 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.

[35]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differ. Equ., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.

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