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November  2020, 25(11): 4479-4492. doi: 10.3934/dcdsb.2020108

Spreading speeds for a class of non-local convolution differential equation

Zhaoquan Xu 1, and  Chufen Wu 2,,

1. 

Department of Mathematics, Jinan University, Guangzhou 510632, China
2. 

School of Mathematics and Big Data, Foshan University, Foshan 528000, China

* Corresponding author: chufenwu@126.com

Received  January 2019 Revised  November 2019 Published  November 2020 Early access  March 2020

Fund Project: The first author is supported by NSF of China grant No. 11701216, NSF of Guangdong Province grant No. 2017A030313015 and the Fundamental Research Funds for the Central Universities. The second author is supported by NSF of Guangdong Province grant No. 2019A1515011648 and NSF of China grant No. 11401096

The spatial spreading dynamics is considered for a class of convolution differential equation resulting from physical and biological problems. It is shown that this kind of equation with monostable structure admits a spreading speed, even when the nonlinear reaction terms without monotonicity. The upward convergence of spreading speed is also established under appropriate conditions.

Keywords: Convolution equation, non-monotone, spreading speed.
Mathematics Subject Classification: Primary: 45J05, 35R10; Secondary: 92A05.
Citation: Zhaoquan Xu, Chufen Wu. Spreading speeds for a class of non-local convolution differential equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4479-4492. doi: 10.3934/dcdsb.2020108
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, 446, Springer, Berlin, 1975, 5–49. doi: 10.1007/BFb0070595.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

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

[4]

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

[5]

Z. X. Chen, B. Ermentrout and B. Mcleod, Traveling fronts for a class of non-local convolution differential quations, Appl. Anal., 64 (1997), 235-253.  doi: 10.1080/00036819708840533.

[6]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73.  doi: 10.1016/0022-0396(79)90080-9.

[7]

O. Diekmann, Thresholds and travelling waves for the geographical spread of an infection, J. Math. Biol., 6 (1978), 109-130.  doi: 10.1007/BF02450783.

[8]

O. Diekmann and H. G. Kapper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9.

[9]

W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.

[10]

B. Ermentrout and J. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.

[11]

J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.

[12]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.

[13]

C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation, J. Math. Anal. Appl., 420 (2014), 1-19.  doi: 10.1016/j.jmaa.2014.05.064.

[14]

S. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[15]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[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]

B. T. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.

[18]

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

[19]

G. Lin, W. T. Li and S. G. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 165-201.  doi: 10.1007/s00285-010-0334-z.

[20]

A. De Masi, T. Gobron and E. Presutti, Travelling fronts in a non-local evolution equation, Arch. Rational Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506.

[21]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[22]

K. Schumacher, Travelling-front solutions for integro-differential equations. Ⅱ, in Biological Growth and Spread, Lecture Notes in Biomath., 38, Springer, Berlin-New York, 1980,296–309. doi: 10.1007/978-3-642-61850-5_28.

[23]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121.  doi: 10.1515/crll.1979.306.94.

[24]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.  doi: 10.1007/BF00279720.

[25]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.

[26]

H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411.  doi: 10.1007/s00285-008-0168-0.

[27]

S. L. Wu, W. T. Li and S. Y. Liu, Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability, Nonlinear Anal. Real World Appl., 10 (2009), 3141-3151.  doi: 10.1016/j.nonrwa.2008.10.012.

[28]

C. Wu, D. Xiao and X.-Q. Zhao, Asymptotic pattern of a migratory and nonmonotone population model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1171-1195. doi: 10.3934/dcdsb.2014.19.1171.

[29]

C. Wu, Y. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019.

[30]

Z. Xu, Asymptotic speeds of spread for a nonlocal diffusion equation, J. Dynam. Differential Equations, 30 (2018), 473-499.  doi: 10.1007/s10884-016-9555-1.

[31]

Z. Xu and D. Xiao, Regular traveling waves for a nonlocal diffusion equation, J. Differential Equations, 258 (2015), 191-223.  doi: 10.1016/j.jde.2014.09.008.

[32]

Z. Xu and C. Wu, Monostable waves in a class of non-local convolution differential equation, J. Math. Anal. Appl., 462 (2018), 1205-1224.  doi: 10.1016/j.jmaa.2018.02.036.

[33]

T. Yi, Y. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, 446, Springer, Berlin, 1975, 5–49. doi: 10.1007/BFb0070595.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

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

[4]

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

[5]

Z. X. Chen, B. Ermentrout and B. Mcleod, Traveling fronts for a class of non-local convolution differential quations, Appl. Anal., 64 (1997), 235-253.  doi: 10.1080/00036819708840533.

[6]

O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations, 33 (1979), 58-73.  doi: 10.1016/0022-0396(79)90080-9.

[7]

O. Diekmann, Thresholds and travelling waves for the geographical spread of an infection, J. Math. Biol., 6 (1978), 109-130.  doi: 10.1007/BF02450783.

[8]

O. Diekmann and H. G. Kapper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737.  doi: 10.1016/0362-546X(78)90015-9.

[9]

W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.

[10]

B. Ermentrout and J. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.

[11]

J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.

[12]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.

[13]

C. Gomez, H. Prado and S. Trofimchuk, Separation dichotomy and wavefronts for a nonlinear convolution equation, J. Math. Anal. Appl., 420 (2014), 1-19.  doi: 10.1016/j.jmaa.2014.05.064.

[14]

S. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[15]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.

[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]

B. T. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.

[18]

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

[19]

G. Lin, W. T. Li and S. G. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 165-201.  doi: 10.1007/s00285-010-0334-z.

[20]

A. De Masi, T. Gobron and E. Presutti, Travelling fronts in a non-local evolution equation, Arch. Rational Mech. Anal., 132 (1995), 143-205.  doi: 10.1007/BF00380506.

[21]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.

[22]

K. Schumacher, Travelling-front solutions for integro-differential equations. Ⅱ, in Biological Growth and Spread, Lecture Notes in Biomath., 38, Springer, Berlin-New York, 1980,296–309. doi: 10.1007/978-3-642-61850-5_28.

[23]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121.  doi: 10.1515/crll.1979.306.94.

[24]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.  doi: 10.1007/BF00279720.

[25]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.

[26]

H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411.  doi: 10.1007/s00285-008-0168-0.

[27]

S. L. Wu, W. T. Li and S. Y. Liu, Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing-monostability, Nonlinear Anal. Real World Appl., 10 (2009), 3141-3151.  doi: 10.1016/j.nonrwa.2008.10.012.

[28]

C. Wu, D. Xiao and X.-Q. Zhao, Asymptotic pattern of a migratory and nonmonotone population model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1171-1195. doi: 10.3934/dcdsb.2014.19.1171.

[29]

C. Wu, Y. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019.

[30]

Z. Xu, Asymptotic speeds of spread for a nonlocal diffusion equation, J. Dynam. Differential Equations, 30 (2018), 473-499.  doi: 10.1007/s10884-016-9555-1.

[31]

Z. Xu and D. Xiao, Regular traveling waves for a nonlocal diffusion equation, J. Differential Equations, 258 (2015), 191-223.  doi: 10.1016/j.jde.2014.09.008.

[32]

Z. Xu and C. Wu, Monostable waves in a class of non-local convolution differential equation, J. Math. Anal. Appl., 462 (2018), 1205-1224.  doi: 10.1016/j.jmaa.2018.02.036.

[33]

T. Yi, Y. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.

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