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July  2020, 19(7): 3651-3672. doi: 10.3934/cpaa.2020161

The dynamics of nonlocal diffusion systems with different free boundaries

Lei Li , Jianping Wang and  Mingxin Wang ,

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author

Received  July 2019 Revised  January 2020 Published  April 2020

Fund Project: The third author is supported by NSFC grants 11771110, 11971128

This paper is concerned with a class of free boundary models with "nonlocal diffusions'' and different free boundaries, which are natural extensions of free boundary problems of reaction diffusion systems with different free boundaries in [M.X.Wang and Y.Zhang, J. Differ. Equ., 264 (2018), 3527-3558] and references therein. These different free boundaries, which may intersect each other as time evolves, are used to describe the spreading front of the species. We prove that such kind of nonlocal diffusion problems has a unique global solution. Moreover, we investigate the long time behavior of global solution and criteria of spreading and vanishing for the classical Lotka-Volterra competition, prey-predator and mutualist models.

Keywords: Nonlocal diffusion, different free boundaries, existence-uniqueness, spreading-vanishing, long time behavior.
Mathematics Subject Classification: Primary: 35K57, 35R20, 35R35, 92D25.
Citation: Lei Li, Jianping Wang, Mingxin Wang. The dynamics of nonlocal diffusion systems with different free boundaries. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3651-3672. doi: 10.3934/cpaa.2020161
References:
[1]

P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[2]

H. Berestycki, J. Coville and H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.  Google Scholar

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J. F. Cao, Y. H. Du, F. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

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J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differ. Equ., 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[6]

Y. H. Du and Z. G. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[7]

Y. H. Du, M. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542v1. Google Scholar

[8]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Differ. Equ., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

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J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

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V. Hutson, S. Martinez, K. Mischaikow and G. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1093/acprof:oso/9780199299126.003.0008.  Google Scholar

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C. Y. Kao, Y. Lou and W. X. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

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L. Li, W. J. Sheng and M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), Art. 123646. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

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S. Y. Liu, H. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differ. Equ., 266 (2019), 4769-4799.  doi: 10.1016/j.jde.2018.10.009.  Google Scholar

[14]

R. Natan, E. Klein, J. J. Robledo-Arnuncio and E. Revilla, Dispersal kernels: review, in Dispersal Ecology and Evolution (eds. J. Clobert, M. Baguette, T. G. Benton, and J. M. Bullock), Oxford University Press, Oxford, UK, (2012), 187–210. Google Scholar

[15]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusions I: global solution, J. Math. Anal. Appl., (2020), Art. 123974. DOI: 10.1016/j.jmaa.2020.123974. Google Scholar

[16]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusions II: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. B, (2020), doi: 10.3934/dcdsb.2020121. Google Scholar

[17]

M. X. Wang, On some free boundary problems of prey-predator model, J. Differ. Equ., 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[18]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differ. Equ., 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar

[19]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.  Google Scholar

[20]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[21]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differ. Equ., 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[22]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equ., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[23]

C. H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differ. Equ., 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[24]

Q. Y. Zhang and M. X. Wang, Dynamics for the diffusive mutualist model with advection and different free boundaries, J. Math. Anal. Appl., 474 (2019), 1512-1535.  doi: 10.1016/j.jmaa.2019.02.037.  Google Scholar

show all references

References:
[1]

P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440.  doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[2]

H. Berestycki, J. Coville and H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017.  Google Scholar

[3]

H. Berestycki, J. Coville and H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.  Google Scholar

[4]

J. F. Cao, Y. H. Du, F. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

[5]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differ. Equ., 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[6]

Y. H. Du and Z. G. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[7]

Y. H. Du, M. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542v1. Google Scholar

[8]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Differ. Equ., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[9]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

[10]

V. Hutson, S. Martinez, K. Mischaikow and G. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1093/acprof:oso/9780199299126.003.0008.  Google Scholar

[11]

C. Y. Kao, Y. Lou and W. X. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[12]

L. Li, W. J. Sheng and M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), Art. 123646. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

[13]

S. Y. Liu, H. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differ. Equ., 266 (2019), 4769-4799.  doi: 10.1016/j.jde.2018.10.009.  Google Scholar

[14]

R. Natan, E. Klein, J. J. Robledo-Arnuncio and E. Revilla, Dispersal kernels: review, in Dispersal Ecology and Evolution (eds. J. Clobert, M. Baguette, T. G. Benton, and J. M. Bullock), Oxford University Press, Oxford, UK, (2012), 187–210. Google Scholar

[15]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusions I: global solution, J. Math. Anal. Appl., (2020), Art. 123974. DOI: 10.1016/j.jmaa.2020.123974. Google Scholar

[16]

J. P. Wang and M. X. Wang, Free boundary problems with nonlocal and local diffusions II: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. B, (2020), doi: 10.3934/dcdsb.2020121. Google Scholar

[17]

M. X. Wang, On some free boundary problems of prey-predator model, J. Differ. Equ., 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[18]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differ. Equ., 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar

[19]

M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.  Google Scholar

[20]

M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467.  doi: 10.1016/j.na.2017.01.005.  Google Scholar

[21]

M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differ. Equ., 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[22]

M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equ., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[23]

C. H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differ. Equ., 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[24]

Q. Y. Zhang and M. X. Wang, Dynamics for the diffusive mutualist model with advection and different free boundaries, J. Math. Anal. Appl., 474 (2019), 1512-1535.  doi: 10.1016/j.jmaa.2019.02.037.  Google Scholar

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