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

The dynamics of nonlocal diffusion systems with different free boundaries

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.

Citation: Lei Li, Jianping Wang, Mingxin Wang. The dynamics of nonlocal diffusion systems with different free boundaries. Communications on Pure and 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.

[2]

H. BerestyckiJ. 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.

[3]

H. BerestyckiJ. 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.

[4]

J. F. CaoY. H. DuF. 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.

[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.

[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.

[7]

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

[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.

[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.

[10]

V. HutsonS. MartinezK. Mischaikow and G. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1093/acprof:oso/9780199299126.003.0008.

[11]

C. Y. KaoY. 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.

[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.

[13]

S. Y. LiuH. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

H. BerestyckiJ. 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.

[3]

H. BerestyckiJ. 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.

[4]

J. F. CaoY. H. DuF. 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.

[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.

[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.

[7]

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

[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.

[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.

[10]

V. HutsonS. MartinezK. Mischaikow and G. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1093/acprof:oso/9780199299126.003.0008.

[11]

C. Y. KaoY. 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.

[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.

[13]

S. Y. LiuH. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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