doi: 10.3934/dcds.2021149
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Two species nonlocal diffusion systems with free boundaries

1. 

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

2. 

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

3. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

4. 

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

*Corresponding author: Mingxin Wang

Received  May 2021 Revised  September 2021 Early access September 2021

Fund Project: Y. Du was supported by the Australian Research Council. M. Wang was supported by NSFC Grant 11771110. M. Zhao was supported by NWNU-LKQN2021-16 and a scholarship from the China Scholarship Council

We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [7], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.

Citation: Yihong Du, Mingxin Wang, Meng Zhao. Two species nonlocal diffusion systems with free boundaries. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021149
References:
[1]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals Ⅰ: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35 pp. doi: 10.1007/s00526-018-1419-6.  Google Scholar

[2]

X. Bai and F. Li, Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small, Discrete Contin. Dyn. Syst., 40 (2020), 3075-3092.  doi: 10.3934/dcds.2020035.  Google Scholar

[3]

X. BaoW. T. Li and W. 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.  Google Scholar

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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

[5]

H. BerestyckiJ. Coville and H. 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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H. BerestyckiJ. Coville and H.-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

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Y. Du and Z. 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

[10]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

[11]

Y. Du and W. Ni, Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407-4448.  doi: 10.1088/1361-6544/ab8bb2.  Google Scholar

[12]

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[13]

Y. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), 36 pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[14]

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

[15]

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

[16]

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[17]

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[18]

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

[19]

L. Li, W. Sheng and M. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), 27 pp. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

[20]

L. LiJ. Wang and M. Wang, The dynamics of nonlocal diffusion systems with different free boundaries, Commun. Pure Appl. Anal., 19 (2020), 3651-3672.  doi: 10.3934/cpaa.2020161.  Google Scholar

[21]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[22]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.  Google Scholar

[23]

R. Nathan, E. Klein, J. J. Robledo-Arnuncio and E. Revilla, Dispersal kernels: Review, Dispersal Ecology and Evolution, J. Clobert, M. Baguette, T. G. Benton and J. M. Bullock, eds., Oxford University Press, Oxford, UK, (2012), 187–210. doi: 10.1093/acprof:oso/9780199608898.003.0015.  Google Scholar

[24]

J. Wang and M. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762.  doi: 10.1002/mma.5189.  Google Scholar

[25]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., 490 (2020), 24 pp. doi: 10.1016/j.jmaa.2020.123974.  Google Scholar

[26]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusion Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4721-4736.  doi: 10.3934/dcdsb.2020121.  Google Scholar

[27]

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

[28]

M. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.  Google Scholar

[29] M. Wang, Nonlinear Second Order Parabolic Equations, Boca Raton: CRC Press, 2021.   Google Scholar
[30]

M. Wang and Q. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst., 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.  Google Scholar

[31]

M. 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

[32]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), 24 pp. doi: 10.1007/s00033-016-0729-9.  Google Scholar

[33]

M. 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

[34]

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

[35]

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

[36]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[37]

Y. Zhang and M. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.  Google Scholar

[38]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar

[39]

M. ZhaoY. ZhangW. T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.  Google Scholar

[40]

Y. Zhao and M. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.  Google Scholar

show all references

References:
[1]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals Ⅰ: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35 pp. doi: 10.1007/s00526-018-1419-6.  Google Scholar

[2]

X. Bai and F. Li, Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small, Discrete Contin. Dyn. Syst., 40 (2020), 3075-3092.  doi: 10.3934/dcds.2020035.  Google Scholar

[3]

X. BaoW. T. Li and W. 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.  Google Scholar

[4]

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

[5]

H. BerestyckiJ. Coville and H. 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

[6]

H. BerestyckiJ. Coville and H.-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

[7]

J. F. CaoY. 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.  Google Scholar

[8]

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

[9]

Y. Du and Z. 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

[10]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

[11]

Y. Du and W. Ni, Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33 (2020), 4407-4448.  doi: 10.1088/1361-6544/ab8bb2.  Google Scholar

[12]

Y. DuM. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

[13]

Y. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), 36 pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[14]

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

[15]

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

[16]

G. HetzerT. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[17]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[18]

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

[19]

L. Li, W. Sheng and M. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), 27 pp. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

[20]

L. LiJ. Wang and M. Wang, The dynamics of nonlocal diffusion systems with different free boundaries, Commun. Pure Appl. Anal., 19 (2020), 3651-3672.  doi: 10.3934/cpaa.2020161.  Google Scholar

[21]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[22]

M. MimuraY. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  doi: 10.32917/hmj/1206130304.  Google Scholar

[23]

R. Nathan, E. Klein, J. J. Robledo-Arnuncio and E. Revilla, Dispersal kernels: Review, Dispersal Ecology and Evolution, J. Clobert, M. Baguette, T. G. Benton and J. M. Bullock, eds., Oxford University Press, Oxford, UK, (2012), 187–210. doi: 10.1093/acprof:oso/9780199608898.003.0015.  Google Scholar

[24]

J. Wang and M. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762.  doi: 10.1002/mma.5189.  Google Scholar

[25]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusions Ⅰ: Global solution, J. Math. Anal. Appl., 490 (2020), 24 pp. doi: 10.1016/j.jmaa.2020.123974.  Google Scholar

[26]

J. Wang and M. Wang, Free boundary problems with nonlocal and local diffusion Ⅱ: Spreading-vanishing and long-time behavior, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4721-4736.  doi: 10.3934/dcdsb.2020121.  Google Scholar

[27]

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

[28]

M. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.  Google Scholar

[29] M. Wang, Nonlinear Second Order Parabolic Equations, Boca Raton: CRC Press, 2021.   Google Scholar
[30]

M. Wang and Q. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst., 38 (2018), 2591-2607.  doi: 10.3934/dcds.2018109.  Google Scholar

[31]

M. 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

[32]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67 (2016), 24 pp. doi: 10.1007/s00033-016-0729-9.  Google Scholar

[33]

M. 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

[34]

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

[35]

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

[36]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[37]

Y. Zhang and M. Wang, A free boundary problem of the ratio-dependent prey-predator model, Appl. Anal., 94 (2015), 2147-2167.  doi: 10.1080/00036811.2014.979806.  Google Scholar

[38]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar

[39]

M. ZhaoY. ZhangW. T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.  Google Scholar

[40]

Y. Zhao and M. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.  Google Scholar

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