March  2022, 42(3): 1127-1162. doi: 10.3934/dcds.2021149

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 Published  March 2022 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 and Continuous Dynamical Systems, 2022, 42 (3) : 1127-1162. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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