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doi: 10.3934/dcdsb.2022062
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Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

*Corresponding author: Ling Zhou

Received  May 2021 Early access March 2022

In this paper we give a classification of the global asymptotic stability for a nonlocal diffusion competition model with free boundaries consisting of an invasive species with density $ u $ and a native species with density $ v $. We not only prove that such nonlocal diffusion problem has a unique global solution and also determine the long-time asymptotic behavior of the solution for three competition cases : (I) $ u $ is an inferior competitor, (II) $ u $ is a superior competitor and (III) the weak competition case. Especially, in case (II), under some additional conditions, we determine the long-time asymptotic behavior of the solution when vanishing happens. Moreover, the criteria for spreading and vanishing are obtained.

Citation: Weiyi Zhang, Ling Zhou. Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022062
References:
[1]

X. L. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals Ⅰ: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 1-35.  doi: 10.1007/s00526-018-1419-6.

[2]

X. L. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.

[3]

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.

[4]

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.

[5]

P. Bates and G. Y. 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.

[6]

R. S. CantrellC. CosnerY. Lou and D. Ryan, Evolutionary stability of ideal free dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38. 

[7]

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.

[8]

J. F. CaoW. T. LiJ. Wang and M. Zhao, The dynamics of a Lotka-Volterra competition model with non-local diffusion and free boundaries, Adv. Differential Equations, 26 (2021), 163-200. 

[9]

J. F. CaoW. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.

[10]

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

[11]

C. Cort$\acute{a}$zarJ. CovilleM. Elgueta and S. Mart$\acute{i}$nez, A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241 (2007), 332-358.  doi: 10.1016/j.jde.2007.06.002.

[12]

C. Cort$\acute{a}$zarF. Quir$\acute{o}$s and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound., 21 (2019), 441-462.  doi: 10.4171/IFB/430.

[13]

C. CosnerJ. D$\acute{a}$vila and S. Mart$\acute{i}$nez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.  doi: 10.1080/17513758.2011.588341.

[14]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.

[15]

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.

[16]

J. CovilleJ. D$\acute{a}$vila and S. Mart$\acute{i}$nez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.

[17]

Y. H. Du and Z. G. 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.

[18]

Y. H. DuM. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, Discrete Contin. Dyn. Syst., 42 (2022), 1127-1162.  doi: 10.3934/dcds.2021149.

[19]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[20]

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.

[21]

C. Y. KaoY. Lou and W. X. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 2047-2072.  doi: 10.3934/dcdsb.2012.17.2047.

[22]

F. LiY. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.

[23]

N. Rawal and W. X. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations, 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.

[24]

N. RawalW. X. Shen and A. J. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.

[25]

W. X. Shen and Z. W. Shen, Existence, uniqueness and stability of transition fronts of non-local equations in time heterogeneous bistable media, European J. Appl. Math., 31 (2020), 601-645.  doi: 10.1017/S0956792519000202.

[26]

W. X. Shen and X. X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations, 259 (2015), 7375-7405.  doi: 10.1016/j.jde.2015.08.026.

[27]

W. X. Shen and A. J. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.

[28]

W. X. Shen and A. J. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.

[29]

W. Y. ZhangZ. H. Liu and L. Zhou, Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment, Discrete Contin. Dyn. Syst. Ser. B., 26 (2021), 3767-3784.  doi: 10.3934/dcdsb.2020256.

show all references

References:
[1]

X. L. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals Ⅰ: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 1-35.  doi: 10.1007/s00526-018-1419-6.

[2]

X. L. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.

[3]

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.

[4]

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.

[5]

P. Bates and G. Y. 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.

[6]

R. S. CantrellC. CosnerY. Lou and D. Ryan, Evolutionary stability of ideal free dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38. 

[7]

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.

[8]

J. F. CaoW. T. LiJ. Wang and M. Zhao, The dynamics of a Lotka-Volterra competition model with non-local diffusion and free boundaries, Adv. Differential Equations, 26 (2021), 163-200. 

[9]

J. F. CaoW. T. Li and M. Zhao, A nonlocal diffusion model with free boundaries in spatial heterogeneous environment, J. Math. Anal. Appl., 449 (2017), 1015-1035.  doi: 10.1016/j.jmaa.2016.12.044.

[10]

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

[11]

C. Cort$\acute{a}$zarJ. CovilleM. Elgueta and S. Mart$\acute{i}$nez, A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241 (2007), 332-358.  doi: 10.1016/j.jde.2007.06.002.

[12]

C. Cort$\acute{a}$zarF. Quir$\acute{o}$s and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound., 21 (2019), 441-462.  doi: 10.4171/IFB/430.

[13]

C. CosnerJ. D$\acute{a}$vila and S. Mart$\acute{i}$nez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.  doi: 10.1080/17513758.2011.588341.

[14]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.  doi: 10.1007/s10231-005-0163-7.

[15]

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.

[16]

J. CovilleJ. D$\acute{a}$vila and S. Mart$\acute{i}$nez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.

[17]

Y. H. Du and Z. G. 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.

[18]

Y. H. DuM. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, Discrete Contin. Dyn. Syst., 42 (2022), 1127-1162.  doi: 10.3934/dcds.2021149.

[19]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.

[20]

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.

[21]

C. Y. KaoY. Lou and W. X. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 2047-2072.  doi: 10.3934/dcdsb.2012.17.2047.

[22]

F. LiY. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.

[23]

N. Rawal and W. X. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations, 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.

[24]

N. RawalW. X. Shen and A. J. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst., 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.

[25]

W. X. Shen and Z. W. Shen, Existence, uniqueness and stability of transition fronts of non-local equations in time heterogeneous bistable media, European J. Appl. Math., 31 (2020), 601-645.  doi: 10.1017/S0956792519000202.

[26]

W. X. Shen and X. X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations, 259 (2015), 7375-7405.  doi: 10.1016/j.jde.2015.08.026.

[27]

W. X. Shen and A. J. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.

[28]

W. X. Shen and A. J. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.

[29]

W. Y. ZhangZ. H. Liu and L. Zhou, Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment, Discrete Contin. Dyn. Syst. Ser. B., 26 (2021), 3767-3784.  doi: 10.3934/dcdsb.2020256.

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