Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries

Weiyi Zhang; Ling Zhou

doi:10.3934/dcdsb.2022062

Article Contents

2022, Volume 27, Issue 12: 7745-7782. Doi: 10.3934/dcdsb.2022062

This issue Previous Article Global solvability to the 3D incompressible magneto-micropolar system with vacuum Next Article Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries

Weiyi Zhang and
Ling Zhou^,

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

^*Corresponding author: Ling Zhou
^*Corresponding author: Ling Zhou

Received: May 2021

Early access: March 2022

Published: December 2022

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35K51, 35R35, 92B05, 35B40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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. Berestycki, J. 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. Berestycki, J. 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. Cantrell, C. Cosner, Y. 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. 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.
[8]	J. F. Cao, W. T. Li, J. 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. Cao, W. 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$\acute{a}$zar, J. Coville, M. 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$\acute{a}$zar, F. 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. Cosner, J. 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. Coville, J. 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. Du, M. 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. 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.
[21]	C. Y. Kao, Y. 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. Li, Y. 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. Rawal, W. 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. Zhang, Z. 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.