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A comparative study of atomistic-based stress evaluation
Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China |
In this paper we study a nonlocal diffusion model with double free boundaries in time periodic environment, which is the natural extension of the free boundary model in [
References:
[1] |
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. |
[2] |
C. Cortázar, F. Quirós and N. Wolanski,
A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound., 21 (2019), 441-462.
doi: 10.4171/IFB/430. |
[3] |
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. |
[4] |
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. |
[5] |
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.
|
[6] |
C. Cortázar, J. Coville, M. Elgueta and S. Martínez,
A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241 (2007), 332-358.
doi: 10.1016/j.jde.2007.06.002. |
[7] |
C. Cosner, J. Dávila and S. Martínez,
Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.
doi: 10.1080/17513758.2011.588341. |
[8] |
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. |
[9] |
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. |
[10] |
J. Coville, J. Dávila and S. Martínez,
Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.
doi: 10.1137/060676854. |
[11] |
X. F. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[12] |
Q. L. Cao, F. Q. Li and F. Wang,
A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment, IMA J. Appl. Math., 82 (2017), 445-470.
doi: 10.1093/imamat/hxw059. |
[13] |
W. W. Ding, Y. H. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions, J. Differential Equations, 262 (2017), 4988-5021.
doi: 10.1016/j.jde.2017.01.016. |
[14] |
W. W. Ding, Y. H. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.
doi: 10.1016/j.anihpc.2019.01.005. |
[15] |
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. |
[16] |
Y. H. Du and Z. M. Guo,
Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[17] |
Y. H. Du, Z. M. Guo and R. Peng,
A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[18] |
Y. H. Du and X. Liang,
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[19] |
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. |
[20] |
Y. H. Du, M. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. Google Scholar |
[21] |
J. P. Gao, S. J. Guo and W. X. Shen, Persistence and time periodic positive solutions of doubly nonlocal Fisher-Kpp equations in time periodic and space heterogeneous media, preprint, arXiv:1808.07162v1. Google Scholar |
[22] |
V. Hutson and M. Grinfeld,
Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.
doi: 10.1017/S0956792506006462. |
[23] |
V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers,
The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
doi: 10.1007/s00285-003-0210-1. |
[24] |
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. |
[25] |
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. |
[26] |
Y. Kaneko and Y. Yanmada,
A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.
|
[27] |
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(1), (2014), 485-497.
doi: 10.1016/j.jmaa.2013.10.071. |
[28] |
L. Li, W. J. Sheng and M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483(2) (2020), 123646.
doi: 10.1016/j.jmaa.2019.123646. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
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. |
[33] |
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. |
[34] |
Z. W. Shen and H. Vo,
Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics, J. Differential Equations, 267 (2019), 1423-1466.
doi: 10.1016/j.jde.2019.02.013. |
[35] |
M. X. 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. |
[36] |
M. X. Wang,
The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.
doi: 10.1016/j.jde.2014.10.022. |
[37] |
M. X. Wang,
A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.
doi: 10.1016/j.jfa.2015.10.014. |
[38] |
M. X. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67(5) (2016), 132, 24 pp.
doi: 10.1007/s00033-016-0729-9. |
[39] |
L. Zhou, S. Zhang and Z. H. Liu,
A free boundary problem of a predator-prey model with advection in heterogeneous environment, Appl. Math. Comput., 289 (2016), 22-36.
doi: 10.1016/j.amc.2016.05.008. |
[40] |
L. Zhou, S. Zhang and Z. H. Liu,
A reaction-diffusion-advection equation with a free boundary and sign-changing coefficient, Acta Appl. Math., 143 (2016), 189-216.
doi: 10.1007/s10440-015-0035-0. |
[41] |
L. Zhou, S. Zhang and Z. H. Liu,
Pattern formations for a strong interacting free boundary problem, Acta Appl. Math., 148 (2017), 121-142.
doi: 10.1007/s10440-016-0081-2. |
show all references
References:
[1] |
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. |
[2] |
C. Cortázar, F. Quirós and N. Wolanski,
A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound., 21 (2019), 441-462.
doi: 10.4171/IFB/430. |
[3] |
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. |
[4] |
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. |
[5] |
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.
|
[6] |
C. Cortázar, J. Coville, M. Elgueta and S. Martínez,
A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241 (2007), 332-358.
doi: 10.1016/j.jde.2007.06.002. |
[7] |
C. Cosner, J. Dávila and S. Martínez,
Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.
doi: 10.1080/17513758.2011.588341. |
[8] |
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. |
[9] |
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. |
[10] |
J. Coville, J. Dávila and S. Martínez,
Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.
doi: 10.1137/060676854. |
[11] |
X. F. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[12] |
Q. L. Cao, F. Q. Li and F. Wang,
A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment, IMA J. Appl. Math., 82 (2017), 445-470.
doi: 10.1093/imamat/hxw059. |
[13] |
W. W. Ding, Y. H. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions, J. Differential Equations, 262 (2017), 4988-5021.
doi: 10.1016/j.jde.2017.01.016. |
[14] |
W. W. Ding, Y. H. Du and X. Liang,
Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.
doi: 10.1016/j.anihpc.2019.01.005. |
[15] |
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. |
[16] |
Y. H. Du and Z. M. Guo,
Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[17] |
Y. H. Du, Z. M. Guo and R. Peng,
A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[18] |
Y. H. Du and X. Liang,
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[19] |
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. |
[20] |
Y. H. Du, M. X. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. Google Scholar |
[21] |
J. P. Gao, S. J. Guo and W. X. Shen, Persistence and time periodic positive solutions of doubly nonlocal Fisher-Kpp equations in time periodic and space heterogeneous media, preprint, arXiv:1808.07162v1. Google Scholar |
[22] |
V. Hutson and M. Grinfeld,
Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232.
doi: 10.1017/S0956792506006462. |
[23] |
V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers,
The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
doi: 10.1007/s00285-003-0210-1. |
[24] |
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. |
[25] |
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. |
[26] |
Y. Kaneko and Y. Yanmada,
A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.
|
[27] |
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(1), (2014), 485-497.
doi: 10.1016/j.jmaa.2013.10.071. |
[28] |
L. Li, W. J. Sheng and M. X. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483(2) (2020), 123646.
doi: 10.1016/j.jmaa.2019.123646. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
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. |
[33] |
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. |
[34] |
Z. W. Shen and H. Vo,
Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics, J. Differential Equations, 267 (2019), 1423-1466.
doi: 10.1016/j.jde.2019.02.013. |
[35] |
M. X. 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. |
[36] |
M. X. Wang,
The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.
doi: 10.1016/j.jde.2014.10.022. |
[37] |
M. X. Wang,
A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.
doi: 10.1016/j.jfa.2015.10.014. |
[38] |
M. X. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67(5) (2016), 132, 24 pp.
doi: 10.1007/s00033-016-0729-9. |
[39] |
L. Zhou, S. Zhang and Z. H. Liu,
A free boundary problem of a predator-prey model with advection in heterogeneous environment, Appl. Math. Comput., 289 (2016), 22-36.
doi: 10.1016/j.amc.2016.05.008. |
[40] |
L. Zhou, S. Zhang and Z. H. Liu,
A reaction-diffusion-advection equation with a free boundary and sign-changing coefficient, Acta Appl. Math., 143 (2016), 189-216.
doi: 10.1007/s10440-015-0035-0. |
[41] |
L. Zhou, S. Zhang and Z. H. Liu,
Pattern formations for a strong interacting free boundary problem, Acta Appl. Math., 148 (2017), 121-142.
doi: 10.1007/s10440-016-0081-2. |
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