American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020256

Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment

Weiyi Zhang , Zuhan Liu and  Ling Zhou ,

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

* Corresponding author: Ling Zhou

Received  January 2020 Revised  July 2020 Published  August 2020

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 [17], where local diffusion is used to describe the population dispersal. We give the existence and uniqueness of global solution and consider the properties of principle eigenvalue of time-periodic parabolic-type eigenvalue problem. With the help of attractivity of time periodic solutions, we establish a spreading-vanishing dichotomy. The sharp criteria for spreading and vanishing are also obtained.

Keywords: Nonlocal diffusion, time periodic, free boundaries, spreading and vanishing.
Mathematics Subject Classification: 35K51, 35R35, 92B05, 35B40.
Citation: Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020256
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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