• Previous Article
    Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China
  • DCDS-B Home
  • This Issue
  • Next Article
    Qualitative analysis of a diffusive SEIR epidemic model with linear external source and asymptomatic infection in heterogeneous environment
doi: 10.3934/dcdsb.2021067
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

* Corresponding author: Zhengce Zhang

Received  October 2020 Revised  January 2021 Early access March 2021

In this paper, we investigate a reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. The primary aim is to study the impact of small advection terms and heterogeneous environment, which is on two species' dynamics via a free boundary. The function $ m(x) $ represents heterogeneous environment, and it can satisfy positive everywhere condition or changeable sign condition. Firstly, on one hand, we provide long time behaviors of the solution in vanishing case when $ m(x) $ satisfies both conditions above; on the other hand, long time behaviors of the solution in spreading case are got when $ m(x) $ satisfies positive everywhere condition. Secondly, a spreading-vanishing dichotomy and several sufficient conditions through the initial data and the moving parameters are obtained to determine whether spreading or vanishing of two species happens when $ m(x) $ satisfies both conditions above. Furthermore, we derive estimates of spreading speed of the free boundary when $ m(x) $ satisfies positive everywhere condition and two species spreading occurs.

Citation: Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021067
References:
[1]

I. E. Averill, The Effect of Intermediate Advection on Two Competing Species, Ph.D thesis, The Ohio State University, 2011.  Google Scholar

[2]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[5]

Q. ChenF. 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

[6]

Y. Du and Z. 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

[7]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar

[8]

Y. DuZ. 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

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar

[12]

Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^{N}$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.  Google Scholar

[13]

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.  Google Scholar

[14]

B. Duan and Z. Zhang, A two-species weak competition system of reaction-diffusion-advection with double free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 801-829.  doi: 10.3934/dcdsb.2018208.  Google Scholar

[15]

B. Duan and Z. Zhang, A reaction-diffusion-advection free boundary problem for a two-species competition system, J. Math. Anal. Appl., 476 (2019), 595-618.  doi: 10.1016/j.jmaa.2019.03.073.  Google Scholar

[16]

H. GuZ. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53.  doi: 10.1016/j.aml.2014.05.015.  Google Scholar

[17]

H. GuZ. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117.  doi: 10.1090/S0002-9939-2014-12214-3.  Google Scholar

[18]

H. Gu and B. Lou, Spreading in advective environment modeled by a reaction diffusion equation with free boundaries, J. Differential Equations, 260 (2016), 3991-4015.  doi: 10.1016/j.jde.2015.11.002.  Google Scholar

[19]

H. GuB. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[23]

D. HilhorstM. Mimura and R. Sch$\ddot{a}$tzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285.  doi: 10.1016/S1468-1218(02)00009-3.  Google Scholar

[24]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.  Google Scholar

[25]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.   Google Scholar

[26]

K. I. KimZ. Lin and Z. Ling, Global existence and blowup of solutions to a free boundary problem for mutualistic model, Sci. China Math., 53 (2010), 2085-2095.  doi: 10.1007/s11425-010-4007-6.  Google Scholar

[27]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, $\acute{E}$tude de l'$\acute{e}$quation de la diffusion avec croissance de la quantit$\acute{e}$ de mati$\grave{e}$re et son application $\grave{a}$ un probl$\acute{e}$me biologique, Bull. Univ. Moskov. Ser. Int. Sect. A, 1 (1937), 1-25.   Google Scholar

[28]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[29]

M. Li and L. Lin, Existence of global solutions to a mutualistic model with double fronts, Electron. J. Differential Equations, (2015), 1–14.  Google Scholar

[30]

M. Li and Z. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2089-2105.  doi: 10.3934/dcdsb.2015.20.2089.  Google Scholar

[31]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[32]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[33]

N. A. Maidana and H. M. Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403-417.  doi: 10.1016/j.jtbi.2008.12.032.  Google Scholar

[34]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[35]

R. Peng and X. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[36]

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.  Google Scholar

[37]

M. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.  doi: 10.3934/dcdsb.2018179.  Google Scholar

[38]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: a diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.  doi: 10.1016/j.jmaa.2014.09.055.  Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

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.  Google Scholar

[42]

C.-H. Wu, Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455.  doi: 10.3934/dcdsb.2013.18.2441.  Google Scholar

[43]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[44]

Q. Zhang and M. Wang, Dynamics for the diffusive mutualist model with advection and different free boundaries, J. Math. Anal. Appl., 474 (2019), 1512-1535.  doi: 10.1016/j.jmaa.2019.02.037.  Google Scholar

[45]

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.  Google Scholar

[46]

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.  Google Scholar

[47]

Y. Zhao and M. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differential Equations, 30 (2018), 743-777.  doi: 10.1007/s10884-017-9571-9.  Google Scholar

[48]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.  Google Scholar

[49]

L. ZhouS. Zhang and Z. 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

show all references

References:
[1]

I. E. Averill, The Effect of Intermediate Advection on Two Competing Species, Ph.D thesis, The Ohio State University, 2011.  Google Scholar

[2]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[5]

Q. ChenF. 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

[6]

Y. Du and Z. 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

[7]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.  Google Scholar

[8]

Y. DuZ. 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

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.  Google Scholar

[12]

Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^{N}$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.  Google Scholar

[13]

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.  Google Scholar

[14]

B. Duan and Z. Zhang, A two-species weak competition system of reaction-diffusion-advection with double free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 801-829.  doi: 10.3934/dcdsb.2018208.  Google Scholar

[15]

B. Duan and Z. Zhang, A reaction-diffusion-advection free boundary problem for a two-species competition system, J. Math. Anal. Appl., 476 (2019), 595-618.  doi: 10.1016/j.jmaa.2019.03.073.  Google Scholar

[16]

H. GuZ. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53.  doi: 10.1016/j.aml.2014.05.015.  Google Scholar

[17]

H. GuZ. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117.  doi: 10.1090/S0002-9939-2014-12214-3.  Google Scholar

[18]

H. Gu and B. Lou, Spreading in advective environment modeled by a reaction diffusion equation with free boundaries, J. Differential Equations, 260 (2016), 3991-4015.  doi: 10.1016/j.jde.2015.11.002.  Google Scholar

[19]

H. GuB. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

D. HilhorstM. IidaM. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180.  doi: 10.1007/BF03168569.  Google Scholar

[23]

D. HilhorstM. Mimura and R. Sch$\ddot{a}$tzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285.  doi: 10.1016/S1468-1218(02)00009-3.  Google Scholar

[24]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.  Google Scholar

[25]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.   Google Scholar

[26]

K. I. KimZ. Lin and Z. Ling, Global existence and blowup of solutions to a free boundary problem for mutualistic model, Sci. China Math., 53 (2010), 2085-2095.  doi: 10.1007/s11425-010-4007-6.  Google Scholar

[27]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, $\acute{E}$tude de l'$\acute{e}$quation de la diffusion avec croissance de la quantit$\acute{e}$ de mati$\grave{e}$re et son application $\grave{a}$ un probl$\acute{e}$me biologique, Bull. Univ. Moskov. Ser. Int. Sect. A, 1 (1937), 1-25.   Google Scholar

[28]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[29]

M. Li and L. Lin, Existence of global solutions to a mutualistic model with double fronts, Electron. J. Differential Equations, (2015), 1–14.  Google Scholar

[30]

M. Li and Z. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2089-2105.  doi: 10.3934/dcdsb.2015.20.2089.  Google Scholar

[31]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[32]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[33]

N. A. Maidana and H. M. Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403-417.  doi: 10.1016/j.jtbi.2008.12.032.  Google Scholar

[34]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[35]

R. Peng and X. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[36]

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.  Google Scholar

[37]

M. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.  doi: 10.3934/dcdsb.2018179.  Google Scholar

[38]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: a diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.  doi: 10.1016/j.jmaa.2014.09.055.  Google Scholar

[39]

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.  Google Scholar

[40]

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.  Google Scholar

[41]

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.  Google Scholar

[42]

C.-H. Wu, Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455.  doi: 10.3934/dcdsb.2013.18.2441.  Google Scholar

[43]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.  Google Scholar

[44]

Q. Zhang and M. Wang, Dynamics for the diffusive mutualist model with advection and different free boundaries, J. Math. Anal. Appl., 474 (2019), 1512-1535.  doi: 10.1016/j.jmaa.2019.02.037.  Google Scholar

[45]

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.  Google Scholar

[46]

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.  Google Scholar

[47]

Y. Zhao and M. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differential Equations, 30 (2018), 743-777.  doi: 10.1007/s10884-017-9571-9.  Google Scholar

[48]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.  doi: 10.1016/j.jde.2013.12.008.  Google Scholar

[49]

L. ZhouS. Zhang and Z. 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

[1]

Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240

[2]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213

[3]

Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121

[4]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[5]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208

[6]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176

[7]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[8]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[9]

Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317

[10]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591

[11]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[12]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785

[13]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116

[14]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[15]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[16]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199

[17]

Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021210

[18]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733

[19]

Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221

[20]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701

2020 Impact Factor: 1.327

Article outline

[Back to Top]