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

Bo Duan; Zhengce Zhang

doi:10.3934/dcdsb.2021067

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doi: 10.3934/dcdsb.2021067

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

Bo Duan and Zhengce Zhang ^,

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

^* Corresponding author: Zhengce Zhang

Received October 2020 Revised January 2021 Published 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.

Keywords: Reaction-diffusion-advection competition system, free boundary, heterogeneous environment, spreading-vanishing dichotomy, spreading speed.

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

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. Bunting, Y. 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
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[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. Du, Z. 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. Du, M. 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. Gu, Z. 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. Gu, Z. 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. Gu, B. 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. Hilhorst, M. Iida, M. 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. Hilhorst, M. 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. Kim, Z. 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. Kolmogorov, I. 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. Zhou, S. 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. Bunting, Y. 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. Chen, K.-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. Chen, F. 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. Du, Z. 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. Du, M. 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. Gu, Z. 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. Gu, Z. 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. Gu, B. 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. Hilhorst, M. Iida, M. 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. Hilhorst, M. 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. Kim, Z. 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. Kolmogorov, I. 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. Zhou, S. 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

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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