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Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system
Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems
1. | School of Science, Chang'an University, Xi'an, Shaanxi 710064, China |
2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
$ \begin{equation*} \frac{{\partial} \mathbf{u}}{{\partial} t}(t,x) = \Delta \mathbf{u}(t,x)+\mathbf{F}(t,\mathbf{u}(t,x)),\quad (x,t)\in {\mathbb{R}}^{N}\times {\mathbb{R}},\, \mathbf{u}\in \Bbb{R}^{K},\, K>1. \end{equation*} $ |
$ \mathbf{F}(t,\mathbf{u}(t,x)) $ |
$ t\in\Bbb{R} $ |
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777–2805.
doi: 10.1090/S0002-9947-99-02134-0. |
[2] |
X. Bao, Transition waves for two species competition system in time heterogenous media, Nonlinear Anal. Real World Appl., 44 (2018), 128–148.
doi: 10.1016/j.nonrwa.2018.04.009. |
[3] |
X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402–2435.
doi: 10.1016/j.jde.2013.06.024. |
[4] |
X. Bao, W.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590–8637.
doi: 10.1016/j.jde.2016.02.032. |
[5] |
X. Bao, W.-T. Li, W. Shen and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), 3048–3091.
doi: 10.1016/j.jde.2018.05.003. |
[6] |
H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949–1032.
doi: 10.1002/cpa.3022. |
[7] |
H. Berestycki and F. Hamel, Generalized traveling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI. 446 (2007), 101–123, .
doi: 10.1090/conm/446/08627. |
[8] |
H. Berestycki and F. Hamel, Generalized transition wave and their properties, Comm. Pure Appl. Math. 65 (2012), 592–648.
doi: 10.1002/cpa.21389. |
[9] |
F. Cao and W. Shen, Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media, Discret. Contin. Dyn. Syst., 37 (2017), 4697–4727.
doi: 10.3934/dcds.2017202. |
[10] |
C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319–343.
doi: 10.1512/iumj.1984.33.33018. |
[11] |
J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678–3704.
doi: 10.1137/140953939. |
[12] |
J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222–4262.
doi: 10.1016/j.jfa.2017.02.028. |
[13] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 335–369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[14] |
F. Hamel, Bistable transition fronts in $\Bbb{R}^{N}$, Adv. Math., 289 (2016), 279–344.
doi: 10.1016/j.aim.2015.11.033. |
[15] |
F. Hamel and L. Rossi, Admissible speeds of transition fronts for nonautonomous monostable equation, SIAM J. Math. Anal., 47 (2015), 3342–3392.
doi: 10.1137/140995519. |
[16] |
B.-S. Han, Z.-C. Wang and Z. Du, Traveling waves for nonlocal Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1959–1983.
doi: 10.3934/dcdsb.2020011. |
[17] |
J. Huang and W. Shen, Spreeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J. Appl. Dyn. Syst., 8 (2009), 790–821.
doi: 10.1137/080723259. |
[18] |
R. A. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1–33.
doi: 10.1137/0518001. |
[19] |
A. Kolmogorov, I. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter and its application to biological problem, Bjul. Moskovskogo, Gos. Univ., 1 (1937), 1–26. Google Scholar |
[20] |
L. Kong, N. Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113–141.
doi: 10.1051/mmnp/201510609. |
[21] |
X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57–77.
doi: 10.1016/j.jde.2006.04.010. |
[22] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1–40.
doi: 10.1002/cpa.20154. |
[23] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Analysis, 259 (2010), 857–903.
doi: 10.1016/j.jfa.2010.04.018. |
[24] |
T. S. Lim and A. Zlatoŝ, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615–8631.
doi: 10.1090/tran/6602. |
[25] |
K. Mischaikow and V. Huston, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987–1008.
doi: 10.1137/0524059. |
[26] |
G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232–262.
doi: 10.1016/j.matpur.2009.04.002. |
[27] |
G. Nadin, Critical traveling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 32 (2015), 841–873.
doi: 10.1016/j.anihpc.2014.03.007. |
[28] |
G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633–653.
doi: 10.1016/j.matpur.2012.05.005. |
[29] |
G. Nadin and L. Rossi, Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coeffcients, Anal. PDE, 8 (2015), 1351–1377.
doi: 10.2140/apde.2015.8.1351. |
[30] |
G. Nadin and L. Rossi, Generalized transition fronts for one-dimensional almost periodic periodic Fisher-KPP equations, Arch. Ration. Mech. Anal., 223 (2017), 1239–1267.
doi: 10.1007/s00205-016-1056-1. |
[31] |
J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoŝ, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217–246.
doi: 10.1007/s00205-011-0449-4. |
[32] |
Z. Ouyang and C. Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 993–1007.
doi: 10.3934/dcdsb.2012.17.993. |
[33] |
S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electron. Res. Arch., 27 (2019), 89–99.
doi: 10.3934/era.2019011. |
[34] |
L. Rossi and L. Ryzhik, Transition waves for a class of space-time dependent monostable equations, Commun. Math. Sci., 12 (2014), 879–900. |
[35] |
W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011–1060.
doi: 10.1007/s10884-004-7832-x. |
[36] |
W. Shen, Traveling waves in time dependence bistable equations, Differential Integral Equations, 19 (2006), 241–278. |
[37] |
W. Shen, Spreading and generalized propagating speeds of discrete KPP models in time varying environments, Front. Math. China, 4 (2009), 523–562.
doi: 10.1007/s11464-009-0032-6. |
[38] |
W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models., Trans. Amer. Math. Soc., 362 (2010), 5125–5168.
doi: 10.1090/S0002-9947-10-04950-0. |
[39] |
W. Shen, Existence of generalized traveling wave in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69–93. |
[40] |
W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent of monostable equations, J. Dynam. Differential Equations, 23 (2011), 1–44.
doi: 10.1007/s10884-010-9200-3. |
[41] |
W. Shen and Z. Shen, Transition fronts in nonlocal Fisher-KPP equations in heterogeneous media, Commun. Pure Appl. Anal., 15 (2016), 1193–1213.
doi: 10.3934/cpaa.2016.15.1193. |
[42] |
W. Shen and Y. Yi, Almost automprphic and almost periodic dynamics in skew-product semiflows, Part Ⅱ. Skew-Product, Mech. Amer. Math. Soc., 136 (1998). Google Scholar |
[43] |
W. Shen and Y. Yi, Almost automprphic and almost periodic dynamics in skew-product semiflows, Part Ⅲ. Application to differential equations, Mech. Amer. Math. Soc. 136 (1998). Google Scholar |
[44] |
T. Tao, B. Zhu and A. Zlatoŝ, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409–2416.
doi: 10.1088/0951-7715/27/9/2409. |
[45] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994. |
[46] |
X.-S. Wang and X.-Q. Zhao, Pulsating waves of a paratially degenerate reaction-diffusion system in a periodic habitats, J. Differential Equations, 259 (2015), 7238–7259.
doi: 10.1016/j.jde.2015.08.019. |
[47] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353–396.
doi: 10.1137/0513028. |
[48] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511–548.
doi: 10.1007/s00285-002-0169-3. |
[49] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for speed in cooperative models, J. Math. Biol., 45 (2002), 183–218.
doi: 10.1007/s002850200145. |
[50] |
Y. Yang, Y. R. Yang and X. J. Jiao, Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence, Electronic Research Archive, 28 (2020), 1–13.
doi: 10.3934/era.2020001. |
[51] |
X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynam. Differential Equations, 29 (2017), 41–66.
doi: 10.1007/s10884-015-9426-1. |
[52] |
G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627–671.
doi: 10.1016/j.matpur.2010.11.005. |
[53] |
A. Zlatoŝ, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. 98 (2012), 89–102.
doi: 10.1016/j.matpur.2011.11.007. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777–2805.
doi: 10.1090/S0002-9947-99-02134-0. |
[2] |
X. Bao, Transition waves for two species competition system in time heterogenous media, Nonlinear Anal. Real World Appl., 44 (2018), 128–148.
doi: 10.1016/j.nonrwa.2018.04.009. |
[3] |
X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402–2435.
doi: 10.1016/j.jde.2013.06.024. |
[4] |
X. Bao, W.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590–8637.
doi: 10.1016/j.jde.2016.02.032. |
[5] |
X. Bao, W.-T. Li, W. Shen and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), 3048–3091.
doi: 10.1016/j.jde.2018.05.003. |
[6] |
H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949–1032.
doi: 10.1002/cpa.3022. |
[7] |
H. Berestycki and F. Hamel, Generalized traveling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI. 446 (2007), 101–123, .
doi: 10.1090/conm/446/08627. |
[8] |
H. Berestycki and F. Hamel, Generalized transition wave and their properties, Comm. Pure Appl. Math. 65 (2012), 592–648.
doi: 10.1002/cpa.21389. |
[9] |
F. Cao and W. Shen, Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media, Discret. Contin. Dyn. Syst., 37 (2017), 4697–4727.
doi: 10.3934/dcds.2017202. |
[10] |
C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319–343.
doi: 10.1512/iumj.1984.33.33018. |
[11] |
J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678–3704.
doi: 10.1137/140953939. |
[12] |
J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222–4262.
doi: 10.1016/j.jfa.2017.02.028. |
[13] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 335–369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[14] |
F. Hamel, Bistable transition fronts in $\Bbb{R}^{N}$, Adv. Math., 289 (2016), 279–344.
doi: 10.1016/j.aim.2015.11.033. |
[15] |
F. Hamel and L. Rossi, Admissible speeds of transition fronts for nonautonomous monostable equation, SIAM J. Math. Anal., 47 (2015), 3342–3392.
doi: 10.1137/140995519. |
[16] |
B.-S. Han, Z.-C. Wang and Z. Du, Traveling waves for nonlocal Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1959–1983.
doi: 10.3934/dcdsb.2020011. |
[17] |
J. Huang and W. Shen, Spreeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J. Appl. Dyn. Syst., 8 (2009), 790–821.
doi: 10.1137/080723259. |
[18] |
R. A. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1–33.
doi: 10.1137/0518001. |
[19] |
A. Kolmogorov, I. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter and its application to biological problem, Bjul. Moskovskogo, Gos. Univ., 1 (1937), 1–26. Google Scholar |
[20] |
L. Kong, N. Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113–141.
doi: 10.1051/mmnp/201510609. |
[21] |
X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57–77.
doi: 10.1016/j.jde.2006.04.010. |
[22] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1–40.
doi: 10.1002/cpa.20154. |
[23] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Analysis, 259 (2010), 857–903.
doi: 10.1016/j.jfa.2010.04.018. |
[24] |
T. S. Lim and A. Zlatoŝ, Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion, Trans. Amer. Math. Soc., 368 (2016), 8615–8631.
doi: 10.1090/tran/6602. |
[25] |
K. Mischaikow and V. Huston, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987–1008.
doi: 10.1137/0524059. |
[26] |
G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232–262.
doi: 10.1016/j.matpur.2009.04.002. |
[27] |
G. Nadin, Critical traveling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 32 (2015), 841–873.
doi: 10.1016/j.anihpc.2014.03.007. |
[28] |
G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633–653.
doi: 10.1016/j.matpur.2012.05.005. |
[29] |
G. Nadin and L. Rossi, Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coeffcients, Anal. PDE, 8 (2015), 1351–1377.
doi: 10.2140/apde.2015.8.1351. |
[30] |
G. Nadin and L. Rossi, Generalized transition fronts for one-dimensional almost periodic periodic Fisher-KPP equations, Arch. Ration. Mech. Anal., 223 (2017), 1239–1267.
doi: 10.1007/s00205-016-1056-1. |
[31] |
J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoŝ, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217–246.
doi: 10.1007/s00205-011-0449-4. |
[32] |
Z. Ouyang and C. Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 993–1007.
doi: 10.3934/dcdsb.2012.17.993. |
[33] |
S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electron. Res. Arch., 27 (2019), 89–99.
doi: 10.3934/era.2019011. |
[34] |
L. Rossi and L. Ryzhik, Transition waves for a class of space-time dependent monostable equations, Commun. Math. Sci., 12 (2014), 879–900. |
[35] |
W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011–1060.
doi: 10.1007/s10884-004-7832-x. |
[36] |
W. Shen, Traveling waves in time dependence bistable equations, Differential Integral Equations, 19 (2006), 241–278. |
[37] |
W. Shen, Spreading and generalized propagating speeds of discrete KPP models in time varying environments, Front. Math. China, 4 (2009), 523–562.
doi: 10.1007/s11464-009-0032-6. |
[38] |
W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models., Trans. Amer. Math. Soc., 362 (2010), 5125–5168.
doi: 10.1090/S0002-9947-10-04950-0. |
[39] |
W. Shen, Existence of generalized traveling wave in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69–93. |
[40] |
W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent of monostable equations, J. Dynam. Differential Equations, 23 (2011), 1–44.
doi: 10.1007/s10884-010-9200-3. |
[41] |
W. Shen and Z. Shen, Transition fronts in nonlocal Fisher-KPP equations in heterogeneous media, Commun. Pure Appl. Anal., 15 (2016), 1193–1213.
doi: 10.3934/cpaa.2016.15.1193. |
[42] |
W. Shen and Y. Yi, Almost automprphic and almost periodic dynamics in skew-product semiflows, Part Ⅱ. Skew-Product, Mech. Amer. Math. Soc., 136 (1998). Google Scholar |
[43] |
W. Shen and Y. Yi, Almost automprphic and almost periodic dynamics in skew-product semiflows, Part Ⅲ. Application to differential equations, Mech. Amer. Math. Soc. 136 (1998). Google Scholar |
[44] |
T. Tao, B. Zhu and A. Zlatoŝ, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409–2416.
doi: 10.1088/0951-7715/27/9/2409. |
[45] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994. |
[46] |
X.-S. Wang and X.-Q. Zhao, Pulsating waves of a paratially degenerate reaction-diffusion system in a periodic habitats, J. Differential Equations, 259 (2015), 7238–7259.
doi: 10.1016/j.jde.2015.08.019. |
[47] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353–396.
doi: 10.1137/0513028. |
[48] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511–548.
doi: 10.1007/s00285-002-0169-3. |
[49] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for speed in cooperative models, J. Math. Biol., 45 (2002), 183–218.
doi: 10.1007/s002850200145. |
[50] |
Y. Yang, Y. R. Yang and X. J. Jiao, Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence, Electronic Research Archive, 28 (2020), 1–13.
doi: 10.3934/era.2020001. |
[51] |
X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynam. Differential Equations, 29 (2017), 41–66.
doi: 10.1007/s10884-015-9426-1. |
[52] |
G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627–671.
doi: 10.1016/j.matpur.2010.11.005. |
[53] |
A. Zlatoŝ, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. 98 (2012), 89–102.
doi: 10.1016/j.matpur.2011.11.007. |
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