Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems

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July 2021, 26(7): 3621-3641. doi: 10.3934/dcdsb.2020249

Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems

Xiongxiong Bao ^1, and Wan-Tong Li ^2,,

1.	School of Science, Chang'an University, Xi'an, Shaanxi 710064, China
2.	School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

^* Corresponding author (wtli@lzu.edu.cn)

Received January 2020 Revised July 2020 Published July 2021 Early access August 2020

The current paper is devoted to the study of the existence and stability of generalized transition waves of the following time-dependent reaction-diffusion cooperative system

$ \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*} $

Here

$ \mathbf{F}(t,\mathbf{u}(t,x)) $

depends on

$ t\in\Bbb{R} $

in a general way. Recently, the spreading speeds and linear determinacy of the above time-dependent system have been studied by Bao et al. [J. Differential Equations 265 (2018) 3048-3091]. In this paper, using the principal Lyapunov exponent and principal Floquent bundle theory of linear cooperative systems, we prove the existence of generalized transition waves in any given direction with speed greater than the spreading speed by constructing appropriate subsolutions and supersolutions. When the initial value is uniformly bounded with respect to a weighted maximum norm, we further show that all solutions converge to the generalized transition wave solutions exponentially in time.

Keywords: Generalized transition wave, reaction-diffusion systems, time dependence, existence, uniqueness, stability.

Mathematics Subject Classification: 35B15, 35K55, 35K57, 92D25.

Citation: Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249

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.
[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).
[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).
[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.
[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).
[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).
[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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