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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

 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  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.
Citation: Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] F. Hamel, Bistable transition fronts in $\Bbb{R}^{N}$, Adv. Math., 289 (2016), 279–344. doi: 10.1016/j.aim.2015.11.033.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] K. Mischaikow and V. Huston, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987–1008. doi: 10.1137/0524059.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] L. Rossi and L. Ryzhik, Transition waves for a class of space-time dependent monostable equations, Commun. Math. Sci., 12 (2014), 879–900.  Google Scholar [35] W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011–1060. doi: 10.1007/s10884-004-7832-x.  Google Scholar [36] W. Shen, Traveling waves in time dependence bistable equations, Differential Integral Equations, 19 (2006), 241–278.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] W. Shen, Existence of generalized traveling wave in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69–93.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] F. Hamel, Bistable transition fronts in $\Bbb{R}^{N}$, Adv. Math., 289 (2016), 279–344. doi: 10.1016/j.aim.2015.11.033.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] K. Mischaikow and V. Huston, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987–1008. doi: 10.1137/0524059.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] L. Rossi and L. Ryzhik, Transition waves for a class of space-time dependent monostable equations, Commun. Math. Sci., 12 (2014), 879–900.  Google Scholar [35] W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011–1060. doi: 10.1007/s10884-004-7832-x.  Google Scholar [36] W. Shen, Traveling waves in time dependence bistable equations, Differential Integral Equations, 19 (2006), 241–278.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] W. Shen, Existence of generalized traveling wave in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69–93.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [49] H. F. Weinberger, M. A. 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