January  2020, 19(1): 253-277. doi: 10.3934/cpaa.2020014

Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system

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

Received  November 2018 Revised  April 2019 Published  July 2019

The existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for the time periodic Lotka-Volterra competition-diffusion system have already been studied in $ \Bbb{R}^{N} $ with $ N\geq 3 $. In this paper, we continue to study the uniqueness and asymptotic stability of such time-periodic pyramidal traveling front in the three-dimensional whole space. For any given admissible pyramid, we show that the time periodic pyramidal traveling front is uniquely determined and it is asymptotically stable under the condition that given perturbations decay at infinity. Moreover, the time periodic pyramidal traveling front is uniquely determined as a combination of two-dimensional periodic V-form waves on the edges of the pyramid.

Citation: Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure and Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014
References:
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X. Bao, Time periodic traveling fronts of pyramidal shapes for periodic Lotka-Volterra competition-diffusion system, Nonlinear Anal. Real World Appl., 35 (2017), 292–311. doi: 10.1016/j.nonrwa.2018.04.009.

[2]

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.

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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.

[4]

X. Bao, W. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 18 (2019), 361–396. doi: 10.3934/cpaa.2019019.

[5]

Z. H. Bu and Z. C. Wang, Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media, Commun. Pure Appl. Anal., 15 (2016), 139–160. doi: 10.3934/cpaa.2016.15.139.

[6]

Z. H. Bu and Z. C. Wang, Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations I, Discrete Contin. Dyn. Syst., 37 (2017), 2395–2430. doi: 10.3934/dcds.2017104.

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A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80–118. doi: 10.1137/S0036141097316391.

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P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (2000), 371–391. doi: 10.1137/S0036139997325497.

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G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237–279. doi: 10.1016/j.jde.2007.01.021.

[10]

X. Chen, J. S. Chen, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable nonlinearity, Ann. Inst. H. Poincare Annal. Lineaire, 24 (2007), 369–393. doi: 10.1016/j.anihpc.2006.03.012.

[11]

M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469–6486. doi: 10.1016/j.na.2011.06.030.

[12]

J. Fang and X. Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. European Math. Soc., 17 (2015), 2243–2288. doi: 10.4171/JEMS/556.

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S. A. Gardner, Existence and stability of traveling wave solutions of a competition model. A degree theoretical approach, J. Differential Equations, 44 (1982), 343–364. doi: 10.1016/0022-0396(82)90001-8.

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C. Gui, Symmetry of traveling wave solutions to the Allen-Cahn equation in ${{\mathbb{R}}^{2}}$, Arch. Rational Mech. Anal., 203 (2012), 1037–1065. doi: 10.1016/j.jde.2012.03.004.

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F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in ${{\mathbb{R}}^{^{N}}}$ with conical-shaped level sets, Comm. Partial Differential Equations, 25 (2000), 769–819. doi: 10.1080/03605300008821532.

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F. Hamel, R. Monneau and J. M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469–506. doi: 10.1016/j.ansens.2004.03.001.

[17]

F. Hamel, R. Monneau and J. M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069–1096. doi: 10.3934/dcds.2005.13.1069.

[18]

F. Hamel, R. Monneau and J. M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75–92. doi: 10.3934/dcds.2006.14.75.

[19]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ${{\mathbb{R}}^{^{N}}}$, Arch. Ration. Mech. Anal., 157 (2001), 91–163. doi: 10.1007/PL00004238.

[20]

F. Hamel and J. M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 101–123. doi: 10.3934/dcdss.2011.4.101.

[21]

P. Hess, Periodic-parabolic boundary value problems and positively, Pitmal Research Notes in Mathematic Series, Vol 247, Longman Scientific and Technical, Wiley, Harlow, Essex, 1991.

[22]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan. J. Indust. Appl. Math., 13 (1996), 343–349. doi: 10.1007/BF03167252.

[23]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031–1054. doi: 10.1017/S0308210510001253.

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

[25]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalyttc chemiacal reaction systems, SIAM J. Math. Anal., 44 (2012), 1474–1521. doi: 10.1137/100814974.

[26]

G. Lv and M. Wang, Stability of planar waves in mono-stable reaction-diffusion equations, Proc. Amer. Math. Soc., 139 (2011), 3611–3621. doi: 10.1090/S0002-9939-2011-10767-6.

[27]

G. Lv and M. Wang, Stability of planar waves in reaction-diffusion system, Sci. China Math., 548 (2011), 1403–1419. doi: 10.1007/s11425-011-4210-0.

[28]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217–2240. doi: 10.1137/080723715.

[29]

W. M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379–395. doi: 10.3934/nhm.2013.8.379.

[30]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204–233. doi: 10.1016/j.jde.2004.06.011.

[31]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819–832. doi: 10.3934/dcds.2006.15.819.

[32]

W. J. Sheng, Time periodic traveling curved fronts of bistable reaction-diffusion equations in ${{\mathbb{R}}^{^{N}}}$, Appl. Math. Letters, 54 (2016), 22–30. doi: 10.1016/j.aml.2015.11.004.

[33]

W. J. Sheng, Time periodic traveling curved fronts of bistable reaction-diffusion equations in ${{\mathbb{R}}^{^{3}}}$, Annali di Matematica Pura ed Applicata., 196 (2017), 617–639. doi: 10.1007/s10231-016-0589-0.

[34]

W. J. Sheng, W. T. Li and Z. C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388–2424. doi: 10.1016/j.jde.2011.09.016.

[35]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319–344. doi: 10.1137/060661788.

[36]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103–2130. doi: 10.1016/j.jde.2008.06.037.

[37]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011–1046. doi: 10.3934/dcds.2012.32.1011.

[38]

M. Taniguchi, An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front in the Allen-Cahn equation, SIAM J. Math. Anal. 47 (2015) 455–476. doi: 10.1137/130945041.

[39]

M. Taniguchi, Convex compact sets in ${{\mathbb{R}}^{^{N-1}}}$ give traveling fronts of cooperative-diffusion system in ${{\mathbb{R}}^{^{N}}}$, J. Differential Equations, 260 (2016), 4301–4338. doi: 10.1016/j.jde.2015.11.010.

[40]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic System, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. Procidence, RI, 1994.

[41]

Z. C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst. 32 (2012), 2339–2374. doi: 10.3934/dcds.2012.32.2339.

[42]

Z. C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equations with bistable nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1053–1090. doi: 10.1017/S0308210515000268.

[43]

Z. C. Wang and Z. H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405–6450. doi: 10.1016/j.jde.2015.12.045.

[44]

Z. C. Wang, W. T. Li and S. Ruan, Existence, uniqueness and stability of pyramidal traveling fronts in bistable reaction-diffusion systems, Sci. China Math., 59 (2016), 1869–1908. doi: 10.1007/s11425-016-0015-x.

[45]

Z. C. Wang, H. L. Niu and S. Ruan, On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ${{\mathbb{R}}^{^{3}}}$, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1111–1144. doi: 10.3934/dcdsb.2017055.

[46]

Z. C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196–3229. doi: 10.1016/j.jde.2011.01.017.

[47]

Y. Wu and X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123–1139. doi: 10.3934/dcds.2008.20.1123.

[48]

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.

show all references

References:
[1]

X. Bao, Time periodic traveling fronts of pyramidal shapes for periodic Lotka-Volterra competition-diffusion system, Nonlinear Anal. Real World Appl., 35 (2017), 292–311. doi: 10.1016/j.nonrwa.2018.04.009.

[2]

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.

[3]

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.

[4]

X. Bao, W. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 18 (2019), 361–396. doi: 10.3934/cpaa.2019019.

[5]

Z. H. Bu and Z. C. Wang, Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media, Commun. Pure Appl. Anal., 15 (2016), 139–160. doi: 10.3934/cpaa.2016.15.139.

[6]

Z. H. Bu and Z. C. Wang, Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations I, Discrete Contin. Dyn. Syst., 37 (2017), 2395–2430. doi: 10.3934/dcds.2017104.

[7]

A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80–118. doi: 10.1137/S0036141097316391.

[8]

P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (2000), 371–391. doi: 10.1137/S0036139997325497.

[9]

G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237–279. doi: 10.1016/j.jde.2007.01.021.

[10]

X. Chen, J. S. Chen, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable nonlinearity, Ann. Inst. H. Poincare Annal. Lineaire, 24 (2007), 369–393. doi: 10.1016/j.anihpc.2006.03.012.

[11]

M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469–6486. doi: 10.1016/j.na.2011.06.030.

[12]

J. Fang and X. Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. European Math. Soc., 17 (2015), 2243–2288. doi: 10.4171/JEMS/556.

[13]

S. A. Gardner, Existence and stability of traveling wave solutions of a competition model. A degree theoretical approach, J. Differential Equations, 44 (1982), 343–364. doi: 10.1016/0022-0396(82)90001-8.

[14]

C. Gui, Symmetry of traveling wave solutions to the Allen-Cahn equation in ${{\mathbb{R}}^{2}}$, Arch. Rational Mech. Anal., 203 (2012), 1037–1065. doi: 10.1016/j.jde.2012.03.004.

[15]

F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in ${{\mathbb{R}}^{^{N}}}$ with conical-shaped level sets, Comm. Partial Differential Equations, 25 (2000), 769–819. doi: 10.1080/03605300008821532.

[16]

F. Hamel, R. Monneau and J. M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469–506. doi: 10.1016/j.ansens.2004.03.001.

[17]

F. Hamel, R. Monneau and J. M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069–1096. doi: 10.3934/dcds.2005.13.1069.

[18]

F. Hamel, R. Monneau and J. M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75–92. doi: 10.3934/dcds.2006.14.75.

[19]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in ${{\mathbb{R}}^{^{N}}}$, Arch. Ration. Mech. Anal., 157 (2001), 91–163. doi: 10.1007/PL00004238.

[20]

F. Hamel and J. M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 101–123. doi: 10.3934/dcdss.2011.4.101.

[21]

P. Hess, Periodic-parabolic boundary value problems and positively, Pitmal Research Notes in Mathematic Series, Vol 247, Longman Scientific and Technical, Wiley, Harlow, Essex, 1991.

[22]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan. J. Indust. Appl. Math., 13 (1996), 343–349. doi: 10.1007/BF03167252.

[23]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031–1054. doi: 10.1017/S0308210510001253.

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

[25]

Y. Li and Y. Wu, Stability of traveling front solutions with algebraic spatial decay for some autocatalyttc chemiacal reaction systems, SIAM J. Math. Anal., 44 (2012), 1474–1521. doi: 10.1137/100814974.

[26]

G. Lv and M. Wang, Stability of planar waves in mono-stable reaction-diffusion equations, Proc. Amer. Math. Soc., 139 (2011), 3611–3621. doi: 10.1090/S0002-9939-2011-10767-6.

[27]

G. Lv and M. Wang, Stability of planar waves in reaction-diffusion system, Sci. China Math., 548 (2011), 1403–1419. doi: 10.1007/s11425-011-4210-0.

[28]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217–2240. doi: 10.1137/080723715.

[29]

W. M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379–395. doi: 10.3934/nhm.2013.8.379.

[30]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204–233. doi: 10.1016/j.jde.2004.06.011.

[31]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dyn. Syst., 15 (2006), 819–832. doi: 10.3934/dcds.2006.15.819.

[32]

W. J. Sheng, Time periodic traveling curved fronts of bistable reaction-diffusion equations in ${{\mathbb{R}}^{^{N}}}$, Appl. Math. Letters, 54 (2016), 22–30. doi: 10.1016/j.aml.2015.11.004.

[33]

W. J. Sheng, Time periodic traveling curved fronts of bistable reaction-diffusion equations in ${{\mathbb{R}}^{^{3}}}$, Annali di Matematica Pura ed Applicata., 196 (2017), 617–639. doi: 10.1007/s10231-016-0589-0.

[34]

W. J. Sheng, W. T. Li and Z. C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388–2424. doi: 10.1016/j.jde.2011.09.016.

[35]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319–344. doi: 10.1137/060661788.

[36]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103–2130. doi: 10.1016/j.jde.2008.06.037.

[37]

M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst., 32 (2012), 1011–1046. doi: 10.3934/dcds.2012.32.1011.

[38]

M. Taniguchi, An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front in the Allen-Cahn equation, SIAM J. Math. Anal. 47 (2015) 455–476. doi: 10.1137/130945041.

[39]

M. Taniguchi, Convex compact sets in ${{\mathbb{R}}^{^{N-1}}}$ give traveling fronts of cooperative-diffusion system in ${{\mathbb{R}}^{^{N}}}$, J. Differential Equations, 260 (2016), 4301–4338. doi: 10.1016/j.jde.2015.11.010.

[40]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic System, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. Procidence, RI, 1994.

[41]

Z. C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dyn. Syst. 32 (2012), 2339–2374. doi: 10.3934/dcds.2012.32.2339.

[42]

Z. C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equations with bistable nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1053–1090. doi: 10.1017/S0308210515000268.

[43]

Z. C. Wang and Z. H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405–6450. doi: 10.1016/j.jde.2015.12.045.

[44]

Z. C. Wang, W. T. Li and S. Ruan, Existence, uniqueness and stability of pyramidal traveling fronts in bistable reaction-diffusion systems, Sci. China Math., 59 (2016), 1869–1908. doi: 10.1007/s11425-016-0015-x.

[45]

Z. C. Wang, H. L. Niu and S. Ruan, On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ${{\mathbb{R}}^{^{3}}}$, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1111–1144. doi: 10.3934/dcdsb.2017055.

[46]

Z. C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196–3229. doi: 10.1016/j.jde.2011.01.017.

[47]

Y. Wu and X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123–1139. doi: 10.3934/dcds.2008.20.1123.

[48]

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.

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