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July  2019, 18(4): 2069-2092. doi: 10.3934/cpaa.2019093

Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

Zhaohai Ma 1, , Rong Yuan 1, , Yang Wang 2, and  Xin Wu 3,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2. 

School of Mathematical Sciences, Shanxi University, Shanxi 030006, China
3. 

School of Sciences, East China Jiao Tong University, Nanchang 330013, China

Received  April 2018 Revised  August 2018 Published  January 2019

In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in $ n $–dimensional space. More precisely, we prove that all planar traveling waves with speed $ c>c^* $ are exponentially stable in $ L^{\infty}(\mathbb{R}^n ) $ in the form of $ t^{-\frac{n}{2\alpha }}{\rm{e}}^{-\varepsilon_{\tau} \sigma t} $ for some constants $ \sigma >0 $ and $ \varepsilon_{\tau} \in (0,1) $, where $ \varepsilon_{\tau} = \varepsilon(\tau) $ is a decreasing function refer to the time delay $ \tau>0 $. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed $ c = c^* $, we show that they are algebraically stable in the form of $ t^{-\frac{n}{2\alpha}} $. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.

Keywords: Multidimensional stability, competitive system, planar traveling waves, weighted energy method, Fourier transform.
Mathematics Subject Classification: 35C07, 92D25, 35B35.
Citation: Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Math., 30 (1978), 33-76. Google Scholar

[2]

P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl., 273 (2002), 45-57. doi: 10.1016/S0022-247X(02)00205-6. Google Scholar

[3]

H. Cheng and R. Yuan, Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1015-1029. doi: 10.3934/dcdsb.2015.20.1015. Google Scholar

[4]

I.-L. Chern, M. Mei, X. Yang and Q. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations, 259 (2015), 1503-1541. doi: 10.1016/j.jde.2015.03.003. Google Scholar

[5]

J. Fang and X. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.1137/140953939. Google Scholar

[6]

G. Faye, Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation, Discrete Contin. Dyn. Syst., 36 (2016), 2473-2496. doi: 10.3934/dcds.2016.36.2473. Google Scholar

[7]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. Google Scholar

[8]

T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764. Google Scholar

[9]

A. Huang and P. Weng, Traveling wavefronts for a Lotka-Volterra system of type-K with delays, Nonlinear Anal. Real World Appl., 14 (2013), 1114-1129. doi: 10.1016/j.nonrwa.2012.09.002. Google Scholar

[10]

R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621. Google Scholar

[11]

R. Huang, M. Mei, K. Zhang and Q. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353. doi: 10.3934/dcds.2016.36.1331. Google Scholar

[12]

L. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Anal. Real World Appl., 12 (2011), 3691-3700. doi: 10.1016/j.nonrwa.2011.07.002. Google Scholar

[13]

L. C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251. doi: 10.1007/s13160-012-0056-2. Google Scholar

[14]

T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1. Google Scholar

[15]

D. Khusainov, A. Ivanov and I. Kovarzh, Solution of one heat equation with delay, Nonlinear Oscil., 12 (2009), 260-282. doi: 10.1007/s11072-009-0075-3. Google Scholar

[16]

A. N. Kolmogorov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Moscow University Bulletin of Mathematics, 1 (1937), 1-25. doi: 10.2307/1968507. Google Scholar

[17]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, Ⅱ, Comm. Partial Differential Equations, 17 (1992), 1901-1924. doi: 10.1080/03605309208820908. Google Scholar

[18]

K. Li, J. Huang and X. Li, Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system, Commun. Pure Appl. Anal., 16 (2017), 131-150. doi: 10.3934/cpaa.2017006. Google Scholar

[19]

C. Lin, C. Lin, Y. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391. Google Scholar

[20]

Z. Ma, X. Wu and R. Yuan, Nonlinear stability of traveling wavefronts for competitive-cooperative Lotka-Volterra systems of three species, Appl. Math. Comput., 315 (2017), 331-346. doi: 10.1016/j.amc.2017.07.068. Google Scholar

[21]

Z. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, Int. J. Biomath., 10 (2017), 1750071. doi: 10.1142/S1793524517500711. Google Scholar

[22]

R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[23]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557. doi: 10.1016/j.jde.2011.08.029. Google Scholar

[24]

H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002. doi: 10.1080/03605300902963500. Google Scholar

[25]

M. Mei, J. So, M. Li and S. Shen, Asymptotic stability of travelling waves for nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. Google Scholar

[26]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Ser. B, 4 (2011), 379-401. Google Scholar

[27]

M. Mei, K. Zhang and Q. Zhang, Global stability of critical traveling waves with oscillations for time-delayed reaction-diffusion equation, Int. J. Numer. Anal. Model., 15 (2019), in press. doi: 10.1080/00036811.2016.1258696. Google Scholar

[28]

J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar

[29]

K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. Google Scholar

[30]

W. Sheng, Multidimensional stability of V-shaped traveling fronts in time periodic bistable reaction-diffusion equations, Comput. Math. Appl., 72 (2016), 1714-1726. doi: 10.1016/j.camwa.2016.07.035. Google Scholar

[31]

W. Sheng, W. Li and Z. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5. Google Scholar

[32]

H. Smith and X. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. Google Scholar

[33]

M. Vidyasagar, Nonlinear Systems Analysis, Society for Industrial & Applied Mathematics (SIAM), Philadelphia, 2002. doi: 10.1137/1.9780898719185. Google Scholar

[34]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, 1994. Google Scholar

[35]

V. A. Volpert and A. I. Volpert, Existence and stability of multidimensional travelling waves in the monostable case, Israel J. Math., 110 (1999), 269-292. doi: 10.1007/BF02808184. Google Scholar

[36]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I, Comm. Partial Differential Equations, 17 (1992), 1889-1899. doi: 10.1080/03605309208820907. Google Scholar

[37]

Z. Yu, F. Xu and W. Zhang, Stability of invasion traveling waves for a competition system with nonlocal dispersals, Appl. Anal., 96 (2017), 1107-1125. doi: 10.1080/00036811.2016.1178242. Google Scholar

[38]

Z. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 260 (2016), 241-267. doi: 10.1016/j.jde.2015.08.037. Google Scholar

[39]

Z. Yu and R. Yuan, Travelling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. Google Scholar

[40]

Z. Yu and R. Yuan, Traveling waves of delayed reaction-diffusion systems with applications, Nonlinear Anal. Real World Appl., 12 (2011), 2475-2488. doi: 10.1016/j.nonrwa.2011.02.005. Google Scholar

[41]

Z. Yu and X. Zhao, Propagation phenomena for CNNs with asymmetric templates and distributed delays, Discrete Contin. Dyn. Syst., 38 (2018), 905-939. doi: 10.3934/dcds.2018039. Google Scholar

[42]

H. Zeng, Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations, Sci. China Math., 57 (2014), 353-366. doi: 10.1007/s11425-013-4617-x. Google Scholar

[43]

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., 96 (2011), 627-671. doi: 10.1016/j.matpur.2010.11.005. Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Math., 30 (1978), 33-76. Google Scholar

[2]

P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl., 273 (2002), 45-57. doi: 10.1016/S0022-247X(02)00205-6. Google Scholar

[3]

H. Cheng and R. Yuan, Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1015-1029. doi: 10.3934/dcdsb.2015.20.1015. Google Scholar

[4]

I.-L. Chern, M. Mei, X. Yang and Q. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations, 259 (2015), 1503-1541. doi: 10.1016/j.jde.2015.03.003. Google Scholar

[5]

J. Fang and X. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.1137/140953939. Google Scholar

[6]

G. Faye, Multidimensional stability of planar traveling waves for the scalar nonlocal Allen-Cahn equation, Discrete Contin. Dyn. Syst., 36 (2016), 2473-2496. doi: 10.3934/dcds.2016.36.2473. Google Scholar

[7]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369. Google Scholar

[8]

T. Gallay, Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764. Google Scholar

[9]

A. Huang and P. Weng, Traveling wavefronts for a Lotka-Volterra system of type-K with delays, Nonlinear Anal. Real World Appl., 14 (2013), 1114-1129. doi: 10.1016/j.nonrwa.2012.09.002. Google Scholar

[10]

R. Huang, M. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621. Google Scholar

[11]

R. Huang, M. Mei, K. Zhang and Q. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations, Discrete Contin. Dyn. Syst., 36 (2016), 1331-1353. doi: 10.3934/dcds.2016.36.1331. Google Scholar

[12]

L. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Anal. Real World Appl., 12 (2011), 3691-3700. doi: 10.1016/j.nonrwa.2011.07.002. Google Scholar

[13]

L. C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251. doi: 10.1007/s13160-012-0056-2. Google Scholar

[14]

T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1. Google Scholar

[15]

D. Khusainov, A. Ivanov and I. Kovarzh, Solution of one heat equation with delay, Nonlinear Oscil., 12 (2009), 260-282. doi: 10.1007/s11072-009-0075-3. Google Scholar

[16]

A. N. Kolmogorov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Moscow University Bulletin of Mathematics, 1 (1937), 1-25. doi: 10.2307/1968507. Google Scholar

[17]

C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, Ⅱ, Comm. Partial Differential Equations, 17 (1992), 1901-1924. doi: 10.1080/03605309208820908. Google Scholar

[18]

K. Li, J. Huang and X. Li, Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system, Commun. Pure Appl. Anal., 16 (2017), 131-150. doi: 10.3934/cpaa.2017006. Google Scholar

[19]

C. Lin, C. Lin, Y. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391. Google Scholar

[20]

Z. Ma, X. Wu and R. Yuan, Nonlinear stability of traveling wavefronts for competitive-cooperative Lotka-Volterra systems of three species, Appl. Math. Comput., 315 (2017), 331-346. doi: 10.1016/j.amc.2017.07.068. Google Scholar

[21]

Z. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, Int. J. Biomath., 10 (2017), 1750071. doi: 10.1142/S1793524517500711. Google Scholar

[22]

R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[23]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557. doi: 10.1016/j.jde.2011.08.029. Google Scholar

[24]

H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002. doi: 10.1080/03605300902963500. Google Scholar

[25]

M. Mei, J. So, M. Li and S. Shen, Asymptotic stability of travelling waves for nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358. Google Scholar

[26]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Ser. B, 4 (2011), 379-401. Google Scholar

[27]

M. Mei, K. Zhang and Q. Zhang, Global stability of critical traveling waves with oscillations for time-delayed reaction-diffusion equation, Int. J. Numer. Anal. Model., 15 (2019), in press. doi: 10.1080/00036811.2016.1258696. Google Scholar

[28]

J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar

[29]

K. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. Google Scholar

[30]

W. Sheng, Multidimensional stability of V-shaped traveling fronts in time periodic bistable reaction-diffusion equations, Comput. Math. Appl., 72 (2016), 1714-1726. doi: 10.1016/j.camwa.2016.07.035. Google Scholar

[31]

W. Sheng, W. Li and Z. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5. Google Scholar

[32]

H. Smith and X. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. Google Scholar

[33]

M. Vidyasagar, Nonlinear Systems Analysis, Society for Industrial & Applied Mathematics (SIAM), Philadelphia, 2002. doi: 10.1137/1.9780898719185. Google Scholar

[34]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, 1994. Google Scholar

[35]

V. A. Volpert and A. I. Volpert, Existence and stability of multidimensional travelling waves in the monostable case, Israel J. Math., 110 (1999), 269-292. doi: 10.1007/BF02808184. Google Scholar

[36]

J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I, Comm. Partial Differential Equations, 17 (1992), 1889-1899. doi: 10.1080/03605309208820907. Google Scholar

[37]

Z. Yu, F. Xu and W. Zhang, Stability of invasion traveling waves for a competition system with nonlocal dispersals, Appl. Anal., 96 (2017), 1107-1125. doi: 10.1080/00036811.2016.1178242. Google Scholar

[38]

Z. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 260 (2016), 241-267. doi: 10.1016/j.jde.2015.08.037. Google Scholar

[39]

Z. Yu and R. Yuan, Travelling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. Google Scholar

[40]

Z. Yu and R. Yuan, Traveling waves of delayed reaction-diffusion systems with applications, Nonlinear Anal. Real World Appl., 12 (2011), 2475-2488. doi: 10.1016/j.nonrwa.2011.02.005. Google Scholar

[41]

Z. Yu and X. Zhao, Propagation phenomena for CNNs with asymmetric templates and distributed delays, Discrete Contin. Dyn. Syst., 38 (2018), 905-939. doi: 10.3934/dcds.2018039. Google Scholar

[42]

H. Zeng, Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations, Sci. China Math., 57 (2014), 353-366. doi: 10.1007/s11425-013-4617-x. Google Scholar

[43]

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., 96 (2011), 627-671. doi: 10.1016/j.matpur.2010.11.005. Google Scholar

Figure 1.  Exact planar traveling wave $ (\phi_1, \phi_2) $ of the system (5.1) with $ b_1 = \frac{1}{2}, b_2 = \frac{19}{2}, r_1 = r_2 = 16, d_1 = 1, d_2 = 7 $
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Figure 2.  The left picture denotes the solution $ u_1 $ of system (1.4) with Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution $ u_1(t, x) $ plots at times $ t = 0, 1, 2, 10, 30, 50 $ and behaves as a stable monotone increasing traveling wave (no change of the waves's shape after a large time in the sense of stability) and travels from right to left.
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Figure 3.  The left picture denotes the solution $ u_2 $ of system (1.4) with Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution $ u_2(t, x) $ plots at times $ t = 0, 1, 2, 10, 30, 50 $ and behaves as a stable monotone increasing traveling wave (no change of the waves's shape after a large time in the sense of stability) and travels from right to left.
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