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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A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production
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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
| 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 |
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 and Applied Analysis,
2019, 18
(4)
: 2069-2092.
doi: 10.3934/cpaa.2019093
![]()
References:
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D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Advances in Math., 30 (1978), 33-76.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
|
| [8] |
T. Gallay,
Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [28] |
J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989.
doi: 10.1007/978-3-662-08539-4. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
M. Vidyasagar, Nonlinear Systems Analysis, Society for Industrial & Applied Mathematics (SIAM), Philadelphia, 2002.
doi: 10.1137/1.9780898719185. |
| [34] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, 1994. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
|
| [8] |
T. Gallay,
Local stability of critical fronts in nonlinear parabolic partial differential equations, Nonlinearity, 7 (1994), 741-764.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [28] |
J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989.
doi: 10.1007/978-3-662-08539-4. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
M. Vidyasagar, Nonlinear Systems Analysis, Society for Industrial & Applied Mathematics (SIAM), Philadelphia, 2002.
doi: 10.1137/1.9780898719185. |
| [34] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, 1994. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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.
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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