Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: 35C07, 92D25, 35B35.

 Citation:

• Figure 1.  Exact planar traveling wave $(\phi_1, \phi_2)$ of the system (5.1) with $b_1 = \frac{1}{2}, b_2 = \frac{19}{2}, r_1 = r_2 = 16, d_1 = 1, d_2 = 7$

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.

•  [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.

Figures(3)