Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

• Yeqing Xue ^[1] ; Zhaohai Ma ^[2] ; Zhihua Liu ^[3]
  1. [1] Taiyuan University of Technology

     Taiyuan University of Technology

     China

     
  2. [2] China University of Geosciences

     China University of Geosciences

     China

     
  3. [3] Beijing Normal University

     Beijing Normal University

     China

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
• Idioma: inglés
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• Resumen

  • In this paper, we consider the multidimensional stability of planar traveling waves for a class of Lotka–Volterra competition systems with time delay and nonlocal reaction term in n–dimensional space. It is proved that, all planar traveling waves with speed c > c∗ are exponentially stable. We get accurate decay rate t − n 2 e−τ σt , where constant σ > 0 and τ = (τ ) ∈ (0, 1) is a decreasing function for the time delay τ > 0. It is indicated that time delay essentially reduces the decay rate. While, for the planar traveling waves with speed c = c∗, we prove that they are algebraically stable with delay rate t − n 2 . The proof is carried out by applying the comparison principle, weighted energy and Fourier transform, which plays a crucial role in transforming the competition system to a linear delayed differential system.

• Referencias bibliográficas
  • 1. Bates, P.W., Chen, F.: Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen–Cahn equation. J. Math. Anal....
  • 2. Chen, X.: Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160...
  • 3. Chen, X., Guo, J.-S.: Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J. Differ. Equ....
  • 4. Chern, I.-L., Mei, M., Yang, X., Zhang, Q.: Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay....
  • 5. Cheng, H., Yuan, R.: Multidimensional stability of disturbed pyramidal traveling fronts in the AllenCahn equation. Discrete Contin. Dyn....
  • 6. Ei, S.I., Ishii, H., Kondo, S., Miura, T., Tanaka, Y.: Effective nonlocal kernels on reaction-diffusion networks. J. Theoret. Biol. 509,...
  • 7. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 355–369 (1937)
  • 8. Fang, J., Zhao, X.: Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. 46, 3678–3704 (2014)
  • 9. Faye, G.: Multidimensional stability of planar traveling waves for the scalar nonlocal Allen–Cahn equation. Discrete Contin. Dyn. Syst....
  • 10. Gourley, S., Ruan, S.: Convergence and travelling fronts in functional differential equations with nonlocal terms: a competition model....
  • 11. Han, B., Wang, Z., Feng, Z.: Traveling waves for the nonlocal diffusive single species model with Allee effect. J. Math. Anal. Appl. 443,...
  • 12. Hamster, C.H.S., Hupkes, H.J.: Travelling waves for reaction–diffusion equations forced by translation invariant noise. Phys. D 401, 132233...
  • 13. Huang, J., Zou, X.: Travelling wave fronts in diffusive and cooperative Lotka–Volterra system with delays. J. Math. Anal. Appl. 271, 455–466...
  • 14. Huang, R., Mei, M., Wang, Y.: Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete Contin. Dyn....
  • 15. Kapitula, T.: Multidimensional stability of planar travelling waves. Trans. Am. Math. Soc. 349, 257–269 (1997)
  • 16. Khusainov, D., Ivanov, A., Kovarzh, I.: Solution of one heat equation with delay. Nonlinear Oscil. 12, 260–282 (2009)
  • 17. Kolmogorov, A.N.: A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem....
  • 18. Levermore, C.D., Xin, J.X.: Multidimensional stability of traveling waves in a bistable reactiondiffusion equation, II. Commun. Partial...
  • 19. Li, D., He, X., Li, X., Guo, J.-S.: Traveling wavefronts in a two-species chemotaxis model with Lotka–Volterra competitive kinetics. Appl....
  • 20. Lv, G., Wang, M.: Traveling wave front in diffusive and competitive Lotka–Volterra system with delays. Nonlinear Anal. Real World Appl....
  • 21. Morita, Y., Tachibana, K.: An entire solution to the Lotka–Volterra competition-diffusion equations. SIAM J. Math. Anal 40, 2217–2240...
  • 22. Ma, Z., Yuan, R.: Nonlinear stability of traveling wavefronts for delayed reaction–diffusion equation with nonlocal diffusion. Taiwanese...
  • 23. Ma, Z., Yuan, R., Wang, Y., Wu, X.: Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive...
  • 24. Ma, S., Zou, X.: Existence, uniqueness and stability of traveling waves in a discrete reaction diffusion monostable equation with delay....
  • 25. Murray, J.D.: Mathematical Biology. Springer, New York (1989)
  • 26. Martin, R., Smith, H.: Abstract functional differential equations and reaction–diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)
  • 27. Matano, H., Nara, M.: Large time behavior of disturbed planar fronts in the Allen–Cahn equation. J. Differ. Equ. 251, 3522–3557 (2011)
  • 28. Matano, H., Nara, M., Taniguchi, M.: Stability of planar waves in the Allen–Cahn equation. Commun. Partial Differ. Equ. 34, 976–1002 (2009)
  • 29. Mei, M., Ou, C., Zhao, X.: Global stability of monostable traveling waves for nonlocal time-delayed reaction–diffusion equations. SIAM...
  • 30. Mei, M., So, J., Li, M., Shen, S.: Asymptotic stability of travelling waves for Nicholson’s blowflies equation with diffusion. Proc. R....
  • 31. Mei, M., Wang, Y.: Remark on stability of traveling waves for nonlocal Fisher-KPP equations. Int. J. Numer. Anal. Model. Ser. B 4, 379–401...
  • 32. Schaaf, K.: Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Am. Math. Soc. 302,...
  • 33. Sheng, W.: Multidimensional stability of V-shaped traveling fronts in time periodic bistable reaction– diffusion equations. Comput. Math....
  • 34. Smith, H., Zhao, X.: Global asymptotic stability of traveling waves in delayed reaction–diffusion equations. SIAM J. Math. Anal. 31, 514–534...
  • 35. Tian, X., Guo, S.: Traveling wave solutions for nonlocal dispersal Fisher-KPP model with age structure. Appl. Math. Lett. 123, 107593...
  • 36. Volpert, V.A., Volpert, A.I.: Existence and stabilty of multidimensional travelling waves in the monostable case. Isr. J. Math. 110, 269–292...
  • 37. Wu, W., Teng, Z.: Traveling wave solutions in a nonlocal dispersal SIR epidemic model with general nonlinear incidence. Acta Appl. Math....
  • 38. Wu, X., Ma, Z., Yuan, R.: Nonlinear stability of traveling waves in a monostable epidemic model with delay. Taiwanese J. Math. 21, 1381–1411...
  • 39. Xin, J.X.: Multidimensional stability of traveling waves in a bistable reaction–diffusion equation, I. Commun. Partial Differ. Equ. 17,...
  • 40. Yu, Z., Xu, F., Zhang, W.: Stability of invasion traveling waves for a competition system with nonlocal dispersals. Appl. Anal. 96, 1107–1125...
  • 41. Yu, Z., Yuan, R.: Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response....
  • 42. Yu, Z., Yuan, R.: Travelling wave solutions in non-local convolution diffusive competitive-cooperative systems. IMA J. Appl. Math 76,...
  • 43. Yu, Z., Zhao, H.: Traveling waves for competitive Lotka–Volterra systems with spatial diffusions and spatio-temporal delays. Appl. Math....
  • 44. Zeng, H.: Multidimensional stability of traveling fronts in monostable reaction–diffusion equations with complex perturbations. Sci. China...
  • 45. Zhao, G., Ruan, S.: Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition...