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

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  • 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:

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  • 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.

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