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

Abstract

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.