Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 02, pp. 1-19.

Title: Traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with nonlocal nonlinearities

Authors: Zhi-Jiao Yang (Northwest Normal Univ., Lanzhou, Gansu, China)
         Guo-Bao Zhang (Northwest Normal Univ., Lanzhou, Gansu, China)
         Juan He       (Northwest Normal Univ., Lanzhou, Gansu, China)
  
Abstract:
This article concerns the traveling wavefronts of a discrete diffusive 
Lotka-Volterra competition system with nonlocal nonlinearities. 
We first prove that there exists a $c_*>0$ such that when the wave speed 
is large than or equals to $c_*$, the system admits an increasing 
traveling wavefront connecting two boundary equilibria by the upper-lower 
solutions method. Furthermore, we prove that 
(i)  all traveling wavefronts with speed $c>c^{*}(>c_*)$ are globally 
stable with exponential convergence rate 
$t^{-1/2}e^{-\varepsilon_{\tau}\sigma t}$,
where $\sigma>0$ and $\varepsilon_{\tau}=\varepsilon(\tau)\in (0,1)$ 
is a decreasing function for the time delay $\tau>0$; 
(ii) the traveling wavefronts with speed $c=c^{*}$ are globally 
algebraically stable in the algebraic form $t^{-1/2}$.
The approaches are the weighted energy method, the comparison 
principle and Fourier transform.


Submitted July 23, 2024. Published January 04, 2025.
Math Subject Classifications: 35K55, 35C07, 92D25.
Key Words: Epidemic system; nonlocal dispersal; bistable traveling waves; stability; time delay.