This paper is concerned with the persistence of wave fronts of the following non-local delay equation \(\frac{\partial u(x,t)}{\partial t}=D\Delta u(x,t)+F\left( u(x,t),\int_{-\tau }^0\int_\Omega du_\tau (\theta ,y)g(u(x+y,t+\theta ))\right) ,\) where \(x\in \mathbb{R}^m,\) \(t\geq 0,\) \( u(x,t)\in \mathbb{R}^n,\) \(D=\text{diag}(d_1,\cdots ,d_n)\) with positive constants \( d_i>0,\) \(\tau \) is a positive constant, \(u_\tau \) is a bounded variation function on \(\left[ -\tau ,0\right] \times \Omega \subseteq \left[ -\tau ,0\right] \times \mathbb{R}^m\) with values in \( \mathbb{R}^{n\times n}\) and normalized so that \(\int_{-\tau }^0\int_\Omega du_\tau (\theta ,y)=1,\) and this measure may be dependent on \(\tau ,\) \(F: \mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}^n\), and \(g:\mathbb{R}^n\rightarrow \mathbb{R}^n\) are \(C^2\)-smooth functions.
By the perturbation argument and Fredholm alternative theory, they proved the persistence of traveling wavefronts for the above reaction-diffusion equations with nonlocal and delayed nonlinearities if the time lag is relatively small. Namely, when \( \tau \) is sufficiently small and the reduced version of an ordinary reaction-diffusion system \(\frac{\partial u(x,t)}{\partial t}=D\Delta u(x,t)+F\left( u(x,t),u(x,t)\right) \) has a travelling wave front, then the above non-local delay equations admit such a solution too, which is true in both monostable and bistable cases and has no requirement on the monotonicity of \(F\) and \(g.\) To illustrate the abstract theory, these results are applied to five famous reaction-diffusion models with non-local delay, including a single species model with age structure, the Fisher model, the spatial spread of rabies by red foxes, a bio-reactor model and a hyperbolic model arising from the slow movement of individuals.