On existence of semi-wavefronts for a non-local reaction-diffusion equations with distributed time delay

We establish the existence of semi-wavefronts solutions for a non-local delayed reaction-diffusion equation with monostable nonlinearity. The existence result is proved for all speeds $c\geq c_\star$, where the determination of $c_\star$ is similar to the calculation of the minimal speed of propagation. The results are applied to some non-local reaction-diffusion epidemic and population models with distributed time delay.