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

We study the problem of existence of semi-wavefront solutions for a non-local delayed reaction-diffusion equation with monostable nonlinearity. In difference with previous works, we consider non-local interaction which can be asymmetric in space. As a consequence of this asymmetry, we must analyze the existence of expansion waves for both positive and negative speeds. In the paper, we use a framework of the general theory recently developed for a certain nonlinear convolution equation. This approach allows us to prove the wave existence for the range of admissible speeds c∈R∖(c⋆−,c⋆+), where the critical speeds c⋆− and c⋆+ can be calculated explicitly from some associated equations. The main result is then applied to several non-local reaction-diffusion epidemic and population models.