Travelling fronts for the KPP equation with spatio-temporal delay

We study an integro-differential equation based on the KPP equation with a convolution term which introduces a time-delay in the nonlinearity. Special attention is paid to the question of the existence of travelling wavefront solutions connecting the two uniform steady states and their qualitative form. Motivated by the analogue between steady travelling fronts and heteroclinic orbits of an associated ordinary differential equation, we prove, using a geometric singular perturbation analysis, that steady travelling wavefront solutions persist when the delay is suitably small, for a class of convolution kernels. These travelling fronts are qualitatively similar to the well known KPP wavefront. The effect of finite and large delay is studied numerically and we find that this introduces qualitative changes to the fronts but that the front remains robust. A numerical integration of the initial-value problem confirms the qualitative shape of these fronts and suggests that - even for large delay - they may be temporally stable. Finally we show that in the discrete delay case the non-zero uniform state can be driven unstable. In this case a travelling wavefront can leave in its wake a periodic travelling wave moving with a different speed.