Travelling fronts in a food-limited population model with time delay

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R-4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.