Non-local effects on travelling waves arising in a moving-boundary reaction–diffusion model

We examine travelling wave solutions of the partial differential equation u t = u xx + u (1 − u ∗ ϕ ) on a moving domain x ⩽ L ( t ), where u ∗ ϕ is the spatial convolution of the population density with a kernel ϕ ( y ). We provide asymptotic approximations of the resulting travelling waves in various asymptotic limits of the wavespeed, the non-local interaction strength, and the moving boundary condition. Crucially, we find that when the moving boundary has a weak interactive strength with the population density flux, there can be two different travelling wave solutions that move at the same wavespeed.