Front slowdown due to nonlocal interactions
Reaction-diffusion equations arise as models of systems in which spreading and growing forces
interact in nontrivial ways, often creating a front (i.e., a moving interface). In many applications
it is natural to consider nonlocal interactions, but, mathematically, the resulting equations have a
number of new features and technical difficulties; in particular, the comparison principle, which
implies that initially ordered solutions remain ordered, no longer applies.
I will survey classical results, present several examples of nonlocal reaction-diffusion equations,
and then focus on a particular one, the cane toads equation, which is inspired by the invasive
species in Australia. In all cases, the emphasis will be on the influence of long-range interactions
due to nonlocal terms on the behavior of fronts. In particular, I will show that, surprisingly, the
cane toads front propagates slower than the standard methods predict.