  In this paper, we study propagation in a nonlocal reaction-diffusion-mutation
model describing the invasion of cane toads in Australia. The population of
toads is structured by a space variable and a phenotypical trait and the
space-diffusivity depends on the trait. We use a Schauder topological degree
argument for the construction of some travelling wave solutions of the model.
The speed $c^*$ of the wave is obtained after solving a suitable spectral
problem in the trait variable. An eigenvector arising from this eigenvalue
problem gives the flavor of the profile at the edge of the front. The major
difficulty is to obtain uniform $L^\infty$ bounds despite the combination of
non local terms and an heterogeneous diffusivity.
