  In this paper, we study the influence of the mortality trade-off in a
nonlocal reaction-diffusion-mutation equation that we introduce to model the
invasion of cane toads in Australia. This model is built off of one that has
attracted attention recently, in which the population of toads is structured by
a phenotypical trait that governs the spatial diffusion. We are concerned with
the case when the diffusivity can take unbounded values and the mortality
trade-off depends only on the trait variable. Depending on the rate of increase
of the penalization term, we obtain the rate of spreading of the population. We
identify two regimes, an acceleration regime when the penalization is weak and
a linear spreading regime when the penalization is strong. While the
development of the model comes from biological principles, the bulk of the
article is dedicated to the mathematical analysis of the model, which is very
technical.
