Propagation phenomena in monostable integro-differential equations: Acceleration or not?

The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of f, i.e. f′(0)=0. First, for algebraic decaying kernels, we prove the exact separation between existence and nonexistence of travelling waves. This in turn provides the exact separation between nonacceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.