  We consider the homogeneous integro-differential equation$\partial \_t
u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we
investigate the existence/non existence of travelling waves, and the
propagation properties of the Cauchy problem.When the dispersion kernel $J$ is
exponentially bounded, travelling waves are known to exist and solutions of the
Cauchy problem typically propagate at a constant speed \cite{Schumacher1980},
\cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a},
\cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is
exponentially bounded, travelling waves are known to exist when $f$ belongs to
one of the three main class of non-linearities (bistable, ignition or
monostable), and solutions of the Cauchy problem typically propagate at a
constant speed \cite{Schumacher1980},
\cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004},
\cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On
the other hand, when the dispersion kernel $J$ has heavy tails and the
non-linearity $f$ is non degenerate, i.e $f'(0)\textgreater{}0$, travelling
waves do not exist and solutions of the Cauchy problem propagate by
accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a
general monostable non-linearity, a dichotomy between these two types of
propagation behaviour is still not known. 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 non existence of travelling waves. This in turn provides
the exact separation between non acceleration 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.
