  Astrophysical reaction rates, which are mostly derived from theoretical cross
sections, are necessary input to nuclear reaction network simulations for
studying the origin of $p$ nuclei. Past experiments have found a considerable
difference between theoretical and experimental cross sections in some cases,
especially for ($\alpha$,$\gamma$) reactions at low energy. Therefore, it is
important to experimentally test theoretical cross section predictions at low,
astrophysically relevant energies. The aim is to measure reaction cross
sections of $^{107}$Ag($\alpha$,$\gamma$)$^{111}$In and
$^{107}$Ag($\alpha$,n)$^{110}$In at low energies in order to extend the
experimental database for astrophysical reactions involving $\alpha$ particles
towards lower mass numbers. Reaction rate predictions are very sensitive to the
optical model parameters and this introduces a large uncertainty into
theoretical rates involving $\alpha$ particles at low energy. We have also used
Hauser-Feshbach statistical model calculations to study the origin of possible
discrepancies between prediction and data. An activation technique has been
used to measure the reaction cross sections at effective center of mass
energies between 7.79 MeV and 12.50 MeV. Isomeric and ground state cross
sections of the ($\alpha$,n) reaction were determined separately. The measured
cross sections were found to be lower than theoretical predictions for the
($\alpha$,$\gamma$) reaction. Varying the calculated averaged widths in the
Hauser-Feshbach model, it became evident that the data for the
($\alpha$,$\gamma$) and ($\alpha$,n) reactions can only be simultaneously
reproduced when rescaling the ratio of $\gamma$- to neutron width and using an
energy-dependent imaginary part in the optical $\alpha$+$^{107}$Ag
potential.......
