  We study stability of stationary solutions for a class of non-local
semilinear parabolic equations. To this end, we prove the Feynman--Kac formula
for a L\'{e}vy processes with time-dependent potentials and arbitrary initial
condition. We propose sufficient conditions for asymptotic stability of the
zero solution, and use them to the study of the spatial logistic equation
arising in population ecology. For this equation, we find conditions which
imply that its positive stationary solution is asymptotically stable. We
consider also the case when the initial condition is given by a random field.
