While stochastic neutron transport theories have been developed in rigorous detail, many applications have historically been investigated using the point-kinetics formulation. In this work we develop a space-dependent model using the diffusion approximation to the Pál-Bell probability generating function equation, resulting in a nonlinear analog of the conventional time-dependent neutron diffusion equation. We investigate a variety of approximate solutions for the time- and space-dependent survival probability in one-dimensional symmetric, one-speed, isotropic, delayed neutron precursor-free systems, and compare them to counterpart point-kinetics results. Following the theoretical developments, we apply the new results in the context of a criticality accident scenario, from which the importance of spatial effects is revealed.