A continuous predator-prey model in which both species diffuse along a spatial gradient is shown to exhibit temporal chaos at a fixed point in space. The model incorporates a nonlinear functional response of the predator and a logistic growth of the prey; the intrinsic growth rate of the prey varies linearly with space. Numerical results demonstrate that low diffusion values drive an otherwise periodic system into aperiodic behaviour with sensitivity to initial conditions. Evidence is provided for a quasiperiodic route to chaos as the diffusion rate decreases. These results suggest that complex temporal dynamics in natural communities may arise through the spatial dimension. Spatially induced chaos may have an important role in spatial pattern generation in heterogeneous environments.