  In many applications involving point pattern data, the Poisson process
assumption is unrealistic, with the data exhibiting a more regular spread. Such
a repulsion between events is exhibited by trees for example, because of
competition for light and nutrients. Other examples include the locations of
biological cells and cities, and the times of neuronal spikes. Given the many
applications of repulsive point processes, there is a surprisingly limited
literature developing flexible, realistic and interpretable models, as well as
efficient inferential methods. We address this gap by developing a modelling
framework around the Mat\'ern type-III repulsive process. We consider a number
of extensions of the original Mat\'ern type-III process for both the
homogeneous and inhomogeneous cases. We also derive the probability density of
this generalized Mat\'ern process. This allows us to characterize the posterior
distribution of the various latent variables, and leads to a novel and
efficient Markov chain Monte Carlo algorithm. We apply our ideas to datasets
involving the spatial locations of trees, nerve fiber cells and Greyhound bus
stations.
