It is well known that sessile bacteria have a strong tendency to exist in a biofilm phenotype, whereby bacterial cells aggregate and produce a gel-like extracellular matrix, which, in an infection scenario, offers a significant barrier to attack by conventional antibiotics and the immune system. In this paper we develop a multi-phase model of a maturing Pseudomonas aeruginosa biofilm, allowing for the production and secretion of exopolysaccharide (EPS). The primary quorum-sensing system of P. aeruginosa (namely the lasR system) is believed to be required for full biofilm development, and we thus take the synthesis of EPS to be regulated by the cognate signal molecule, 3-oxo-C12-HSL. We also take EPS and signal production, along with bacterial growth, to be limited by oxygen availability, thus factoring in the nutrient poor conditions deep inside the biofilm. We use simulations to examine the role played by quorum sensing in the biofilm maturation process, and to investigate the effect of anti-quorum sensing and antibiotic treatments on EPS concentration, signal level, bacterial numbers and biofilm growth rate. In addition, we undertake analysis of the associated travelling-wave behaviour.