Pumps do not put out a constant pressure. Fixed displacement pumps put out an almost constant flow. There are pressure compensated pumps that vary the stroke to attempt to maintain pressure but the control mechanism is a spring so there is only a proportional band. The pressure will droop as flow increases. What you should look at is pump and systems curves. There are two curves. In the case of the pressure compensated pump this is nearly a flat line from 0 flow and system pressure down to full flow and some 15% to 20% less than system pressure. The system curve starts a 0 pressure and no flow and increases non-linearly at a faster rate as flow increases. Pressure can increase linearly in some sections and proportional to the flow squared in other sections. Where the the pump curve and system curve intersect is the operating point. Basically you solve for two unknowns with two equations.
Peter, thanks for sharing your insights on pumps. I typically use something like P_pump = P_pump_max - alpha * q^2 for centrifugal pumps and fix q = constant for positive displacement pumps. The pressure compensated pumps are interesting. Have you had any experience with VFDs on pumps that maintain a desired flow or pressure?
