Graham, your Suzuki DF115, like my DF70, doesn't need a primer bulb. Although I've got one as it helps refill the "filter/water separator" when I change the filter.
If the primer bulb is being sucked in, it must be trying to pull a vacuum, which suggests there's something restricting the fuel flow from the tank to the bulb:
1. Is the fuel tank breather clear?
2. Is there a filter on the bottom of the fuel pickup tube in the tank and if so is it clear?