If the main brands in a category invest in advertising on an on-going basis, any brand that wants to compete will almost certainly also need to invest (see The Long and the Short of it). As a rule of thumb, a brand should invest enough so that its share of advertising spend is roughly in line with its share of market just to stand still (see this illustration).
If a brand stops spending on advertising, it is at risk of decline. Larger, well-established brands are likely to cope better when spending stops (especially if they were on a growth trajectory) but after a year or two without advertising, sales tend to start declining. Even just 12 months without advertising can be hard/expensive to recover from (see this illustration).
Small brands are less resilient still. Even up-and-coming small brands tend to suffer badly if advertising support is withdrawn.
You can learn more about this topic from this report by the Ehrenberg-Bass Institute. This discussion on LinkedIn also airs some useful debates.