This is not the first time that Procter & Gamble has faced activist pressure: The billionaire William A. Ackman pushed the board to oust Robert A. McDonald as chief executive in 2013, prompting Mr. McDonald’s predecessor, Alan G. Lafley, to return to the post from retirement.

Companies with mature consumer brands — like Procter & Gamble, with its Tide detergent and Gillette razors — can be ripe targets for activists because their days of surging growth are often well behind them.

While companies may want to keep the brands, enjoying the cash they can generate, activist investors tend to be more aggressive in their approach. They often look for ways to sell the brands and reinvest in new technologies or brands that can generate better growth opportunities, said Damien Park, a managing director at Spotlight Advisors, a firm that consults with companies and funds on activist campaigns. (Mr. Park is not involved in Mr. Peltz’s crusade with Procter & Gamble.)

That Procter & Gamble has become the quarry for activist investors is a sign of how far the once-iconic giant has fallen.

For years, Procter & Gamble attracted the best-and-brightest marketing M.B.A.s from the nation’s top schools. It spurred many a Harvard Business School study of its internal methodologies and disruptive innovations. And it was the home to many big consumer hits, like the Swiffer duster and Crest Whitestrips.

But thanks in no small part to lackluster economic growth overseas, combined with a strong dollar, Procter & Gamble has had 13 consecutive quarters of sales declines. Its shares have lagged behind the Standard & Poor’s 500-stock index.