White fuzzy lumps on paddle cactus indicate the presence of a parasite that pierces the plant's skin and consumes its juices. A bit of cochineal (coach-en-ee-al) scale is no big deal, but it does tend to spread and may eventually kill the plant.
Your first line of defense is to blast what appears to be bits of cotton with a hose to dislodge them. However, in crevices and where a hose can't reach, they'll proliferate. I take preventive action and scrub my paddle (opuntia) cacti at the first sign of infestation with insecticidal soap. I also remove any badly infested pads and those growing quite close to each other (slice them off at the joints).
Then I treat the plants with Safer insecticidal soap. So named because it's safe for the environment, it controls soft-bodied insects without harming plants like harsher soaps may. (Don't use dishwashing liquid on succulents.) I mix 4 or 5 tablespoons of concentrated Safer soap to a gallon of water, then dip a long-handled, soft-bristle brush into the solution and scrub the pads.
This removes the insects (which are not mealybugs, a common misconception), won't scratch the pads, and leaves a soapy residue that inhibits the pests' regeneration. I do this "as needed"---about every six weeks in the summer.
Cochineal scale does have the odd benefit of entertaining garden visitors, especially kids. Squish a white lump and you'll get marvelously realistic "blood." (Use cold water to remove any stains on clothing.) Of course if the opuntia has spines---as most do---real blood also is a possibility. If so, simply use a stick to squash and smear the cottony bumps.
Insects nestled within the waxy white fuzz are rich in carminic acid, a red liquid that repels ants, birds and other predators. Used by ancient Aztecs to color woven fabric, cochineal scale was highly prized by Spanish conquistadors, who considered it second in value only to New World silver.
Because harvesting cochineal dye is labor intensive, it was replaced by synthetic dyes in the 1800s. But due to cancer-causing concerns, natural red dye is now back in favor with food and cosmetic manufacturers. Next time you pick up something that's edible, red and packaged, read the label. If ingredients list "carmine," "cochineal," "E 120," or "natural red 4," it's been colored---ironically, to make it more appealing---with insects.
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