By having diversity among the plants in your landscape, you will have more species of birds and beneficial insects that will enjoy noshing on leafminers or their eggs. The Bird Predators of leafminers include Chickadees, Purple finches, and Robins. Possible Insect Predators include Ladybug beetles and lacewings who eat the eggs. Certain chalcid and braconid wasps may parasitize leafminers.