Answer: I suspect you are dealing with the Catalpa Sphinx Moth. The larva of the catalpa sphinx moth can eat large quantities of leaves. The caterpillar is yellow with black markings. You can use the botanical Bt (bacillus thuringensis) which is a stomach poison, as soon as you see the damage in the spring. Or, you can use sprays of Orthene when the caterpillars are first seen. Waiting until the insects are full grown before spraying will not be beneficial.
Spider mites like hot, dusty conditions so regularly spraying your evergreens with water will help deter them. If you need more control, graduate to the insecticidal soaps ? like Safer Insecticidal Soap, by Ringer. The key is to use them at a time of day that allows them to stay wet as long as possible. Spray the plant thoroughly being careful to get good coverage on the undersides of the leaf where the mites are. Spray early in the day so the soap solution stays wet as long as possible ? it kills on contact when it is wet. One note of caution, if your shrub is not listed on the soap label, test a small area first before spraying the entire plant.
If the soap is not working, then I would graduate to the horticultural oils. I personally prefer Sunspray Horticultural Oil ? it is paraffin based and seems to be less detrimental to the plants particularly during periods of warm weather. Use according to the instructions and again be sure to get into the undersides of the plant. Be sure that you are using a summer rate ? some of the oils can also be used during the dormant season. One note, some lacebugs overwinter on the plant in a protected area ? those little crooks and crannies into which an insect could crawl ? be sure the spray penetrates into these areas. Other brand names will include Ortho's Volk Oil or Lilly Millers' Supreme Oil. These are specially formulated for use on plants, and when used according to the directions should not be harmful to the plant. A couple of tips, be sure the plant is watered and is not stressed before you spray it. I have seen a lot of plants damaged when sprayed when they were dry and under drought stress.
Best wishes with your landscape!
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