Answer: First of all, it's really important to identify the pest before using any pesticide -- "natural" or not. Always try cultural methods first -- pruning, weeding, thinning, cleaning up garden debris, etc. Also, there is a misconception that "natural" pesticides are safer than "chemical" ones. Note that many so-called "natural" pesticides are extremely toxic. For example, nicotine was once used to kill insects, but this is no longer recommended because of its high toxicity.
There are certain biological controls, such as Bt, that target only one type of pest. Bt is a bacteria that is used to control caterpillars, among other pests. It is very important to choose the correct control -- for example, different strains of Bt are used to control different pests. Also, remember that the Bt that controls caterpillars in the garden will also kill butterfly larvae, so use it only when absolutely necessary.
Others pesticides, such as horticultural oil, suffocate a variety of pests. Horticultural oil may be made from petroleum or vegetable oils; note that you should always use one made specifically for plants, since they have been highly refined for this purpose. It is often used while plants are dormant, to kill overwintering scale and mites.
Finally, there are "broad spectrum" controls, like pyrethrum and rotenone. Even though these are derived from plants, they must be used with great caution, if used at all, because they will kill benefical insects as well as pests -- and these beneficials can often keep pests in control by themselves! The one benefit they have over synthetic insecticides is that they break down quickly in the environment, so they don't persist like some synthetics. (However, this also means that they may need to be reapplied more often.)
Some gardeners make their own using a liquid dish soap like Ivory Liquid at a 1 or 2 percent concentration in water. Avoid spraying soap in the heat of the day and remember that contact with the insect is important if the soap is to be effective. Once dry, soap sprays are no longer effective. Be advised that to make a homemade concoction is to do so at your own risk (and your plant's). Commercial soap products are specifically formulated to be effective on insects with a minimum of damage to plants.
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