what are some natural ways of eliminating these bugs without using pesticides?

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The quickest way is to take a spray bottle, add several drops of dish detergent about one quart of water, shake and spray on plants.

This is much stronger and works on almost all bugs.
Make The Garlic Oil Solution
Soak 3 ounces of minced garlic in 2 teaspoons of mineral oil for 24 hours
Strain out the garlic and add 1 pint of water and 1 teaspoon of liquid cheap dish soap to the remaining liquid
Mix together well
To Spray Plants
Mix 1 to 2 tablespoons of the garlic-soap mixture with 1 pint water and spray plants
Because homemade mixtures have no preservatives, you're better off dumping any you have left over, and simply mixing a fresh batch every time you want to spray. ...Show more

Almost all organic methods use plants for the killing substance. The problem I have found is that all plants have insects that eat them, so when you use garlic or marigolds, or other plants, even nicotine from tobacco, for a potion, you attract the bugs that eat those plants.

My experience has been Organic sprays only work in areas where most of the bugs have already been killed off by other means. In areas where there are fields, forest and natural places for bugs to breed, they just keep on coming.

Dipel (Thuricide) attacks the catapillar stage of the bug cycle only. When Bees pollinate a veggie sprayed or sprinkled with Dipel, It is carried back to the bee colony and kills the bees in the grub stage of development.

I have had limited success using soap and ashes. It is the lye in the soap and ashes that the bugs don't like, but that isn't really organic since lye is a toxic chemical.

The best organic method is to plant enough for the bugs and you, and accept the fact some of your veggies will have bug bites out of them.

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I save my friends' cigarette butts. When I get a cup or so, I add one quart of water, a minced garlic clove, and a minced hot pepper or a half teaspoon on chili oil. Marinate for a couple of hours in the sun, and spray the resulting "tea" on the affected plants. I wear gloves so I don't get any on my hands--it can cause a heck of burn.

Since these are collard greens, make sure you wash them in two changes of water when you pick them, just in case. ...Show more

Thrurcide (bacillus thuringensis) or Dipel.Its a biological spray made from a type of mum in Africa and is safe and non harmful to people..pets or wildlife. Its used to controll leaf eatin caterpillars...Show more

No photo I assume they are cabbage loopers. Bt works well, smashing them is very satisfying...Show more