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One reason is that the life cycle of most insects is very short. There may be 1-15 generations in a year (As opposed to 1 every 14-20ish years in humans). Another reason is many insects have hundreds to thousands of offspring every reproductive cycle. Let's assume we're talking about an insect with 10 generations per year, that they exist in a 1:1 male to female ratio (which is common in insects), that each mating pair can produce 1,000 eggs and that we start with 1 mating pair.

At generation 1, we have 1 mating pair. These 2 insects make 1000 babies.

At Generation 2 we have 500 mating pairs (1000/2 because of 1:1 ratio). These 500 mating pairs make 500x1000 (500,000) babies.

At Generation 3, which isn't even 6 months into the first year, we have 250,000 mating pairs that have 250000x1000 babies. Since each egg may contain mutations, you can see why insect populations can evolve quickly.

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