Final answer:
The German cockroach produces the most eggs the fastest among the listed species. It can carry 30-40 eggs in an egg capsule and produce six generations per year, resulting in a high reproduction rate.
Step-by-step explanation:
The German cockroach (Blattella germanica) is considered the most common and problematic species when it comes to reproduction rates and infestation in human habitats. These cockroaches are known for their rapid reproductive cycle and the ability to produce more eggs at a faster rate than the other species listed. A single German cockroach female can produce up to 30 to 40 eggs at a time, which she carries in an egg capsule, or ootheca, until the day before they hatch. Over her lifetime, which can span about 6 to 9 months, she can produce more than six generations per year, each with hundreds of offspring. In comparison, the American cockroach's egg capsule usually contains around 16 eggs, while the oriental and Asian species tend to produce fewer eggs per capsule.