The Western Mediterranean.
After the fall of Carthage, the Romans gained dominance over the Mediterranean. It was at this time that they could actually be called a super power.
The Romans persevered over Carthage .
They defeated Carthage
Carthage - both Rome and Carthage sought commercial control over the Mediterranean .
The Romans won three wars against Carthage over a century, and finally sold its resilient people into slavery to neutralise them in 146 BCE. The Romans progressively absorbed the Greek states into their Empire over a period of 150 years, starting with Macedonia in 197 BCE, which had supported the Carthaginians against them in the second war with Carthage.
The Romans , throughout the Punic Wars (264 to 146 BC) , fought against Carthage (Notably Hannibal.) for hegemony in the Mediterranean . The Romans persevered over their Carthaginians opponents and , at the end , razed the city of Carthage and it's citizens were sold into slavery .
Yes, they established a colony for retired soldiers on the site.
Because Carthage was its one remaining big competitor for power over the Western Mediterranen at the time. It was richer than Rome and controlled large areas in North Africa, Spain, southern France and Sicily. It took three wars because the Romans were out to completely destroy Carthage as a nation and as a competitor.
The Romans , after winning the Punic Wars over their rival Carthage , ruled over the Mediterranean Sea .
Carthage did not win over the Romans. The Second punic War was an invasion of Italy by Hannibal Barca. He Routed the armies of the Romans and her Italic allies three times and won some other battles. However, he lost the war. He was eventually cornered into mountainous Calabria (the toe of Italy) and was stuck there for three years until he was recalled to fight the Romans in Tunisia (Carthage's homeland). Moreover, this was Hannibal's war, not Carthage's. His family had conquered southern Spain and ruled it as a private domain. Hannibal conducted a war for revenge. Although many people in Carthage supported him, many objected to it. In fact, Carthage did not provide any aid to Hannibal when he was in need.
Carthage did not give anyone to Rome. Rome captured prisoners of war, but they were not actually given to her by Carthage. Rome demanded that Carthage surrender Hannibal to her, but he fled into exile in the eastern Mediterranean.
Rome and Carthage clashed over control of Sicily, Rome developed naval power to rival Carthage's, and defeated it, gaining power in the Western Mediterranean.