By sea.
We call it the Peloponnesian War today because it was between the Athenian Empire and the Peloponnesian League, of which Sparta was a member and leader. At the time, they referred to it the war between Sparta and its allies and Athens and its allies.
The advantage that Athens had during the Peloponnesian wars was that they had a bigger army. They also had a larger Navy.
During the Peloponnesian war.
Sparta had the largest area after it conquered the southern Peloponnese, while Athens had limited land in Attica, until it created an empire for 50 years of the GREEK cities in the eastern Aegean Sea littoral.
Sparta .
Sicily.
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Sparta and its allies and Athens and its allies fought for domiance. The Spartan side won with Persian assistance.
Sparta and its allies.
Phase 1 - the Peloponnesian League invested Athens, which sent out amphibious expeditions targeting the Peloponnesian League cities. Phase 2 - a phony peace during which Athens tried to capture Syracuse in Sicily. Phase 3 - more direct hostilities during which Persia supplied the funds for the Peloponnesian League to mount a fleet which could match the Athenian one. Athens defeated at sea, invested and surrendered.
The city-states of Athens and Sparta as well as their allies were involved in the war that started at 431 and ended at 404 BCE. The Spartans and their allies were the winners of the war that later was named "the Peloponnesian war"
Each had allies which augmented its power. Athens had an empire of about 180 city-states spread around the eastern Aegean Sea,. It had strong walls, which extended down to the sea, which enabled it to be resupplied during a siege. It also had a superior navy, which gave it the ability to threaten and raid opposing cities of the Peloponnesian League led by Sparta, and to ensure resupply of the city by sea from its empire. The Peloponnesian League had the strongest land forces, and was able to besiege Athens. By whittling don Athens' allies, it progressively reduced Athens' power and resupply, until finally after 27 years of fighting, the Athenian fleet was destroyed (Persia gave money to pay for a Peloponnesian fleet) and Athens was starved into submission.