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"
Its Peloponnesian League opponents were superior on land. Athens' strategy was to defend the city walls and attack the opposition cities with its superior navy. This worked until persia financed a comparable navy for the Peloponnesian League.