Excitable wave patterns consisting of large coherent spirals are obser
ved during the aggregation of Dictyostelium amoebae. These emerge from
an initially disordered state which arises via random cell firing. In
this work, we show that this phenomenon can be understood as being du
e to a specific spiral competition instability which occurs in certain
three-component excitable medium models. This instability can be unde
rstood as symmetry breaking of a spiral pair leading to one spiral sup
pressing and expelling another.