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The nonlinear dynamics of a dual-beam optically injected semiconductor laser are studied. Most of the observed dual-beam injection dynamics can be understood from competitions of single-beam injection dynamics that are associated separately with each injection beam. Three general scenarios are identified and mapped for various combinations of detuning frequencies and injection strengths of the two injection beams. In Scenario A, simultaneous injection of two beams completely suppresses the dynamics due to separate single-beam injection and replaces them with the dynamics resulting from strong nonlinear interactions of the two injection beams. In Scenario B, simultaneous injection of two beams does not suppress the dynamics due to single-beam injection of one of the two beams but modifies such dynamics through nearly degenerate four-wave mixing with the other injection beam associated with the suppressed dynamics. In Scenario C, simultaneous injection of two beams does not suppress the dynamics due to single-beam injection of either of the two beams but allows the two dynamics to coexist and mix nonlinearly.