The two brothers should not vie with each other but work in co-operation.

Fight : take part in a violent struggle involving the exchange of physical blows or the use of weapons.
Conspire : make secret plans jointly to commit an unlawful or harmful act.
Compete : strive to gain or win something by defeating or establishing superiority over others.
Envy : a feeling of discontented or resentful longing aroused by someone else's possessions, qualities, or luck. Vie : compete eagerly with someone in order to do or achieve something.

Synonym of Vie is Compete