Well, of course, our countries will not let go without a fight: the US continues to play a major role in Latin America, through Free Trade Agreement or even a strong military presence (here).
As for France, its oil firms (Total) and other private companies (such as Bouygues or more impressively, Boloré whose CEOs are friends of the French president - here and here) continue to play a major economic role. Politically, the latest developments in Ivory Coast have also shown that Sarkozy is willing to keep France as a key player.
One last point, the vast majority of French and American people are either clueless about what the policies of their countries in those 'backyards' of theirs. Well, they don't know and don't want to know. They will not see, for instance, that the immigration that some French or Americans fear is also in part the result of French and American plunder of the local economies where the immigrants live. After all, the Africans have been 'independent' for 50 years and the Hispanics have their own countries - the say.
As a conclusion, one can look at the events unfolding in Ivory Coast with a different perspective - this country that used to be the jewel in the crown of the post-colonial African empire of France for decades.
Even though Gbagbo has some nasty politics (such as his xenophobic concept of “Ivoirité”, or Ivorianness), even though he has also instigated a coup, AND even though Ouattara (his opponent) is probably the legitimate elected president, France should not mingle. precisely because it has lots of interests at stake, 15,000 nationals in the country and zero credibility when it comes to supporting African democracies.
Of course, Gbagbo has been using anti-French sentiment, indeed: