Josiah Strong, an American advocate of "Anglo-Saxon" supremacy in the late 19th century, at least, did portray the competition between the races as a competition, one in which the strongest race would win out and populate the entire world to the displacement of the others. He did insist, however, that this was indeed a contest, and like all contests, if "Anglo-Saxons" didn't give it their all, they might be on the losing end.

http://www1.assumption.edu/users/McClymer/hi119net/Josiah_Strong1

This whole essay of his is fascinating and I'll only quote short bits.

The unoccupied arable lands of the earth are limited, and will soon be taken. The time is coming when the pressure of population on the means of subsistence will be felt here as it is now felt in Europe and Asia. Then will the world enter upon a new stage of its history-the final competition of races, for which the Anglo-Saxon is being schooled. Long before the thousand millions are here, the mighty centrifugal tendency, inherent in this stock and strengthened in the United States, will assert itself. Then this race of unequaled energy, with all the majesty of numbers and the might of wealth behind it-the representative, let us hope, of the largest liberty, the purest Christianity, the highest civilization-having developed peculiarly aggressive traits calculated to impress its institutions upon mankind, will spread itself over the earth. If I read not amiss, this powerful race will move down upon Mexico, down upon Central and South America, out upon the islands of the sea, over upon Africa and beyond. And can any one doubt that the result of this competition of races will be the "survival of the fittest?" "Any people," says Dr. Bushnell, "that is physiologically advanced in culture, though it be only in a degree beyond another which is mingled with it on strictly equal terms, is sure to live down and finally live out its inferior. Nothing can save the inferior race but a ready and pliant assimilation. Whether the feebler and more abject races are going to be regenerated and raised up, is already very much of a question. What if it should be God's plan to people the world with better and finer material?"

Note how he puts this: the Anglo-Saxons are going to seize and displace the native inhabitants of all of Latin America and Africa, stopped only by climate if by anything. This is a challenge, though, and one that can only be accomplished if America holds true to the values he mentions. It is a race war that he sees as winnable, but not yet won.

Have one more passage:

Thus the Finns were supplanted by the Aryan races in Europe and Asia, the Tartars by the Russians, and thus the aborigines of North America, Australia and New Zealand are now disappearing before the all-conquering Anglo-Saxons. It seems as if these inferior tribes were only precursors of a superior race. Every civilization has its destructive and preservative elements. The Anglo-Saxon race would speedily decay but for the salt of Christianity. Bring savages into contact with our civilization, and its destructive forces become operative at once, while years are necessary to render effective the saving influences of Christian instruction. Moreover, the pioneer wave of our civilization carries with it more scum than salt. Where there is one missionary, there are hundreds of miners or traders or adventurers ready to debauch the native. Whether the extinction of inferior races before the advancing Anglo-Saxon seems to the reader sad or otherwise, it certainly appears probable. I know of nothing except climatic conditions to prevent this race from populating Africa as it has peopled North America. And those portions of Africa which are unfavorable to Anglo-Saxon life are less extensive than was once supposed. The Dutch Boers, after two centuries of life there, are as hardy as any race on earth. The Anglo-Saxon has established himself in climates totally diverse-Canada, South Africa, and India-and, through several generations, has preserved his essential race characteristics.

So he argues that the only thing that stands in the way of Anglo-Saxon global domination is climate, and that even climate isn't as much of a boundary as could be imagined. He also argues that whether you pity the displaced and doomed natives or not is immaterial, their fate is the result of the coming conflict, not something that pity can alter.

A lot of this late 19th century triumphalist tone will take on a tinge of panic after 1906 when the Japanese decisively beat the Russians in the Russo-Japanese War. The years between 1906 and World War I would see the rise of "Yellow Peril" panic, the fear that Japan would lead a Pan-Asian coalition to destroy European imperialism, or, to the Social Darwinists, the fear that the coming race war they desired might not necessarily have a positive outcome.