Whenever I notice that certain words tend to appear several times in certain context, I have often found it worthwhile to study it more closely. This is the case with the word "desolate". It often seems to appear when there is a curse, destruction or scattering, especially in the Old Testament and in the Book of Mormon. Given the meaning of the word, that shouldn't come as a big surprise, but it seems that the authors sometimes go out of their way to include the word.
Here are some examples
Alma 16 tells about the destruction of Ammonihah. In verse 11 we read
And now so great was the scent thereof that the people did not go in to possess the land of Ammonihah for many years. And it was called Desolation of ; for they were of the profession of Nehor, who were slain; and their lands remained desolate.
This is what happened to a people that Alma called "highly favored" (which I believe to signify a covenant people) that had fallen into transgression.
Nephites at the time of Christ
And highways shall be broken up, and many cities shall become desolate. (Helaman 14:24)
And now, my beloved brethren, behold, I declare unto you that except ye shall repent your houses shall be left unto you . (Helaman 15:1)
And many and notable cities were , and many were , and many were shaken till the buildings thereof had fallen to the earth, and the inhabitants thereof were slain, and the places were left desolate. (3 Nephi 8:14)
And who knoweth but the Lord will carry us forth into a land which is above all the earth? And if it so be, let us be faithful unto the Lord, that we may receive it for our inheritance. (Ether 1:38)
Ether 2 and Ether 13 are also full of promised land/covenant language. The Jaredites are a chosen people who should serve God. Eventually they fall, just like the Nephites. The Book of Ether tells us about their complete destruction. This happens in the land that the Nephites later referred to as the land northward. What was this land called? Desolation!
And it bordered upon the land which they called , it being so far northward that it came into the land which had been peopled and been destroyed, of whose we have spoken (Alma 22:30)
It started out as a promised land and an inheritance for a favored people, but because of wickedness it became Desolation.
This is an echo of something Jesus had told his disciples a little more than a week earlier.
In Matthew, Jesus goes on to talk about the destruction of Jerusalem and scattering of the Jews by the Romans. Keep in mind the other examples shown here that all ended with destruction. Jesus also goes on to talk about the last days culminating in his second coming. Notice the addition in 3 Nephi: "until the time of the fulfilling of the covenant to your fathers". This leads me to the following insight: When desolation is used in the scriptures, it often describes a promised land not inhabited by its original covenant people. It does not necessarily mean that it is empty. It only means that the covenant people that God established there broke the covenants and were cursed, bringing destruction upon themselves. The scattering of Israel happened in several promised lands with several branches at different times. But in the latter days, they will be restored. That is "the time of the fulfilling of the covenant to your fathers".
There is a lot of desolation language in the Old Testament, often connected to the scattering of Israel as a result of disobedience or breaking covenants. The prophet Jeremiah states
Psalm 69 is another example. This is a Messianic Psalm and contains prophecies about the Jews being scattered for rejecting the Messiah
A few verses later, the Psalmist describes "the time of the fulfilling of the covenant to your fathers".
For God will save , and will the cities of Judah: that they may dwell there, and have it in possession.
This is a latter-day reversal of the various "desolations". We find plenty of this language in Isaiah too, both the desolation as a covenant curse/scattering and the regaining possession of the promised land/desert blossom as a rose/building Zion etc. in the latter days as a renewal of the covenants. But I think the examples given here suffice for now. I will let Nephi have the final word, though.