Latvia has been losing its population for the last 30 years, and continues to steadily lose around 2% of its population every year. And that is with steady economic growth and no wars or crises (like Venezuela or Syria), so there are opinions that such trend will not stop.
Until of course country has no population left to sustain the country. What would happen to it in such case? Would its land be somehow taken by another country? How it would be decided?
Nature abhors vacuum. If a population of certain country is so decadent that it cannot ensure natural replacement even if there is no natural or human made catastrophe ( earthquake, war, disease etc ...) nature will take its course. Population from other countries would gradually seize opportunity to settle in relatively prosperous conditions. Would they do it legally or illegally is irrelevant. Over time their culture, language and even genetic makeup would become dominant. Original inhabitants would slowly disappear along with their customs. And, at certain point, if borders remain same, original name of the country would disappear too.
There is no general answer to your question. Some states have very low population like Iceland, Monaco, Andorra.
We may draw off a few hypotheses though.
The country won't likely be taken over by another. It is a member of NATO and EU.
Many countries in Europe are declining in terms of population. So it is possible that the government will see this as natural and won't do a thing.
You may also expect a natalist politic; some countries like Denmark are doing it. They even has ads in order to incite people to make love.
Another way, would be a more lenient immigration policy. It is an immediate solution, but it may cause problems if immigrants are not well integrated. Germany is the greatest example. France to an extent, but it also has pro natality policies.
Another possibility while less realistic, would be to merge with one or both the other Baltic countries. They have been united in the past (even without accounting for USSR or Russian Empire).
Last possibility, EU could become a giant state, but this is becoming less and less realistic.
There are answers dealing with the things that might be done to avoid this scenario or stating that this would never happen. Therefore this answer will assume that these things didn't occur or failed to work and will address what might happen if this scenarios succoured.
Firstly it may be important to think about what makes a nation state. there are two main things mentioned in other answers on this site that make a nation state. Usually both things would be the case but occasionally only one of them is required
The wikipedia page on states with limited recognition lists two theories about what makes a state. As this question is about a pre-existing state all of those points should be covered. It's worth noting that the requirement of a permanent population. The declarative theory states a requirement for a government however in theory this could just be a single person carryout the functions of government
A number of possibilities exist
This entirely depends on what happens. The options laid out above all involve someone deciding what to do although the relevant decision maker would depend on which one happened.
Potentially the deciding factor would be which one of the above the international community would allow to happen, there are some options/results that may be diplomatically impossible to achieve.
The historical examples of near complete depopulation of a territory are extremely rare, probably Easter Island stands chiefly among them, for a combination of ecological and external factors:
It is believed that Easter Island's Polynesian inhabitants arrived on Easter Island sometime in the 12th century AD. They created a thriving and industrious culture, as evidenced by the island's numerous enormous stone moai and other artifacts. However, land clearing for cultivation and the introduction of the Polynesian rat led to gradual deforestation.[3] By the time of European arrival in 1722, the island's population was estimated to be 2,000–3,000. European diseases, Peruvian slave raiding in the 1860s and emigration to other islands, e.g.Tahiti, further depleted the population, reducing it to a low of 111 native inhabitants in 1877.[4] [...] The 2017 Chilean census registered 7,750 people on the island, of whom 3,512 (45%) considered themselves Rapa Nui.
Otherwise, there have been cycles of population boom and bust (of lesser magnitude) practically everywhere. (Not till long ago, the main concern was that population was growing too fast, and might still be the case world-wide.) The concept of population cycle is better understood in biology, for the obvious reason that the factors affecting non-human populations are simpler to study.
So it's pretty improbable that the downward trend is going to reach zero in some countries instead of a non-zero minima (and possibly rebounding). Except for the massive ecological disaster scenario, say global warming out of control, rendering some countries uninhabitable, everything else is too speculative in terms of complete depopulation, i.e. has no historical equivalent. But if we take Easter Island as example, a combination of fertility rebound and/or immigration seem to be the likely scenario.
What would happen politically is even more speculative. Easter Island was annexed, but it's not clear that that would happen to a larger country whose borders are guaranteed by international alliances etc.
A sufficiently scary scenario is many such countries becoming like Greece, i.e. long-term economic disaster zones, despite preserving their independence. However, that's not of immediate concern; despite topping the depopulation charts, most countries in Eastern Europe also top them in economic growth (at EU level), so it's not even clear when population crunch will start to affect their growth.
