There isn't one.
I think there are, however, two primary "mechanisms" that produce what, to many people, appears to be declines, albeit temporary ones, and neither of them have much to do with Morgoth, at least directly. (Indeed none of the artifacts described would have been made if not for Morgoth. In fact, as Eru tells Melkor that all his doings will ultimately only redound to the greater glory of Eru, and make the World "so much the more wonderful and marvellous." )
Set in the Music, the Valar (and Maiar) play a large role in the beginning, which is then ceded to the Children. Similarly, Men are to "take over" after the Elves. So if we're looking at the world in terms of "what can the Valar do" and "what can the Elves do", then obviously we're going to see less and less of that as time goes on. But notice in the actual music, there is no decline at all: the later themes are greatest.
Many things could only ever be made one time, because only one person could have ever done it and/or the personal cost is so high that there's nothing left in the tank to make another. This often also involves often also involves secret lore, and the death of the maker precludes the secret being revealed. But this does not prevent other people in the future from making other, new things.
So, I'll go into detail about each of these, using the examples you provided. We'll talk about the demographics of the Hobbits and Dwarves at the end.
First, let's establish there is no continuous decline. Even considering only the time period of the Silmarillion and Lord of the Rings (remembering we ourselves live in Middle Earth), we could argue there are declines, but there is definitely no continuous decline. We need only consider the following:
The Elves peaked just before and during the First Age, after thousands to tens of thousands of years.
Men within the time of the story peaked at the end of the Second Age, after at least thousands of years. Well, if they lived in Numenor or its colonies. For most Men, they were better off in the Third Age. Also don't forget that fairly quickly after their awakening, all Men became Morgoth-worshippers, so pretty much all the Mannish populations we actually see are "greater" than they were in their beginning in the moral sense.
The Two Trees themselves, if we regard that as Peak Valar, did not come until after ages and ages of ceaseless labor.
So next, we need to consider that the Ainur are immensely powerful, and their main purpose was carrying out the shaping and forming of the world. Many great works could, obviously, only be done by these lower-g gods, or angels, or whatever you want to call them. But, as time goes on, the Valar are more and more hindered by the very nature of their power from acting directly, because it's so Big. This is why the Valar seem continually hesitant to act directly in Middle-Earth after the awakening of the Elves: they can't do so without causing massive destruction. Even in the War of Wrath, the Valar themselves don't come, but Beleirand is still destroyed. In tandem with this, they "know" their role in the world, and won't act unless it aligns with the will of Eru.
The Valar 'fade' and become more impotent, precisely in proportion as the shape and constitution of things becomes more defined and settled. The longer the Past, the more nearly defined the Future, and the less room for important change (untrammelled action, on a physical plane, that is not destructive in purpose). The Past, once 'achieved', has become part of the 'Music in being'. Only Eru may or can alter the 'Music'. The last major effort, of this demiurgic kind, made by the Valar was the lifting up of the range of the Pelori to a great height. [...] The Valar were like architects working with a plan 'passed' by the Government. They became less and less important (structurally!) as the plan was more and more nearly achieved. (HoME, Myths Transformed)
This of course doesn't hinder Morgoth, who doesn't mind destruction. Fortunately at the end of the First Age, he was executed and his spirit ejected from Arda, so until he "rebuilds", we're safe from Valaric action.
So, clearly, any doing or making that requires demiurgic levels of power won't happen anymore, but it's not because the world is in a decline, it's because the structure of the world is more-or-less complete and can play out without them. So let's now consider the Elves.
We need to remember that the Elves are in some ways "better" (or at least different) than Men in terms of creative potential. They have much more variance spiritually, and unlike Men, they can "invest" themselves into what they make. (Tolkien waffled on this last point a little.) But at the end of the First Age, the Elves have already risen and fallen (in Middle-Earth, not necessarily Aman), and it was part of their Purpose in the world to usher in the time of Men.
The doom of the Elves is to be immortal, to love the beauty of the world, to bring it to full flower with their gifts of delicacy and perfection, to last while it lasts, never leaving it even when 'slain', but returning – and yet, when the Followers come, to teach them, and make way for them, to 'fade' as the Followers grow and absorb the life from which both proceed. [...] Men came in [the story[ inevitably: after all the author is a man, and if he has an audience they will be Men and Men must come in to our tales, as such, and not merely transfigured or partially represented as Elves, Dwarfs, Hobbits, etc. But they remain peripheral - late comers, and however growingly important, not principals. (Letters)
So the story is written at a time of transition: but we see anyway that as the world is architected in such a way that Men take over, and Elves leave or fade. So any works done by the Elves in Middle-Earth will cease. Naturally, it will appear to the Elf-Friends and the Elves that this end of Elves in Middle-Earth is a "decline" - but we know that in many respects that Men rose much higher in later Ages than they did in the Third Age.
Secondly, then, you might ask: why no more Silmarils? Why no more Rings? Why no more Trees? The answer is because when something really great or powerful is made, the creator puts part of themselves into it. You could say of this, as Yavanna did, that:
Even for those who are mightiest under Ilúvatar there is some work that they may accomplish once, and once only. The Light of the Trees I brought into being, and within Eä I can do so never again. (The Silmarillion)
or as Feanor said:
But Fëanor spoke then, and cried bitterly: Tor the less even as for the greater there is some deed that he may accomplish but once only; and in that deed his heart shall rest. It may be that I can unlock my jewels, but never again shall I make their like; and if I must break them, I shall break my heart, and I shall be slain; first of all the Eldar in Aman.’ (The Silmarillion)
or as said of the Ring:
And much [in LotR a similar passage says: the best part, e.g., most] of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow. (The Silmarillion)
Well, in your question, you mention the "power to duplicate them no longer exists." But the power to duplicate them never existed, by their makers. Yavanna could not have made four trees instead of two. Feanor could not have made unlimited Silmarils while the Trees shone. And Sauron could not have made multiple One Rings. Their power was bound in them, and could not be re-spent. The power of the Trees was transmuted into empowering Ungoliant, and in the Sun and the Moon - and while maybe the Trees were more beautiful, the Sun and the Moon were a lot more powerful and gave a lot more benefit to the whole world. Is that really a decline for anyone but the Elves and Yavanna?
The Trees required demiurgic power to make, and likely no other had the power and "attunement" to make anything like them. But the same is probably not true for the Silmarils or the One Ring. These were not made solely out of will and power of archangelic powers but:
Then [Feanor[ began a long and secret labour, and he summoned all his lore, and his power, and his subtle skill; and at the end of all he made the Silmarils.
Feanor was jealous and proud and never taught anyone else all his secret lore and skill. It is conceivable that, if he had, some other Elf (or, given Feanor's innate strength, one of the Maiar may have been required) could have made Silmarils, while the Trees lasted. But it may well be the case that the Elves in Aman continued to make wonderful thngs their lore and skill evolved and we just don't know about it - Tolkien remarks that the great talent of the Elves was for "making and for discovery". Similarly, it may well be that someone could make a One Ring, if they learned all the Ring-lore necessary - Saruman certainly attempted it, and did not think it impossible. Indeed, Tolkien tells us if the narrative had been inspired by WWII, then Saruman would have ended up in Mordor, where:
Saruman, failing to get possession of the Ring, would in the confusion and treacheries of the time have found in Mordor the missing links in his own researches into Ring-lore, and before long he would have made a Great Ring of his own with which to challenge the self-styled Ruler of Middle-earth. (LotR, foreword)
Of course, it's doubtful Saruman's Ring would have been quite up to par with the One Ring - perhaps someone like Eonwe or a Vala would have had to make it. But it's not impossible, and we can see that nothing stopped the creation of more Great Rings except historical contingencies, not inevitable decline.
At any rate, Men continued to develop and advance in fits and starts. Although Tolkien would no doubt regard this comparison as very crude, and it is, the Elves never had nukes.
Finally, let's talk demographics. You point out the Hobbits and the Dwarves did not thrive, at least into our time, making the comment that "in our world, if there are no predators, the prey will always flourish". But that isn't true at all. Humans have no predators, and yet the total fertility rate in many countries is below what's required to maintain the population As the Wikipedia article notes, this is not a new phenomenon, it was also observed in ancient Greece and Rome.
First, we should mention that for dwarves, they have an unusual gender ratio, and most never have children:
It is because of the fewness of women among them that the kind of the Dwarves increases slowly, and is in peril when they have no secure dwellings. For Dwarves take only one wife or husband each in their lives, and are jealous, as in all matters of their rights. The number of dwarf-men that marry is actually less than one-third. For not all the women take husbands: some desire none; some desire one that they cannot get, and so will have no other. As for the men, very many also do not desire marriage, being engrossed in their crafts. (Lord of the Rings, Appendix A)
The real wonder is that the Dwarves were able to maintain any population until the Fourth Age at all! (For comparison, consider that women are approximately half of our population, and 85-90% of women have children by the end of childbearing age, even now. For Tolkien only the married have children, so this means that at best dwarves have only one-third, proportionally, the reproductive population as humans.) We know there must not be very many dwarven settlements, because Tolkien also notes they are very loathe to migrate, and we know that the Dwarves were involved in many very costly and ruinous wars, and in the Third Age had lost Moria and the Kingdom-under-the-Mountain. You could easily see how the War of the Orcs and Dwarves, under these conditions, may have set off a terminal population decline.
Durin's Folk gathered all their host, and they were joined by great forces sent from the Houses of other Fathers; for this dishonour to the heir of the Eldest of their race filled them with wrath. [...] So began the Battle of Azanulbizar (or Nan-duhirion in the Elvish tongue), at the memory of which the Orcs still shudder and Dwarves weep. [...] They took the head of Azog and thrust into its mouth the purse of small money, and then they set it on a stake. But no feast nor song was there that night; for their dead were beyond the count of grief. Barely half of their number, it is said, could still stand or had hope of healing. (LotR, Appendix A)
Of course, it needn't have been that or then that the Dwarves began terminally declining into extinction: later disasters may have been the cause. Disease and wars did not end with Sauron. No doubt the dwarves warred against Men.
Hobbits are still around, simply less numerous:
Hobbits are an unobtrusive but very ancient people, more numerous formerly than they are today; (LotR, Prologue)
Given the multiplication of Big People, aside from the possibility of demographic decline on its own, there's no mystery why Hobbits are skittish of Big People. No doubt many of them would kill or enslave them.
Finally, lets remember that there is still magic in the world today: the most powerful magical artifact made was Morgoth's Ring: Morgoth infused his power and magic into all matter, influencing and corrupting the world and the people in it to this day:
Melkor 'incarnated' himself (as Morgoth) permanently. He did this so as to control the hroa,2 the 'flesh' or physical matter, of Arda. He attempted to identify himself with it. A vaster, and more perilous, procedure, though of similar sort to the operations of Sauron with the Rings. Thus, outside the Blessed Realm, all 'matter' was likely to have a 'Melkor ingredient',3 and those who had bodies, nourished by the hroa of Arda, had as it were a tendency, small or great, towards Melkor: they were none of them wholly free of him in their incarnate form, and their bodies had an effect upon their spirits. [...]
Morgoth's vast power was disseminated. The whole of 'Middle-earth' was Morgoth's Ring, though temporarily his attention was mainly upon the North-west. [..]
No such eradication of Morgoth was possible, since this required the complete disintegration of the 'matter' of Arda. Sauron's power was not (for example) in gold as such, but in a particular form or shape made of a particular portion of total gold. Morgoth's power was disseminated throughout Gold, if nowhere absolute (for he did not create Gold) it was nowhere absent. (It was this Morgoth-element in matter, indeed, which was a prerequisite for such 'magic' and other evils as Sauron practised with it and upon it.) (HoME, Morgoth's Ring)
There is no reason to preclude the possibility that even today, there are secret Sauron-cults practicing some kind of "magic".
Lastly, let's not forget that the world ends in the utter end of everything evil and bad, the eternal triumph of Eru, and Arda Healed.
