How do I identify the fermentation process in a wild ferment? TL;DR - How do I go about to identify the microbiological processes in a wild ferment?
Background
I have a recipe for rose hip wine based on wild (spontaneous) fermentation, that I got from a Swedish wine/beer makers forum. The recipe is simple (I've translated and summarized it a bit to make it fit better in the Q&amp;A forum style). I've used the recipe above five times now with good results. The wine becomes very sweet and it has a quite low alcoholic content. It has a strong character of dried fruit and rose hips.
The recipe used

Take 2.5 liters of rose hips. Pick off bad parts but don't clean them with water or desinfect them in any way - make sure the wild microbes are kept!
Make a syrup of 3 liters (0.66 gal) water and 1.5 kg (2.2lbs) white sugar.
Add rose hips and syrup to a big ceramic jar with a plate as a lid (optionally use a proper fermentation lock). Let ferment like this for 3-6 months.
Strain and age on a carboy for 3 years. The taste is terrible initially, quite okay after 1 year, good after 2, excellent after 3.

This differ from most recipes online only on a few parts. (a) this is the only recipe with wild fermentation that I've found (b) the amount of water in other recipes is rather 7-10 liters for the same amount of hips and sugar (c) the first fermentation is 3-6 months instead of the typical 1-3 weeks seen in inocculated wine making (d) other recipes typically have some acid introduced in the beginning. lemons or lemon juince and possible lactic acid.
This is similar to other recipes online in the proportion of hips to sugar, and the recommendation to age 3 years is the same.
The current ferment
This year is the first that I use a fermentation lock. I altered the recipe slightly by admitting oxygen freely the first week, stirring occasionally to get the ferment going. After that, I put the lock on.
The fermentation is slow, but carbon dioxide is released continously. A bubble every minute or so, and has now been going so for 3 weeks. The specific gravity is still very high, so there is sugars in abundance. The smell from the bubbles is a little yeasty. The smell is also a little sour, but not at all like vinager. This is the typical smell this ferment makes!
I've read online that rose hips are high in malic acid, so I guess the long aging is for malolactic fermentation to take place.
The question
I would like to know what fermentation is currently going. Is it yeast or LAB? Could there be other carbon dioxide releasing processes of significance?
This very slow fermentation indicates to me that there might be not enough nutrition to sustain a larger yeast population. It would be nice to do some kind of before/after test if I would ad nutrition to see what effect that has.
I have pH measurements and gravity measurements currently. I consider getting other equipment for up ca $200, such as a microscope.
How does one go about to analyze the fermentation process at this hobby level?
In addition to @kitukwfyer's answer. There's quite a bit of biology that goes on here...
Not only are you influencing the competition, you are actively selecting for organisms that grow well under the conditions. This is the real reason brewing (and bread etc) works better with a starter culture. There are a few things influencing this:

With a starter culture you are putting in a high number of organisms, which results in a shorter lag phase. Wild ferments rely on a relatively low abundance of naturally occurring organisms - so have a long lag.
With a starter culture you are also adding organisms that are already at least semi-adapted to the type of culture, so they will grow better.
It takes a while for the wild organisms that are there to actually adapt to the conditions. These are related to abundance of water, salt(s), nutrients, oxygen, CO2, pH etc; all of which may be on the boundaries of what is tolerable for the organisms that are there. Cultured organisms are at least partially selected for these characteristics already and selected against ones you don't want, indeed that is actually part of the reason people add things like acid to fermentations - it inhibits gut microflora like Escherichia coli that can make you quite ill.
In the ferment you are selecting for strains that grow well under the conditions; this will be a small proportion of those that are there, so it results in a very small inoculum, and a long lag phase.
Finally, wild organisms often have slow growth rates compared to cultured ones, because we generally want things quickly, so we select for strains that grow faster under the conditions we want - you probably did this subconsciously a couple of times already - you didn't see any signs of fermentation in a brew, so you threw it out and made a fresh batch; or something along those lines.

As to how you can identify the organisms - your best bet would be to get or make a bunch of bacterial (aim for lactic-acid fermentation ones here) and yeast plates (use some filter sterile fermentation broth as the liquid base if possible) and start streaking out your cultures. You most likely won't be able to culture all the organisms in your ferment, but you probably can culture those with the highest abundance (you already know they grow well under some conditions). You can then pick off individual colonies from our plates and grow them in liquid culture to see which have characteristics (flavour profiles) you like by themselves or in combination (try smelling them before drinking...). This is not a trivial task (list of techniques you might need) and relies on a bit of good sterile technique and a knowledge of what a &quot;normal&quot; organism looks like, not a contaminant from your own microflora.