There are several things which need to be considered:
- Human impacts on ecosystem: Humans alter ecosystems in tremendous ways. In your case, the smaller the island and the larger the population, the higher the impact. Impacts can include:
a) Tree logging: Generally, it can be assumed that trees (if present) will be gone pretty fast. Timber has always been used excessively due the its importance for so many different things (house/ship etc. building, firewood). One example are the Easter Islands (https://en.wikipedia.org/wiki/Easter_Island). This being said, all animals which depend on forest/trees will be gone probably before the last tree has been logged. This includes for example tree nesting birds, many insects etc.
b) Hunting: Hunting is a big issue, especially if the need of the human population for food exceeds the reproductive capability of the animal populations. Usually, predators will be gone first as they compete for food sources, may often be harmful to humans and usually have the lowest population size. They may also be negatively impacted by the decreasing numbers of prey available (see next sentence). After that, animals which are slow, naive (not afraid of humans), large and tasty will be gone next. Examples are the Moa , Dodo and some large tortoise species found on the Galapagos Islands. When those are gone, humans will start hunting for smaller, less nutritious animals (again, see Easter Islands)
c) Invasive species: Humans usually bring along (knowingly or unknowingly) species which might impact the local flora and fauna. The best example are rats, which often have devastating impacts, such as the impact on the Kakapo. Similar impacts can be expected by introduced pigs, cats and snakes; the brown tree snake on Guam is a good example (https://en.wikipedia.org/wiki/Guam). Both are devastating bird and small mammal (also small reptile) populations.
d) Agriculture: Agriculture leads to the loss of natural habitats and therefore the loss of flora and fauna. The extend and intensity of the agriculture will decide how severe the impact will be. Of course, the larger the human population gets, the larger the impacts will be. If, for example, crops which are mainly wind pollinated, such as wheat or barley, replace natural habitats, this will have a strong negative impact on pollinator populations.
- Island biogeography: Island biogeography tells you how species rich your island was before humans arrived and to which degree organisms will be able to recolonize the island. Also, read metapopulation theory in this context.
a) Distance from mainland: The further away your island is from the mainland the less species can be found on it.
b) Size of Island: The larger your island, the more species will be found on it (species-area-curve).
c) Age of Island: The longer the island existed the more species are likely to be found on it (even though this might not be 100% true). The idea is, the longer the island existed, the more time organisms had to arrive on that island and establish a population and/or diversify (evolve into new and different species. One example is the island dwarfism, which has already been mentioned. But I do not consider this an issue in your case). There is also a specific order: Usually plants are the first organisms to establish populations, followed by herbivores followed by predators. The higher the trophic level, the later the organism will establish a population. Seabirds, however, might use islands as breeding grounds in the meantime.
- Inbreeding: Inbreeding and the already mentioned bottlenecks might negatively impact populations in the long term. However, I do not consider these to be to severe as there are many counter examples where organisms still survive besides bottlenecks in the past (e.g., Californian condor, cheetah). Other scientists might disagree on this point, but this is my personal opinion. Read about metapopulation theory in this context.
Further things to consider are e.g. pollution and topography, e.g. habitats at higher elevations usually stay pristine for a longer period of time as they are less accessible.
So as a short prediction from my side: Large vertebrate animals will be gone soon, followed by smaller vertebrate animals. The only "larger" vertebrate animals which will be present in the long run are rats, if they have been introduced.
P.S. Sorry I had to remove most of the links as I am only allowed to post two links with my reputation. But I am sure you can find the examples and theories on wikipedia.