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Most countries have a list of endangered species and international organisations too issue such lists of vulnerable and endangered species.

What mathematical or statistical models are those estimates based on?

Regardless of which techniques the lists use, what are some of the most popular mathematical models to find out whether a certain species is endangered or vulnerable given the population and/or other environmental data in the academic literature? What is the state of the art currently?

Especially, if I want to learn the models at an intermediate and then advanced level (I already have training as an Economics student on mathematical and econometric modelling), what are some books/ other sources you would recommend?

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Based on the IUCN Criteria for endangered status there are two broad categories of models to help in determining endangered status. Note that no one methodology is used to determine the status of any species. Rather, a panel of experts consider all available evidence (including numerous modelling studies) and essentially make a determination based on their own judgement.

  1. Range Estimates
    Estimating the range of a species uses two primary techniques. The first is an expert range maps drawn by people very familiar with the species distribution, which involves essentially no modelling. The 2nd is ecological niche modelling, aka. species distribution modelling. See 1 and 2 for introductions on the latter. Note that the IUCN guidelines have criteria based on total range size, and well as the amount of fragmentation and patchiness of the range.

  2. Population Size Estimates
    Since it is usually impossible to count all individuals of a species, models are used to estimate total populations. These usually fall in the realm of mark-recapture models. There are numerous variations on the basic model form which account for things like an open population, varying birth and death rates, and varying capture probabilities.

1: Elith, Jane, and John R. Leathwick. "Species distribution models: ecological explanation and prediction across space and time." Annual review of ecology, evolution, and systematics 40 (2009): 677-697. https://doi.org/10.1146/annurev.ecolsys.110308.120159
2: Araújo, Miguel B., et al. "Standards for distribution models in biodiversity assessments." Science Advances 5.1 (2019): https://doi.org/10.1126/sciadv.aat4858.

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  • $\begingroup$ Thanks a lot for your answer, it was really insightful! If it's not an inconvenience, I also wanted to know if panel data on the population estimates are or can be used to predict when would a species likely go extinct in a region. I have heard of predator-prey models in game theory, but have no idea if or how any such forecasting is done empirically. Also if any other techniques are used for forecasting $\endgroup$ – Ishan Kashyap Hazarika 18 hours ago
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Wikipedia has an article on the IUCN Red List Categories, There are some formulae there that show how the category is determined.

For example: https://en.m.wikipedia.org/wiki/Endangered_species_(IUCN_status)

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