# Mathematics identify species extinction risks

It’s now possible to predict how likely an endangered species is to go extinct, with mathematical models acting as windows into the future. These models help scientists foresee how a population is likely to react to changes in the environment and therefore how likely it is to die out.

As the sixth mass extinction event rumbles on, this represents a powerful tool in the arsenal of conservationists. However, accurate information about species in the wild is still crucial to inform these models and divine the fate of wildlife populations.

A species’ risk of extinction depends on how many individuals in its populations can reproduce and how long they can survive. To design management programmes that can prevent extinction, it’s essential to understand how survival and reproductive rates change within a population as the environment changes.

We know that the number of individuals surviving and reproducing – known collectively as a species’ demographic rates – vary each year in response to environmental conditions. These include the availability of food and water, rainfall or extreme temperatures.

As global temperatures increase and disrupt local weather, how variable these environmental conditions become will increase the extinction risk for many species.

To predict if a population is likely to go extinct, we need to predict how changes in these environmental conditions will affect the population’s demographic rates and how the number of individuals in the population will change annually.

Models can take the likelihood of individuals in the population to survive and reproduce at different ages, and “shock” them in the same way the environment does.

It would be impossible to directly model how species respond to changes in rainfall or other conditions in the environment, as different species thrive under different conditions.

However, these “shocks” in the model – which disturb a species’ survival and reproductive rates – reproduce the random nature of the environment. By running the model many times to generate multiple outcomes, we can calculate what percentage of the populations will go extinct as the environment changes, and how long it will take them to die out.

## How models are getting smarter

In our recent study we found that simplifying the information used to construct these models can distort the predictions. Survival and reproductive rates vary in individuals as they age and each species has a very particular trend associated with their age.

Until recently, researchers and wildlife managers assumed it was enough to represent these rates as constant as individuals age. This would assume that juveniles and adults all respond uniformly to changes in the environment.

However, in our study we showed that reducing a population into general age classes can greatly distort the population’s predicted growth rate and our understanding of a population’s chances of avoiding extinction.

Grouping individuals into general age classes can give the illusion that a population’s chances of extinction are slim when in reality, it will fast go extinct.

We also found that it’s important to consider trade-offs between survival and reproduction. A good year for reproduction may result in a bad year for survival. This is because more energy invested in reproducing means there’s less left over for the healthy upkeep of the body. Despite increasing evidence of these trade-offs in nature, models tend to ignore them.

We showed that ignoring these seemingly minor issues can harm a model’s accuracy in predicting the fate of populations in their natural environment. This reduces their capacity to anticipate the impacts of climate change, invasive species or habitat loss.

There is still hope for many endangered species, but preparation needs to be made. For this, we need to continue developing more sophisticated models with more accurate information about each species and their environment. With reliable foresight, we can give them the fighting chance they need in an uncertain future.

Fernando Colchero, Associate Professor in Mathematics and Computer Science, University of Southern Denmark and Dalia A. Conde, Associate Professor of Biology, University of Southern Denmark