**December 12, 2017**

**Lowell Fritz**

**Biologist**

We permanently mark Steller sea lions to estimate **vital rates** of the population, which are:

**Survival**(from year to year)**Reproduction**(how often females give birth to a pup)**Dispersal**(where marked sea lions are observed at each age)

Why is estimating **vital rates** important?

By seeing marked animals through time, we can determine which vital rate is most likely responsible for this decline.

For a population that’s declining, like Steller sea lions in the Aleutian Islands, estimating **survival**, **reproduction** and **dispersal** can help us determine what factors might be affecting the population. For instance, we know Steller sea lions in the western Aleutian Islands are declining at an alarming rate of about 7% per year. If they continue to decline at this rate, they could be go extinct in this region within the next 50 years. Which is why we need your help to classify images on Steller Watch.

Because the number of Steller sea lions (or abundance) is going down in the western Aleutian Islands we know that either they are dying faster than new pups are being born or they are abandoning this area and settling elsewhere.

By seeing marked animals through time, we can determine which vital rate is most likely responsible for this decline. For example, suppose we discover that **survival** during the first 2 years in the western Aleutians is similar to areas where the species is currently increasing. This would suggest that factors that directly kill young sea lions, such as entanglement in fishing nets or predation by killer whales, are likely not affecting the western Aleutian population any more than in parts of the range where the population is increasing. If we knew this, then we could focus our research and management attention on other pieces of the puzzle, such as factors that would affect **reproduction** (e.g., disease, nutritional stress) and adult **survival **(e.g., illegal shooting). In addition, because we know a lot about each of the pups that were marked, we can determine whether males and females are affected differently, or whether the weight of the pup (which is an indication of the health and age of its mother) was a factor.

We began marking Steller sea lion pups in the western Aleutian Islands in 2011, so as of December 2017, the oldest marked animals from this region are only about 6½ years old. Given that female Steller sea lions can live to be about 30 years old and don’t start having pups until they are 4-6 years old, this means we don’t yet have enough years of sightings to estimate **reproduction** or adult** survival**. However, we are closer to being able to estimate juvenile** survival**.

I’m going to provide a short introduction into how we estimate **survival**, in this case, of juveniles. This will get a little messy and into the muddy math so skip ahead to the last two paragraphs if you want to skip this part. To set the stage, let’s look at a table of a simplified version of our experiment with ‘pretend’ data:

Imagine we marked 100 pups in 2011 and set them free. In each of the following years, there are really only 2 options for us as researchers: we either see them alive that year or we don’t. Let’s say in the 2nd year (in 2012) we observed only 50 of these 100 marked individuals. In the 3^{rd} year (2013), we only saw 30. We want to try to estimate the percent of animals that survived to the 2^{nd} and 3^{rd} years (and beyond) which we call **survival**. In the most simplistic terms, **survival** is 50% to year 2 and 30% to year 3.

But it’s not that simple! What complicates this is **sighting probability**, or the chance that we will actually observe a live marked animal. While we have remote cameras at several locations and visit the Aleutian Islands at least once a year, we know that we do not see every single marked sea lion that is alive in the population. This means we have to account for the probability of observing a live marked animal, and how that might change over time, for instance, as the animals age or with different levels of sighting effort. Another reason we might not see a marked animal is that it completely left our study area never to be seen by us again. We collaborate with researchers in Russia and look for marked animals in other parts of Alaska, but we still try to account for this possibility, however slim. For these reasons, we use the term “apparent” **survival** to describe what we are actually estimating since we can’t distinguish death from permanent emigration. But for this blog, we’ll just call it **survival**.

So, how do we account for **sighting probability** and how it might vary between years so we can estimate **survival**? This is where some math comes into play and why collecting data over many years is so valuable.

The table to the right is what we call a **capture history** of how many marked sea lions were seen (Y) or not seen (N) in Year 2. Of course, all of the 100 sea lions marked in 2011 were “seen” in the first year, which is why they have a “Y” listed for the first year. Then in year 2 (2012) there were 50 marked sea lions seen so their capture history is “YY”, and the other 50 were not seen which means their capture history is “YN”.

Pretty simple for year 2, right? They were either seen or not seen.

Let’s add sightings collected during Year 3 (2013), and you can see that this is when it starts to get complicated. In the first table, you can see we saw only 30 marked animals in Year 3. Of those 30 marked animals seen, 10 were seen all three years so they have a capture history of YYY. The other 20 were not observed in year 2, so their capture history is: YNY.

Seventy of the original 100 marked sea lions were not observed in year 3 but 10 of these were seen in year 2 so they have a capture history of YYN. That leaves the remaining 60 who were not seen in year 2 and 3, and these have a capture history of YNN.

How is this sighting data by year used to estimate **sighting probability** (**P**) and **survival** (**S**) in years 2 and 3? We use a mathematical model that finds the values of **P** and **S** that best fit the following equations. Let’s start from the top by examining the number of sea lions that had each type of capture history in year 3 and equations that express the probabilities for each one.

In our data, 10% of the original marked group of 100 pups has a capture history of “YYY” in year 3. This can also be expressed as:

Pr[YYY] = 0.1 = [**P2** * **S2**] * [**P3** * **S3**]

Our data indicate that the probabilities of both being seen (**P2**) and surviving (**S2**) to year 2 multiplied by the probabilities of both being seen (**P3**) and surviving (**S3**) to year 3 is equal to 0.1 or 10%.

That tells us a little bit but not too much about the individual values of each of the 4 parameters. Some more information will come from examining the equations associated with the other capture histories.

We not only have **sighting probability** and **survival** in our model, but we also have their opposites: the probability of NOT surviving (or dying) and of NOT being seen. Let’s say that we estimated that **S** = 0.6 for a particular year. The opposite of that, or the probability that an animal did NOT survive that year, would be (1 – **S**) = 0.4. In other words, if an animal had a 60% chance of surviving, it also had a 40% chance of dying. Similarly, if a marked animal had a 70% chance of being observed (P = 0.7), it also had a (1 – P) = 0.3, or 30% chance of NOT being observed. So for the **capture history** of “YNY” we would use the equation below:

Pr[YNY] = 0.2 = [(1-**P2**) * **S2**] * [**P3** * **S3**]

For these 20 animals, we know they survived through year 2 because they were observed alive in year 3. Therefore, during year 2, the probability of being NOT seen (1-**P2**) is multiplied by the probability of surviving (**S2**), while for year 3, the terms are exactly the same as for the animals with capture histories of “YYY” since they were seen alive in year 3.

Pr[YYN] = 0.1 = [**P2** * **S2**] * [(1-**S3**) + (**S3** * (1-**P3**))]

OK, now it’s starting to look ugly, right? Let’s just break it down term by term. Since these 10 animals were all seen alive in Year 2, the equation has the same terms for year 2 as the “YYY”s. But year 3 is where it really starts to change, and this is because we don’t know if they didn’t survive to year 3 or they were alive but just not observed that year. Data obtained in year 4 and beyond will help us untangle this, but at this point in the analysis of these example data, we do not know. Therefore, the year 3 term takes into account both possibilities: the probability that these 10 animals did NOT survive to year 3 (1-**S3**) and the probability that they survived to year 3 (**S3**) but were NOT observed (1-**P3**).

And now the messiest of all is the equation for the probability of having a capture history of “YNN”.

Pr[Y N N] = 0.6 = [(1-**S2**) + (**S2** * (1-**P2**))] * [(1-**S3**) + (**S3** * (1-**P3**))]

These 60 animals were marked in year 1 and never seen again, but we don’t know if they survived to year 3 (**S2** and **S3**) but were just not observed either year [(1 – **P2**) and (1 – **P3**)]; if they didn’t survive to year 2 (1 – **S2**) and were not available to be seen in year 3; or if they survived to year 2 (**S2**) and were not observed (1 – **P2**) and then died in year 3 (1 – **S3**).

Without going into the gory detail, finding the values of **sighting probability** (**P**) and **survival** (**S**) for each year that best fit the data is quite a process, and luckily there’s a program called MARK that performs this task (and many more!) with remarkable speed.

For this example, **survival** to year 2 (**S2**) is estimated to be 0.82. In other words, we estimate that 82% of the marked sea lion pups survived to celebrate their first birthday. **Sighting probability** during year 2 (**P2**) was estimated to be pretty low, only 0.19. In other words, there was a 19% chance of seeing a marked animal during year 2. You can see how adding sighting probability significantly changed our perception of survival, given that our first ‘guess’ for survival during year 2 was 50% when we only considered how many we actually saw alive in year 2. At this point in the data collection, **P3** and **S3** are not estimable with much precision because it is the last year of data in the analysis and we don’t have enough information to know whether a marked animal that was not seen in year 3 was alive or not. For each additional year of sightings, the number of years for which survival can be estimated usually increases, and the number of unique capture histories doubles. So you can see that the equations expressing the probabilities get very complicated very quickly! Add some other variables (also called co-variates) to the mix, such as sex, cohort (different island rookeries, different birth years), and weight at the time of marking, and you’ve got yourself quite a sophisticated model.

And that’s Survival 101!

*I have been studying Steller sea lions since 1990 with NOAA Fisheries Alaska Fisheries Science Center in Seattle. My primary research interests are sea lion population dynamics, demographics, and interactions with commercial fisheries. I’ve also worked on fish during my career with NOAA, particularly species eaten by sea lions, like Atka mackerel, walleye pollock (you may know them as fish sticks and imitation “krab”), and Pacific cod. I graduated from Bucknell University (B.A. Biology, 1976) and College of William and Mary (M.S. Marine Science, 1982), and started my science career in 1982 at Rutgers University as a Research Associate. At Rutgers, I worked at the Haskin Shellfish Research Laboratory in Bivalve, NJ (down the road from Shellpile… you can’t make this up) studying the shells of mollusks living in habitats ranging from freshwater lakes and streams to deep-sea hydrothermal vents. I even had the opportunity to go down in the Alvin submersible!*