From Ecological Model to Conservation Action

From Ecological Model to Conservation Action
Lead Poisoning of Laysan Albatrosses at Midway Atoll: From Ecological Model to Conservation Action Today we’re going to discuss how we use mathematical models in ecology. We’ll focus on the case study of lead poisoning of Laysan albatrosses at Midway Atoll. ( 2013 Kristin McCully & Myra Finkelstein)

Objectives Learn how to use structured population models to assess and protect populations Appreciate the importance and uses of mathematical models in ecology and conservation Our objectives for today are to: Learn how scientists use structured population models to assess and protect populations Appreciate the importance and many uses of mathematical models in ecology and conservation

Outline Introduction to case study: Laysan albatrosses at Midway Atoll
Introduction to mathematical models Structured population models Assumptions Model structure Mathematical model: Importance of models to ecology and conservation First, I’ll introduce the case study. Then we’ll discuss mathematical models and specifically structured, or matrix, population models. Then we’ll work through the mathematical model of Laysan albatrosses. We’ll end with a discussion how mathematical models are used by and important to both ecology and conservation.

Introduction to mathematical models Structured population models Assumptions Model structure Mathematical model: Importance of models to ecology and conservation I’ll begin by introducing the albatross case study.

Case Study: Lead Poisoning of Laysan Albatrosses at Midway Atoll
Research by Myra Finkelstein In this lecture, we discuss conceptual and mathematical models by focusing on the case study of lead poisoning of Laysan albatrosses at Midway Atoll. This research was conducted by Myra Finkelstein during her Ph.D. dissertation at UC Santa Cruz. She is now an environmental toxicology researcher at UCSC. This lecture focuses on these two papers she published.

Threats to Albatross Populations
Longline fishing bycatch Introduced mammals Marine debris Lead poisoning Because albatrosses live long lives and raise at most one chick per year, their populations are very susceptible to human disturbances. Although Laysan albatrosses are only ranked “Near Threatened” on the IUCN Red List, other albatrosses are more threatened or even endangered. For example, the Does anyone know of any threats to albatross populations? One of the biggest threats to albatrosses is longline fishing, which kills thousands of albatrosses and other seabirds as bycatch each year when they dive for bait, get dragged underwater, and drown. In populated areas such as the island of Kauai, feral dogs often kill albatrosses and rats often attack and eat their eggs and chicks. A recent concern for albatrosses is marine debris because they ingest food from the surface of the ocean, which is often small pieces of plastic, and then often die of starvation or dehydration when their stomachs fill with plastic. Here you see the carcass of a chick whose stomach was full of plastic marine debris. Finally, as we’ll discuss today, chicks also get lead poisoning from old buildings at Midway. We’ll use a mathematical model to determine how much lead poisoning is impacting the population and whether managers should clean up the lead paint. Brenda Zaun/USFWS

Where is Midway Atoll? Papahānaumokuākea Marine National Monument
Laysan albatrosses fly all over the northern Pacific ocean to feed, but breed on only a few islands of Hawaii and off Mexico. The largest breeding colony, with over 70% of the world’s population, is at Midway Atoll in the Northwestern Hawaiian Islands. Midway Atoll is at the northern end of the Hawaiian Archipelago and is part of the Papahanaumokuakea Marine National Monument. Midway Atoll

Midway Atoll Keith Castellano
You may have heard of Midway because it was a naval base through World War II and the Cold War from 1940 to It was attacked both on Dec (Pearl Harbor Day) and during the Battle of Midway (June ). Now it’s a national wildlife refuge and emergency airport for planes flying over the Pacific.

Wisdom: World’s Oldest- Known Wild Bird
A Laysan albatross known as “Wisdom” – believed to be at least 62 years old – is the world’s oldest-known wild bird and hatched a chick on Midway Atoll National Wildlife Refuge for the sixth consecutive year (in 2013). She was at least 5 years old when she was originally banded in 1956 and has probably raised at least 30 chicks in her lifetime! [Check the FWS Midway website ( to determine if she’s still breeding and/or alive. If so, search online for news articles – she’s often mentioned in newspapers every year when she hatches a chick.] Check her out on Facebook: Pete Leary/USFWS

Peripheral Neuropathy
“Droopwing” syndrome Peripheral Neuropathy When Myra Finkelstein first went to Midway around 2000, she observed albatross chicks with paralyzed wings like this one around buildings. This problem is known as droopwing syndrome. The medical term is peripheral neuropathy, which refers to damage to the peripheral nerves which run between the spinal cord and the rest of the body. This can be due to many factors, including diabetes, traumatic injuries, and exposure to toxins. Chicks with droopwing will never fly and as a result will die either at the end of the breeding season (when their parents stop feeding them) or sooner from other complications. National Institutes of Health

Problem Lead poisoning of Laysan chicks near buildings
- Sileo and Fefer (1987) J Wildlife Disease - Sileo et al. (1990) J Wildlife Disease - Work et al (1996) Env Sci & Toxicology She searched the scientific literature and found that previous studies at Midway showed that lead poisoning caused droopwing syndrome, but there was little recent data and no evidence for the source of the poisoning. Because chicks with droopwing syndrome were most common near buildings with lead paint and adults didn’t exhibit symptoms, she hypothesized that the chicks were ingesting lead paint chips. Military base → many sources of lead Need evidence to link source with poisoning

> 80% of chicks near bldgs. >100 ug/dL
Step 1: Quantify Lead Exposure 1600 > 80% of chicks near bldgs. >100 ug/dL 1400 1200 1000 Lead (μg/dL whole blood) 800 Droopwing or other symptoms 600 400 First, Myra quantified blood lead levels from albatrosses (adults, chicks with droopwing or not near and away from the buildings with lead paint. Blood lead concentrations between groups were significantly different from each other. [These box plots show the median (horizontal line within the box), 25th and 75th percentiles (lower and upper margin of the box), and 10th and 90th percentiles (lower and upper hash marks) as well as outliers. Blood lead concentrations between groups were significantly different from each other (one-way ANOVA, F3,42 )130 on log-transformed data,p<0.001; Tukey,p<0.001 for all comparisons). Similar to Figure 2 from Finkelstein et al. (2003) Environmental Science & Technology] 200 Away from bldgs n =15 Near bldgs n = 21 Finkelstein et al. (2003) Environmental Science & Technology

Step 2: Identify Source of Lead
Fingerprint source using isotope analysis Isotopes = variants of chemical element with different numbers of neutrons Lead exists as four isotopes: 204Pb, 206Pb, 207Pb, 208Pb Lead isotope ratios in organism reflects lead isotope ratios in exposure source 3 isotopes of hydrogen: Next, Myra decided to collect evidence to identify the source of the lead, which could be the paint chips from the buildings, soil, or other sources. She did this by essentially “fingerprinting” the lead chemically, similar to when police identify criminals using DNA fingerprinting. Atoms of a particular chemical element all have the same number of protons, but may have a different number of neutrons and thus a different atomic mass. These different forms are called isotopes. Some isotopes can spontaneously lose protons and/or neutrons and thus are radioactive. Stable isotopes do not and keep the same atomic mass. Because the ratios of these isotopes vary among different sources, we can use stable isotopes of nitrogen (N) and carbon (C) to determine what organisms are eating and use stable isotopes of lead (Pb) to determine where organisms are getting lead. Image: Dirk Hünniger (Wikimedia)

→ nest soil is not source of poisoning
Step 2: Identify Source of Lead Near Buildings: r 2 = 0.363 p = 0.152 Chick Blood 207Pb/206Pb Away from Buildings: r 2 = 0.224 p = 0.074 Myra compared the ratio of lead isotopes in the chicks’ blood to the ratio of lead isotopes in the soil and paint chips in each chick’s nest. She found that the ratio of lead isotopes in the chicks’ blood was not related to the ratio of lead isotopes in the soil either near or away from buildings, so she concluded that nest soil was the source of the poisoning. Nest Soil 207Pb/206Pb → nest soil is not source of poisoning Finkelstein et al. (2003) Environmental Science & Technology

→ lead-based paint is source of poisoning
Step 2: Identify Source of Lead Chick Blood 207Pb/206Pb r 2 = 0.853 p = 0.003 However, the ratio of stable isotopes in the blood of chicks was highly correlated to the ratio of stable isotopes in lead paint chips found in their nests. The r2 value means that 85% of the variation in lead isotope ratio in chicks’ blood is explained by the lead isotope ratio in paint chips. This shows that albatrosses are getting lead poisoning by ingesting lead paint chips near their nests. Nest Paint Chips 207Pb/206Pb → lead-based paint is source of poisoning Finkelstein et al. (2003) Environmental Science & Technology

Step 3: Show Population Consequences
Mathematical Model This study wasn’t sufficient to convince managers to spend millions of dollars to clean up the lead paint, so Myra decided to show how lead poisoning is impacting the albatross population at Midway and whether removing the lead would actually help the population, since most people thought long-lining bycatch of adults impacted the population far more than any impact to chicks could. She couldn’t run an experiment poisoning chicks to determine the consequences, so instead she created and analyzed a mathematical model.

Introduction to mathematical models Structured population models Assumptions Model structure Mathematical model: Importance of models to ecology and conservation Now, we’ll move on to discussing mathematical models and how Myra selected which type of model to use.

Discussion: What is a Model?
Simplified representation of object or process Requires assumptions Essentially, all models are wrong, but some are useful. - George Box, 1987 Think for a minute about what defines a model, mathematical or not, and then turn to the person next to you and discuss the question for another minute. Here are some examples of models to help you come up with your own definition. They include a map, a model building, a model organism (fruit fly) used for genetic studies, a conceptual model known as a food web, and a mathematical model. Think about what is common to all of them. [Think-Pair-Share] All models are simplified representations of some object or process and require assumptions and simplifications, whether they’re a map of a place, physical model of a building, a model organism, a mathematical population model, or a conceptual model of a food web. We’ll focus today on conceptual and mathematical models. Models require the modeler to decide what aspects or elements of the situation or process are important and which can be safely ignored when trying to answer the question. For example, when we see a physical model of a building like the one here, we don’t know what color the carpet or paint is inside or what exactly is in all the rooms because the modeler decided that the most important thing to show is the outside of the building. USGS

Mathematical Population Models
Exponential Growth Logistic Growth N time N time K For example, you are probably familiar with at least these 2 classical mathematical models for ecological populations that are usually plotted as population size (N) vs. time. Q1: What’s the first one? A1: Exponential growth, which is what populations of organisms can do when resources are unlimited. It has a characteristic J-shape. We often see this when population are small, perhaps when an endangered species is recovering, such as Northern Elephant Seals in California, or when a species is invading a new location. Q2: What’s the second one? A2: Logistic growth, which is what populations of organisms do (more or less) when resources, such as food, space, and light, are limited in some way. Logistic growth has a characteristic S-shape. Populations experience negative density dependence – the growth rate dN/dt declines as density or population size increases and finally reaches 0 at the carrying capacity K. Malthus (1798) Verhulst (1838)

Steps to a Mathematical Model
Ask research questions Make assumptions Develop conceptual model Formulate mathematical model Assign values to parameters Use model to answer questions In order to create a mathematical model, a scientist must follow these general steps. Parameter = terms in a model specified by the model, usually as a constant. For example, r and K in the exponential and logistic growth models are parameters. Based on: Soetaert & Herman 2008 A Practical Guide to Ecological Modeling (Fig. 1.7)

Introduction to mathematical models Structured population models Assumptions Model structure Mathematical model: Importance of models to ecology and conservation Next, we’ll move on to discussing structured population models, which are also sometimes called matrix population models. We’ll discuss what assumptions we make and how we develop these models.

1. Ask Research Questions
Is the population growing? Which stage(s) and demographic processes should managers focus monitoring and conservation effort on? How might threats or management actions impact the population? Q: What kinds of research questions do wildlife managers and conservationists have about populations of organisms? A: Wildlife managers often develop population models to determine whether the population is growing, how threats or management actions might impact the population, and which stages they should focus their monitoring and conservation efforts on.

2. Make Assumptions Constant environment (Deterministic)
time Constant environment (Deterministic) Unlimited resources Constant birth & death rates No immigration or emigration No time lags All individuals are identical Exponential Growth Next, managers have to decide what assumptions are reasonable to make for their particular population – which features of the population are important and which ones they can safely ignore for the purposes of their questions. Let’s start with the assumptions for exponential growth. Can you think of any? If the environment is totally constant, either through time or depending on density, the model will not include any random variation, which we call stochasticity. It’s pretty easy to add stochasticity to any population model though. Also, the population growth rate doesn’t depend on the population size or density, so resources must be unlimited. There’s no immigration into the population or emigration out of the system. This assumption doesn’t make sense for some organisms, such as corals with a pelagic (open ocean) larval stage. Since this is a continuous-time model, with a straight line, we’re assuming there’s no time lag. There is a similar model, called geometric growth, that does have a time lag for organisms with discrete breeding seasons, such as most birds. Each individual experiences the same probability of birth and death, so we must assume that all individuals are identical. Q: Which types of organisms do these assumptions make sense for? A: Pretty much only bacteria! From Gotelli (2001) A Primer of Ecology

Complex Life Histories → structured population model
2. Make Assumptions Constant environment (Deterministic) Unlimited resources Constant birth & death rates No immigration or emigration No time lags All individuals are identical All individuals in each stage are identical vital rates Complex Life Histories Q: Do all of these assumptions make sense to model the life history of more complex organisms, such as albatrosses that don’t reproduce for the first 6-9 years of their lives or lionfishes that have a pelagic larval stage? Which assumptions should we change? A: For example, lionfish larvae can’t reproduce and experience a much higher mortality rate than adults, so it doesn’t make sense to assume that all individuals are identical. Similarly, juvenile albatrosses can’t reproduce and adult albatrosses aren’t affected by lead poisoning, so we can’t assume that all individuals are identical. Instead we define stages for each population based on age, sex, size, or development and assume that all individuals in each stage are identical. Since reproduction, survival and growth vary between these stages, we’ll instead say that vital rates are constant. Vital rates are rates of demographic processes, such as fecundity, survival rate, and growth rate of individuals. Because these rates are constant for each stage, the population still have unlimited resources and can grow exponentially. These assumptions lead us to choose a structured, or matrix, population model. It’s fairly easy to relax the assumptions about unlimited resources and constant environment by adding variation to the vital rates that are constant in the basic structured population model we’ll discuss today. → structured population model From Gotelli (2001) A Primer of Ecology

Structured Population Model
Structured by • Age • Size • Sex • Developmental stage Transition Matrix n1 P1 F2 F3 F4 F5 F6 n2 G1 P2 n3 = G2 P3 n4 G3 P4 n5 G4 P5 n6 G5 P6 Structured population models track the number of individuals in each stage as a function of their vital rates (i.e., survival rate, fecundity, growth rate). The population can be structured by, or individuals can be divided into stages by, age, size, sex, or developmental stage. We have to decide how to structure the model based on the life history of the organism and what information we can collect about the population. The mathematical structure for a structured population model is a transition matrix, in which each element refers to the contribution of individuals in a stage to another, or the same, stage. We keep track of the number of individuals in each stage using vectors (essentially a list of numbers) where each number, shown here as variables, is the number of individuals in a particular stage in a particular year. The transition matrix always has as many rows and columns as the population has stages, so this population has 6 stages. We can project the population in a particular year by multiplying the transition matrix by the population vector in the previous year. Now we’ll discuss how to develop that matrix. t+1 t

Structured Population Model
P = Persistence G = Growth F = Fecundity FROM Transition Matrix = 1 2 3 4 5 6 G1 G2 G3 G4 G5 n1 P1 F2 F3 F4 F5 F6 n2 G1 P2 n3 = G2 P3 n4 G3 P4 n5 G4 P5 n6 G5 P6 1 2 3 4 5 6 = 1 2 3 4 5 6 P1 P2 P3 P4 P5 P6 1 2 3 4 5 6 F2 F3 F4 F5 F6 TO The transition matrix describes changes in population size due to mortality, reproduction, and growth, so it is a matrix of vital rates (survival, growth, fecundity). I think it’s easiest to look at the matrix as a table with the stage numbers written on the top and side. Each element in the transition matrix represents the contribution of the column stage to the row stage. For example, the number in column 2 row 3 is the contribution of stage 2 to stage 3. Now let’s work through each type of transition probability. First, this main diagonal here represents the probability of persistence – the likelihood that an individual in the stage will survive and remain in the stage. For example, P2 is the probability that an individual in stage 2 will survive and remain in stage 2. If each stage represents an age class, then individuals cannot persist in a stage for more than one year and all the Ps are zero. Second, the first subdiagonal represents the probability of individual growth – the likelihood that an individual in the stage will survive and grow into the next stage. For example, G4 is the probability that an individual in stage 4 will survive and grow into stage 5. There’s no G6 because individuals in that stage must either persist in the same stage or die. Third, the first row represents fecundity – the average number of offspring produced by an individual in the stage in one time period. Only elements in this row generally have values greater than 1, because an individual can have more than one offspring but can’t have a probability of survival greater than 1. For example, F5 is the fecundity of stage 5 – an individual in stage 5 will, on average, have F5 offspring in one time period. Depending on the organism’s life history and how the stages are defined, some of these Fs and Ps may be zero if individuals in a particular stage can’t reproduce or can’t remain in that stage. For example, a 2-year old albatross can’t reproduce or remain 2 for another year. This matrix can also get more complicated than this for organisms with more complex life histories, such as corals that can get smaller or split into 2 individuals. t+1 t

Simpler Example: Lionfish (Pterois volitans)
Larvae Juveniles Adults Todd Gardner NOAA SE Fisheries Science Center G = Growth FA L J A PL GL GJ PA L J A FA P = Persistence Let’s start with a simpler example – lionfish, a coral reef fish that has become invasive in the Atlantic Ocean. They have a pelagic larval stage and then settle down to a particular reef as a juvenile and then grow into a mature adult. It makes sense to structure the population as developmental stages (larvae, juveniles, and adults) and to use time periods of months because the pelagic larval duration is about a month. Let’s develop our conceptual model first and then our mathematical model. Larvae can grow into juveniles and juveniles can grow into adults, so both larvae and juveniles have a G value. Larvae can’t remain larvae for more than a month, so they don’t have a P arrow and the P value is 0. Juveniles and adults can remain in the same stage for more than a month, so they have P values. Only adults can reproduce, so only adults have a fecundity value greater than 0. GL GJ F = Fecundity PJ PA Time step: 1 month Morris et al. (2011) Biological Invasions

Simpler Example: Lionfish (Pterois volitans)
G = Growth FA L J A P = Persistence GL GJ F = Fecundity PL PA Time step: 1 month From Stage Larvae Juveniles Adults Now let’s translate this conceptual model into a mathematical model. Let’s do persistence first because it’s in the main diagonal (0,0; 1,1; 2,2; etc.). Then growth is in the subdiagonal just below the main diagonal. Fecundity is in the top row. FA To Stage GL PL GJ PA

Ex2: Life Cycle Diagram → Transition Matrix
F = Fecundity G = Growth P = Persistence F3 Mature Plants Seeds P1 P3 G1 G2 Now I’d like each of you to practice converting a life cycle diagram to a transition matrix. First, take a second to label the Fs, Gs, and Ps. Q: In which stages can individuals reproduce and therefore have an arrow that should be labeled F? These stages contribute to the first stage, seeds. A: Only mature plants Q: What options does a seed have? A: It can grow into an immature plant (G1) or persist as a seed (P1) or die. Q: What options does an immature plant have? A: It can grow into a mature plant (G2) or persist as an immature plant (P2) or die. Q: What options does a mature plant have? A: It can persist as a mature plant (P3) or die. This stage can also reproduce (F3). [Alternatively, although animations are set for other order: Q: In which stages can individuals survive and grow into another stage? These stages have an arrow which should be labeled G. A: Seeds and Immature Plants Q: In which stages can individuals survive and persist in the same stage? These stages have an arrow which should be labeled P. A: All stages] Immature Plants Seeds Immature Plants Mature Plants P2

Ex2: Life Cycle Diagram → Transition Matrix
Stage-Based Model F = Fecundity G = Growth P = Persistence From Stage Seeds Immature Plants Mature Plants F G1 P1 P2 P3 G2 P2 G2 F P3 Seeds Immature Plants Mature Plants Mature Plants P1 G1 To Stage Now, let’s work on converting this life cycle diagram into a transition matrix. Take a minute to think about how this works and fill it in yourself. Then check your table with your neighbor and discuss any differences. [Think-Pair-Share] Q:First, what options does a seed have? A: It can grow into an immature plant (G1), which is from seed to immature plant and should be in column Seeds and row Immature Plant, or persist as a seed (P1), which is from seed to seed (or die). Q: Next, what options does an immature plant have? A: It can grow into a mature plant (G2) or persist as an immature plant (P2) (or die). Q: Finally, what options does a mature plant have? A: It can persist as a mature plant or die. You may be asked on the exam to convert a life cycle diagram into a transition matrix or a transition matrix into a life cycle diagram, so make sure you practice this during lab and are ready to do it for the exam.

Ex3: Transition Matrix → Life Cycle Diagram
F3 F4 G1 P2 G2 P3 G3 P4 F3 F4 This time, practice converting a transition matrix into a life cycle diagram. First, write “From” and “To” in the right places. This isn’t given in the papers you’ll read for lab, so make sure you know “From” is columns and “To” is rows. Take a minute to sketch the life cycle diagram and add in arrows yourself. You can even label the transition probabilities if you want. Then discuss your sketch with your neighbor. Q: In which stages can individuals reproduce? A: Stages 3 and 4 [click to show F arrows at top] Q: In which stages can individuals survive and grow into the next stage? A: All stages [click to show G arrows] Q: In which stages can individuals survive and persist in the same stage? A: Stages 2,3,4 [click to show P arrows] G1 G2 G3 1 2 3 4 P2 P3 P4

Introduction to mathematical models Structured population models Assumptions Model structure Mathematical model: Importance of models to ecology and conservation Now, let’s get back to using this model with Laysan albatrosses at Midway Atoll.

Ask research questions Make assumptions Develop conceptual model Formulate mathematical model Assign values to parameters Use model to answer questions We’ll follow all these steps for the Midway albatross population. Based on: Soetaert & Herman 2008 A Practical Guide to Ecological Modeling (Fig. 1.7)

1. Determine Research Questions
How does lead poisoning of chicks affect population growth? Will lead remediation help protect the population? Remember that the point of the model is to show the population consequences of lead poisoning. We can phrase that as “How does lead poisoning of chicks affect population growth?” Myra also wanted to show whether lead remediation would help protect the population. Finkelstein et al Animal Conservation

Complex Life Histories → structured population model
2. Make Assumptions Constant environment (Deterministic) Unlimited resources Constant birth & death rates All individuals are identical No immigration or emigration No time lags All individuals in each stage are identical vital rates Complex Life Histories Again, these are the assumptions we’re making for the albatross population. We decided that a structured, or matrix, population model best fits the complex life history of the albatross → structured population model From Gotelli (2001) A Primer of Ecology

Video: Albatross Life History
3. Develop Conceptual Model Video: Albatross Life History In order to develop the conceptual model, we need to know about the life history of the organism we’re modeling, so here’s a video. This is what it looks and sounds like at Midway for most of the year – birds every meter or so and making noise all day and night long. First, an adult female lays an egg in late November and the egg hatches two months later in early February. The parents take turns feeding at sea and return to feed the chick by regurgitating squid. The chick gets bigger and starts losing its down and growing its adult feathers. It heads to the beach and starts learning how to fly. Eventually, if it’s not eaten by a shark, it spends about 3-5 years at sea and returns to try to find a mate with this elaborate courtship ritual. By about age 8, most adults have successfully mated and laid their own egg.

3. Develop Conceptual Model
8 E 5 4 2 1 3 6 7 8 2 1 3 5 4 6 7 E Eggs Juvenile Immature Subadult Adults Pete Leary/USFWS Gina Ruttle Chicks The model given in the paper uses one year time steps, so it puts all the egg and fledgling stages into one stage. Then albatrosses spend about 3 years completely at sea as juveniles and return to Midway to try to find a mate for about 4 years when they are considered immature and subadults. Finally at about 8 years of age, albatrosses begin to successfully raise chicks. They live up to at least 60 years old and produce a chick every year or two. This last stage is the only stage in which albatrosses can remain for more than one year and the only stage to reproduce. Time step: 1 year Fisher,1975

4. Formulate Math Model F Gone Gj Gj Gj Gi Gi Gsa Gsa Pa 8 2 1 3 5 4 6
7 E Gone Gj Gj Gj Gi Gi Gsa Gsa Pa Next, we have to formulate the math model from the conceptual model, just like we did for the lionfish. We add Gs for every stage except the last one. We assume that all juveniles, immatures, and subadults respectively experience the same probability of survival and transitioning into the next stage, so we only have 5 G parameters. Since an albatross can only be 1 or 2 or 3 years old once, stages E to 7 do not have a P value – only individuals in stage 8 can survive and remain in the same stage, or persist. Only individuals in stage 8 can reproduce, so only stage 8 has an F value, which we can simply call F. G = Growth P = Persistence F = Fecundity

4. Formulate Math Model F Gone Gj Gi Gsa Pa E 5 4 2 1 3 6 7 8 E 1 2 3
Now, let’s work together to add the Ps, Gs, and F to the transition matrix. [exit presentation view]

Fisher,1975

F Gone Gj Gi Gsa Pa Every other cell contains a zero. Fisher,1975

5. Assign Values to Parameters
Andrew Derocher (IUCN) NOAA SEFSC Pete Leary/USFWS Q: Next, we need to put actual numbers into the mathematical model. How do we figure out what those vital rates actually are? First, how do we do it for albatrosses? Here’s a hint! A: We often track individuals over their lifetimes to see what percentage of individuals die in each stage, grow into the next stage, and the rate at which they reproduce. For albatrosses, we do this by banding all the birds in permanent plots which biologists study every year and band fledglings. It’s relatively easy for albatrosses because, if they survive, birds will return to breed within a few meters of where they were born. Scientists tag polar bears by putting a tattoo on the inside of their lip! Sometimes we can identify all individuals in a population using photographs: for example, we can identify all individuals in an orca population because they have a distinctive dorsal fin coloration and corals don’t move so we can mark their location. Sometimes we have to determine the age or size distribution of the population at one point in time, for example by measuring sharks fishermen catch, and then use math to estimate the vital rates or even just take vital rates from a related species. NOAA

F 8 = 0.366 2 1 3 5 4 6 7 E Gone Gj Gj Gj Gi Gi Gsa Gsa Pa 0.482 0.84 0.84 0.84 0.906 0.906 0.945 0.945 0.916 G = Growth P = Persistence F = Fecundity Myra was able to find survival and fecundity values in the literature for Midway albatrosses from the 1930s to the 1970s, before lead poisoning probably impacted the population very much because the paint on the building was maintained before the Navy left. Fisher,1975

F E 5 4 2 1 3 6 7 8 Gone Gj Gi Gsa Pa E 1 2 3 4 5 6 7 8 F Gone Gj Gi Gsa Pa Next, we put these numbers into the transition matrix. Fisher,1975

F E 5 4 2 1 3 6 7 8 Gone Gj Gi Gsa Pa E 1 2 3 4 5 6 7 8 0.366 0.482 0.840 0.906 0.945 0.916 All the other values in the transition matrix equal 0. Fisher,1975

Ask research questions Make assumptions Develop conceptual model Formulate mathematical model Assign values to parameters Use model to answer questions Now, we’re finally ready to use the model to answer our research questions. Based on: Soetaert & Herman 2008 A Practical Guide to Ecological Modeling (Fig. 1.7)

6. Use Model to Answer Questions
Is the population growing? Which vital rate(s) should managers focus monitoring and conservation effort on? Which stage(s) should managers focus monitoring and conservation effort on? Which stage(s) contain most of the population? How might management actions impact population? Typical research questions for a structured population model include these. Of course, Myra was mostly interested in the last one.

Is the population growing?
Transition Matrix n1 P1 F2 F3 F4 F5 F6 n2 G1 P2 n3 = G2 P3 n4 G3 P4 n5 G4 P5 n6 G5 P6 We can determine if the population is growing by projecting the population forward in time. Remember, we can calculate the number of individuals in each stage in time t+1 by multiplying the transition matrix by the number of individuals in each stage in time t. In math terms, this is matrix multiplication. At the bottom of many of the next slides, it will say in small text what the math is for calculating these terms, for those of you who have taken matrix algebra. It will also say how to calculate these terms using the PopTools add-on to Microsoft Excel that we’ll be using in lab. [These are written here for your reference in lab and later in case you ever want to know. You don’t need to know the math or PopTools ways to answer these questions for the exam.] t+1 t Math: multiplication of transition matrix & initial population vector PopTools: Matrix Tools → Matrix Projection

Population Size Once you project the population into future, you can add up the number of individuals in each stage and determine the total population size. Then you can graph the population size against time and determine if the population is growing or not. Time

Population Growth Rate (λ)
Is the population growing? Population Growth Rate (λ) λ = Nt+1 / Nt If λ = 1 , Nt+1 = Nt → population size If λ > 1, Nt+1 > Nt → population size If λ < 1, Nt+1 < Nt → population size Math: dominant eigenvalue of transition matrix PopTools: = DomEig(TM) is constant However, we have a much easier way to tell if the population is growing than projecting the population into the future. We can calculate the population growth rate using the transition matrix. You may remember this population growth rate (λ = lambda) from learning exponential growth. It is simply the ratio between the number of individuals in one year to the number of individuals in the previous year (although again, this only works after many years). Q: If λ = 1 , Nt+1 = Nt → population size A: is constant Q: If λ > 1, Nt+1 > Nt → population size A: is increasing Q: If λ < 1, Nt+1 < Nt → population size A: is decreasing For those of you that have studied math algebra, the mathematical way to the population growth rate is to take the dominant eigenvalue of the transition matrix. is increasing is decreasing

# Lionfish As you might already know, the lionfish population in the Atlantic is growing exponentially. The authors of this paper used this model to suggest that, in order to reduce the lionfish population, managers must remove monthly 27% of adult lionfish. Q: If the lionfish is growing exponentially, as we saw in the previous graph, is the population growth rate less than 1, equal to 1, or greater than 1? [RAISE HANDS!!] A: Greater than 1. Indeed, the authors calculated it was λ = 1.134 Morris et al. (2011) Biological Invasions

Is the population growing? [After many years with these vital rates]
λ = 0.995 Here I projected the albatross population matrix 50 years into the future using the number of albatrosses currently at Midway Atoll. Q: Is the population size increasing, constant, or decreasing? A: Decreasing just a little, so lambda is just under 1. If we project the population using different numbers of individuals in each stage in the initial population (but same total number), the model does weird things for a few years, but eventually always settled out to the same population growth rate. The population growth rate depends completely on the transition matrix, not on the initial population distribution at all. Thus, what we really want to know is “Is the population growing after many years with these vital rates?”. With initial population vector based on stable stage distribution With initial population vector not based on stable stage distribution

Which vital rate(s) should managers focus monitoring and conservation effort on?
Elasticity = Effect that a change in each one of the vital rates has on population growth rate λ Used to determine which vital rate changes the population growth rate λ most To determine which vital rate(s) managers need to protect most and have the best estimate to understand what’s happening with the population, we calculate the tool elasticity. Elasticity is essentially the effect that a change in each one of the vital rates (or transition probabilities) has on the population growth rate. Managers should focus monitoring and conservation effort on the vital rate with the highest elasticity. Math: , where <> denotes the scalar product PopTools: Matrix Tools → Elasticity

Elasticity: The elasticity may be output as a table like this. It corresponds exactly to the transition matrix. Q: Where is the highest elasticity? A: In the last column, last row, which corresponds to the transition matrix cell 8→8, which is Pa. Corresponds to Pa

Elasticity Gone Gj Gi Gsa Ga F You can also present the elasticities in a table or graph like this which are easier to read! Gone Gj Gi Gsa Ga F

# of individuals of age x
Which stage(s) should managers focus monitoring and conservation effort on? Reproductive Value = expected number of offspring that remain to be born to each individual of a stage Essentially which stages are most valuable for future population growth Math: left eigenvector of transition matrix PopTools: Matrix Tools → Age Distribution v(x) = # of offspring produced by individuals of age x or older (discounted by likelihood of surviving to reproduce) # of individuals of age x To determine which stage(s) managers should focus monitoring and conservation effort on, we need to know the stage in which an individual contributes most to future population growth. In order to do this, the individual must have babies and must survive to have them. We call this reproductive value. It isn’t necessarily the stage or age at which individuals are having the most babies, although it often is. If the adults that are having the most babies aren’t likely to survive very long, another stage may have the highest reproductive value, even that stage isn’t reproducing at all!

Which stage(s) should managers focus monitoring and conservation effort on?
Reproductive Value Egg 0.035 1 0.0718 2 0.0851 3 0.1009 4 0.1195 5 0.1314 6 0.1443 7 0.1520 8 0.1601 Q: Here’s the reproductive value distribution for the albatross population. Which stage should managers focus monitoring and conservation effort(s) on? A: The stages with the highest reproductive value, so the older stages. However, reproductive value is similar across all the stages, so it’s important to protect all stages, with just a little more effort to the adult stages.

Reproductive Value Humans? RV
(based on Daly and Wilson 1988 Homicide) To maximize sustainable harvest yield, harvest stages with reproductive value To maximize impact of restoration, transplant individuals with reproductive value Natural selection will act most strongly on stages with reproductive value RV Age low Q: What does reproductive value look like in humans? A: It peaks in young adults, but peaks earlier and declines faster in females than males because males are capable of reproduction at later ages than females. It’s relatively small for children because they may not live to reproduce. Q: To maximize sustainable harvest yield, should we harvest stages with low, medium, or high reproductive value? That is, stages with low, medium, or high contributions to future population growth? A: As we should protect most the stages with high reproductive value, we definitely should only harvest stages with low reproductive value because they have lower contributions to future population growth. Q: To maximize the conservation impact of restoring a population, should we transplant from a greenhouse or introduce from captive breeding individuals with low, medium, or high reproductive value? A: We can help the population recover faster by transplanting or introducing individuals with high reproductive value because they will contribute the most to future population growth. Q: Will natural selection act most strongly on stage with high, medium, or low reproductive value? A: Natural selection will act most strongly on individuals with high evolutionary fitness, which is actually very closely related to reproductive value. Thus, natural selection will act most strongly on stages with high reproductive value because these stages have highest evolutionary fitness and contribute most to future population growth. high high Gotelli 2008 A Primer of Ecology, Caswell 1980 Ecology

Which stage(s) contain most of the population?
To determine how many individuals are in the population are a particular time, we need to pick an initial population vector and then project the population into the future using the transition matrix. [click] As you can see, after a few years, the population settles into one distribution. Because this population overall is declining slowly, the number of individuals in each stage is also declining slowly. If the population was growing quickly, each stage would also increase quickly, but keep the same proportion of individuals in each stage after a few years. Remember, as we discussed with population growth rate, what happens in the first few years depends on the initial population vector, but after that it depends on the transition matrix. PopTools: Matrix Tools → Matrix Projection

Stable Stage Distribution
Which stage(s) contain most of the population? [After many years with these vital rates] Stable Stage Distribution Q: How is this graph different from the previous one? A: The y-axis is proportion, not number, of individuals, so it’s the number of individuals in each stage divided by the total number of individuals in the whole population. Again, what happens in the first few years is due to the initial population vector, but the population eventually [click] settles into having the same proportion of individuals in each stage, which is what we call the stable stage distribution [click].

Which stage(s) contain most of the population? [After many years with these vital rates] Stable Stage Distribution = constant proportion of individuals in each stage (derived from constant vital rates) Math: Right eigenvector of transition matrix PopTools: Matrix Tools → Age Distribution w(x) = # of individuals of age x in SSD total # of individuals The stable stage distribution is a vector with a constant proportion of individuals in each stage, which each population reaches eventually if it has constant vital rates for each stage. Instead of picking an initial population distribution and graphing, we can also directly calculate this stable stage distribution.

Which stage(s) contain most of the population? [After many years with these vital rates] Stage Stable Stage Distribution Egg 0.167 1 0.081 2 0.068 3 0.058 4 0.049 5 0.044 6 0.041 7 0.038 8 0.455 As in the graph with the projection of the different stages, after many years, most of the population is in stage 8. This makes sense because stage 8 includes all albatrosses of ages 8 and up to their maximum lifespan, which can be at least 60 years old.

How might management actions impact population?
SIMULATION – What if . . . How does lead poisoning of chicks affect population growth? Will lead remediation help protect the population? Remember, our research questions were How does lead poisoning of chicks affect population growth? Will lead remediation help protect the population?

How does lead poisoning of chicks affect population growth?
8 2 1 3 5 4 6 7 E To determine how threats impact the population, we have to determine how to implement them in the model. Lead poisoning primarily impacts chicks, which we put in the E stage. Myra estimated lead-induced chick mortality using surveys of the chicks around buildings. We want to compare lead poisoning to the other main threat to albatrosses at Midway, longlining bycatch, which primarily impacts adults. The number of birds killed annually in global fishing operations is unknown.

How does lead poisoning of chicks affect population growth?
Lead-induced chick mortality 0% 3.5% 7% 14% Q: When we project the population using the historical vital rates Myra found, without even including lead poisoning, is the population size increasing, constant, or decreasingA: Decreasing slightly Q: What does this mean about the population growth rate lambda? ? [RAISE HANDS!!] A: It’s slightly less than 1, actually Although Myra estimated the rate of lead-induced mortality as well as possible, there’s a lot of uncertainty, so she ran the model with 3 different levels to see how the level of lead-induced chick mortality impacts the population. Q: How does lead poisoning affect population growth? A: It decreases the population even more. Finkelstein et al Animal Conservation

Will lead remediation help protect the population?
1% Reduction in Adult Bycatch Mortality starting in 21 years Lead Remediation 1% Reduction in Adult Bycatch Mortality No Action starting in 21 years No Action Finally, we can determine how different conservation actions can affect the population by running a simulation – a “what if” scenario. First, Myra wanted to know what would happen if managers were able to eliminate all lead-induced chick mortality. Q: So, how does eliminating lead-induced chick mortality impact the population? A: The population still declines, although more slowly. However, immediate cessation of lead-based chick mortality will create a buffer for the population by increasing the number of birds in 50 years by about 100, ,000. [click] Although the number of birds killed annually in global fishing operations is unknown, we can project the impacts of reducing adult mortality by 1%. Q: How does the effect of reducing bycatch compare to the effect of lead remediation? A: This has a bigger impact than lead remediation, so reducing bycatch is really the most important conservation action we can take. However, reducing bycatch is really hard because these albatrosses feed and are getting caught in long-lines all over the North Pacific Ocean, in the waters belonging to many countries and in international waters, so all those countries would have to come to an agreement and enforce the rules, so unfortunately that’s not going to happen for many years, if at all. Lead Remediation Finkelstein et al Animal Conservation

Result: Conservation Action
Pete Leary / USFWS USFWS After Myra published that work in 2010, the Center for Biological Diversity threatened to sue the U.S. government for violating the Migratory Bird Treaty Act and other laws, citing primarily Myra’s work. Immediately, FWS began a study of feasibility and cost. In 2012, FWS and the Center for Biological Diversity entered a settlement that requires FWS to complete clean-up of lead-based paint by FWS estimates it will take about $12 million to clean-up 86 buildings. Lead-paint remediation, which involves either removing the paint or covering with more layers of a special encapsulating paint, started in 2012. For more info:

Introduction to mathematical models Structured population models Assumptions Model structure Mathematical model: Importance of models to ecology and conservation Finally we’ll end with a quick discussion of how models are used and why they’re important to ecology and conservation.

Discussion: How do we use and why do we need mathematical models in ecology and conservation?
Take a minute to think about how we use and need mathematical models in ecology, based on both your previous experience and this lecture. What can we do with mathematical models that we can’t do with field or lab work? Then discuss your thoughts with your neighbor. [Think-Pair-Share] Main points: Analyze and synthesize monitoring data Determine which types of data are most useful and important to collect in the future Identify key life stages or demographic processes as management targets Predict impacts of management actions Soetaert & Herman 2008 – A Practical Guide to Ecological Modeling: Generate testable predictions to guide experiments Analyze systems that are impossible or difficult to experiment with (when spatial or temporal scales are too large, against regulations or morals, too expensive, dangerous, take too long – think astronomy or endangered or invasive species!) Force scientists to think very clearly & be explicit about every assumption & term → identifying gaps in knowledge that may give direction to research priorities & new experiments Extrapolate or interpolate studies in time & space Quantify immeasurable processes (e.g., rate of oxygen flux across sediment-water interface) Guide management by “performing experiments” that are inappropriate in the real system and assessing consequences of our actions in advance Gotelli 2008 A Primer of Ecology: Science of ecology is study of distribution and abundance → need tools of math & statistics We need models because natural is so complex Mathematical models act of simplified road maps, giving us some direction and idea of exactly what things we should be trying to measure in nature Models also generate testable predictions Highlight distinction between patterns we see in nature and the different mechanisms that might cause those patterns

Objectives Learn how scientists use structured population models to assess and protect populations Appreciate the importance and uses of mathematical models in ecology and conservation

Lab: Cooperative Activity Using PopTools
Dwayne Meadows, NOAA/NMFS/OPR For the lab associated with this lecture, each group of students selects one of these organisms and reads the associated paper. During lab, each group will together work through the 6 steps to a mathematical model using their research paper, using a handout similar to the lecture guide you have today, using an add-on to Microsoft Excel called PopTools. You will want to bring your lecture guide and lecture notes to lab! John Snow Chris Muenzer

Using PopTools Add-on to MS Excel
We’ll do this using an add-on to Microsoft Excel called PopTools. If you want to use your computer in lab, it must be a PC with Microsoft Excel and you should install the add-on by following the instructions at the website.

Resources Crouse, D. T., Crowder, L. B., & Caswell, H. (1987). A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology 68(5): Finkelstein, M. E., Doak, D. F., Nakagawa, M., Sievert, P. R., & Klavitter, J. (2010). Assessment of demographic risk factors and management priorities: impacts on juveniles substantially affect population viability of a long-lived seabird. Animal Conservation, 13(2), Gotelli, N. (Various) A Primer of Ecology. Chapter 3: Age-Structured Population Growth. Sinauer. If you need more help on understanding structured population models, these can serve as resources.

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From Ecological Model to Conservation Action

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