FW364 Ecological Problem Solving

1. FW364 Ecological Problem Solving Class 12: Age Structure October 16, 2013

2. Outline for Today Continue to make population growth models more realistic by adding in age structure Objectives for Today: Introduce age structure Objectives for Next Two Classes: Matrix algebra primer Introduce the Leslie Matrix Complete in class examples of age structured population growth Text (optional reading): Chapter 4: Sections 4.1 – 4.5 Not covering life table information (sections 4.6+)

3. Age Structure Introduction Continue to improve population growth models by taking into account variation due to ages of individuals in the population Distribution of individuals of each age is called age structure Most populations consist of individuals of different ages For these individuals, age often affects: Likelihood of reproducing Number of offspring produced Chance of dying or emigrating

4. Age Structure Introduction For example: Younger individuals may not reproduce at all # young may increase with age (due to body growth) Older individuals might compete more effectively for mates Older individuals might hold territories better Younger individuals may be more susceptible to predation Younger individuals may be more likely to emigrate

5. Age Structure Introduction Because of the differing vital rates, the age structure of a population is important to incorporate in models of population growth For example: A fish population that consists of young individuals (small bodies, lower fecundity) may grow slower than a population that consists of older individuals (large bodies, higher fecundity) The effect of age structure on population growth is why fisheries management uses on size limits

6. Age Structure Introduction Because of the differing vital rates, the age structure of a population is important to incorporate in models of population growth In models: Age structure is incorporated as a type of deterministic variation (like density dependence) For populations, we can determine how birth, death, and emigration rates vary with age, and we can add parameters to models that encompass that variation • Age 0 • Age 1 • Age 2 • Age 3

7. Age Structure Introduction Because of the differing vital rates, the age structure of a population is important to incorporate in models of population growth In models: Age structure is incorporated as a type of deterministic variation (like density dependence) This is different from demographic stochasticity! Demographic stochasticity results from “roll of the dice” for individuals i.e., chance affecting whether individuals reproduce or die Age structure is predictable variation e.g., predictable changes in reproductive output that occur with age

8. Helmeted Honeyeaters – Case study Critically endangered subspecies of the yellow-tufted honeyeater Endemic to Victoria, Australia Populations face extinction due to habitat destruction and wild fires Current population consists of ~20 breeding pairs Only three wild populations remain: 1 natural population in a reserve 2 introduced populations in a state park Bird emblem of Victoria

9. Helmeted Honeyeater Age Structure Hypothetical helmeted honeyeater age structure across four years: Table 4.1 Population has been counted and aged for four successive years Age is categorized by years For some organisms (e.g., inverts), months or days would be better Modelers pick the age step based on life history of organisms (often = frequency of reproduction)

10. Helmeted Honeyeater Age Structure Hypothetical helmeted honeyeater age structure across four years: Naming of age classes: Table 4.1 Age 0: Individuals before 1st birthday Age 1: Individuals that have had 1st birthday Age 2: Individuals that have had 2nd birthday …Like humans

11. Helmeted Honeyeater Age Structure Hypothetical helmeted honeyeater age structure across four years: Notation for age-structured population size: Table 4.1 ageNtime 2N1 = 12 OR 2N1991 = 12 OR 7N3 = 2 7N1993 = 2 Text uses: N7(3) = 2 !! N2 = 83 OR N1992 = 83

12. Helmeted Honeyeater Age Structure Hypothetical helmeted honeyeater age structure across four years: Goal: Describe population growth of an age structured population We need: Age specific survival (1 - death) Age specific fecundity Table 4.1 Similar to how we modeled population growth before: Nt+1 = Ntλ , λ = 1 + b’ – d’ But our birth and survival rates will be specific to each age class

13. Determining Age Specific Rates How do we determine age specific survival and fecundity rates? We follow abundance of cohorts Cohort: A group of individuals with the same birth year Anyone born the same year as you is in your cohort Cohorts change age class each year as the cohort ages • For example: • All people born in 1991 are part of the same cohort • The age class of the 1991 cohort is 21 in 2012 • The age class of the 1991 cohort will be 22 in 2013

14. Determining Age Specific Survival We will assume we are working with a closed population (no immigration or emigration) In closed populations, the abundance of a cohort can only decline over time (never increase) 1991 1992 1993 Age 2 Age 3 Age 4

15. Determining Age Specific Survival Challenge: What is the survival rate of this cohort from: 1991 to 1992? 1992 to 1993? 1991 1992 1993 Age 2 Age 3 Age 4

16. Determining Age Specific Survival Challenge: What is the survival rate of this cohort from: 1991 to 1992? = 8 / 12 = 0.667 1992 to 1993? = 6 / 8 = 0.750 1991 1992 1993 Age 2 Age 3 Age 4

17. Determining Age Specific Survival We can make the same calculation for a cohort using the age structure table Survival Rates: 1991 to 1992 = 8 / 12 = 0.667 1992 to 1993 = 6 / 8 = 0.750 1993 to 1994 = 5 / 6 = 0.833 Table 4.1 This is survival rate within a cohort between years For our population growth models, we want to know average survival rates for a specific age class across years

18. Determining Age Specific Survival We can make the same calculation for a cohort using the age structure table Survival of Age 0 individuals: Table 4.1 1991 to 1992: 17/26 = 0.654 1992 to 1993: 20/28 = 0.714 1993 to 1994: 20/27 = 0.741 Average survival across years: 0S = (0.654 + 0.714 + 0.741)/3 0S = 0.703 ± 0.045 SD For our population growth models, we want to know average survival rates for a specific age class across years General formula is (for age class, x): xSt = x+1Nt+1 / xNt Then average across years if data are available

19. Determining Age Specific Survival We can make the same calculation for a cohort using the age structure table Survival of Age 1 individuals: Table 4.1 1991 to 1992: ? 1992 to 1993: ? 1993 to 1994: ? Average survival across years? General formula is (for age class, x): xSt = x+1Nt+1 / xNt Then average across years if data are available

20. Determining Age Specific Survival We can make the same calculation for a cohort using the age structure table Survival Rates: 1991 to 1992 = 8 / 12 = 0.67 1992 to 1993 = 6 / 8 = 0.75 1993 to 1994 = 5 / 6 = 0.83 Survival of Age 1 individuals: Table 4.1 1991 to 1992: 11/16 = 0.688 1992 to 1993: 13/17 = 0.765 1993 to 1994: 14/20 = 0.700 Average survival across years: This is survival rate between years within a cohort 1S = (0.688 + 0.765 + 0.700)/3 1S = 0.718 General formula is (for age class, x): xSt = x+1Nt+1 / xNt Then average across years if data are available

21. Determining Age Specific Survival At-home Challenge: Try calculating the survival rates for the other age classes We can make the same calculation for a cohort using the age structure table Table 4.2 Textbook lists survival rates for all age classes Average (xS) We just calculated two of them: 0S = 0.703 1S = 0.718 (some rounding error) General formula is (for age class, x): xSt = x+1Nt+1 / xNt Then average across years if data are available

22. Determining Age Specific Survival Note: No survival estimate for age 9 (previous figure had age 9 data) There were no data for age 10, so we assume 9S = 0 Table 4.2 • Notice any patterns? • Survival slightly increases from age 0 to age 2 • But overall, survival is pretty consistent… • Somewhat typical of • birds & • mammals • but not fish Average (xS)

23. Determining Age Specific Fecundity That’s how we get age-specific survival… … now we need age-specific fecundities Age-specific fecundities are the average offspring per individual for the different age classes i.e., the contribution of each adult age class (measured as average # offspring per individual, i.e., a per capita rate) to the production of offspring For helmeted honeyeaters, we need: 1F, 2F… 9F Note: No 0F because honeyeaters mature at age 1 (therefore, 0F = 0)

24. Determining Age Specific Fecundity • The textbook makes a simplifying assumption for the honeyeaters: • Assumes that fecundity is the same for all adult age-classes • 1F = 2F =… = 9F Is this reasonable for birds? Not too bad For fish? No For humans? Yes during reproductive years No across whole life

25. Determining Age Specific Fecundity We can determine honeyeater fecundity from age-structured table Given the assumption that 1F = 2F =… = 9F Can calculate yearly fecundity as: adult xFt = 0Nt+1 / adult xNt Table 4.1 Fecundity for all adult age classes: 1991 to 1992: 28/59 = 0.475 1992 to 1993: 27/55 = 0.491 Quick exercise: 1993 to 1994: = ? Average fecundity across years?

26. Determining Age Specific Fecundity We can determine honeyeater fecundity from age-structured table Given the assumption that 1F = 2F =… = 9F Can calculate yearly fecundity as: adult xFt = 0Nt+1 / adult xNt Table 4.1 Fecundity for all adult age classes: 1991 to 1992: 28/59 = 0.475 1992 to 1993: 27/55 = 0.491 1993 to 1994: 29/61 = 0.475 Average fecundity across years: adult xF = (0.475 + 0.491 + 0.475)/3 adult xF = 0.480 ± 0.009 SD

27. Forecasting Age-Structured Growth • We now have estimates of age-specific survival and fecundity • We can use these estimates to build a model • to forecast population size! Recall how we forecasted population size previously: Nt+1 = Ntλ where λ = 1 + b’ – d’ OR λ = b’ + s’ (because s’ = 1 – d’) (note: not including density dependence) We have all of these components for our age-structured model…

28. Forecasting Age-Structured Growth Nt+1 = Nt (b’ + s’) Table 4.1 adult xF = 0.48 1995? 0N1995 1N1995 (xS) 2N1995 3N1995 4N1995 5N1995 6N1995 7N1995 8N1995 9N1995 N1995

29. Forecasting Age-Structured Growth Nt+1 = Nt (b’ + s’) Table 4.1 adult xF = 0.480 The trick to all of this is building equations that account for age-structure Step 1: Start simple and build separate equations to forecast population size for each age class (we have most of the pieces to do this already) Step 2: Put all the equations together and forecast population size using matrix algebra (which will take a bit of explanation) 1995? 0N1995 1N1995 (xS) 2N1995 3N1995 4N1995 5N1995 6N1995 7N1995 8N1995 9N1995 N1995

30. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 0N1995 1N1995 (xS) 2N1995 3N1995 To make things even more simple, we’ll pretend there are only four age classes (0, 1, 2, 3) We’ll build both general equations and honeyeater specific equations

31. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 0N1995 1N1995 (xS) 2N1995 3N1995 How many age 0 individuals will there be in 1995? 0Nt+1 = General equation: + 1F * 1Nt + 2F * 2Nt + 3F * 3Nt 0F * 0Nt General equation accounts for possibility of all age classes producing young and all age classes having different fecundities For honeyeaters, we are assuming: 0F = 0 and 1F = 2F = 3F = 4F = adult xF 0N1995 = adult xF * 1N1994 + adult xF * 2N1994 + adult xF * 3N1994 0N1995 = 0.48 * 20 + 0.48 * 14 + 0.48 * 10 = 21 age 0 honeyeaters

32. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 21 1N1995 (xS) 2N1995 3N1995 How many age 1 individuals will there be in 1995? 1Nt+1 = General equation: 0S * 0Nt Note: Age 0 individuals “this year” will be age 1 “next year” For honeyeaters, 1N1995 = 0S * 0N1994 1N1995 = 0.703 * 29 = 20 age 1 honeyeaters

33. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 21 20 (xS) 2N1995 3N1995 How many age 2 individuals will there be in 1995? 2Nt+1 = General equation: 1S * 1Nt For honeyeaters, 2N1995 = 1S * 1N1994 2N1995 = 0.717 * 20 = 14 age 2 honeyeaters

34. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 21 20 (xS) 14 3N1995 How many age 3 individuals will there be in 1995? 3Nt+1 = General equation: 2S * 2Nt For honeyeaters, 3N1995 = 2S * 2N1994 3N1995 = 0.751 * 14 = 11 age 3 honeyeaters

35. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 21 20 (xS) 14 11 Items to Note: Did not need age 3 survival In general, do not need survival of last age class  Typically assume all individuals die in last age class… … we have a survival for age 3 in this example because we chopped-off half our data set… … usually survival of last age class is 0

36. Building Equations for Each Age Class Table 4.1 adult xF = 0.48 1995? 21 20 (xS) 14 11 Summary: We just used four different equations to forecast population size from one year to the next This wasn’t too bad given four age classes, but this method gets cumbersome given more age classes There is a method to manipulate and organize equations that is very helpful in this situation…. Next class! MATRIX ALGEBRA!

37. Looking Ahead Next 2 Classes: Continue with age structure Matrix algebra primer