FW364 Ecological Problem Solving

1. FW364 Ecological Problem Solving Class 6: Population Growth September 18, 2013

2. Outline for Today Goal for Today: Continuing introduction to population growth – Discrete Growth Last Class: Derived a simple model of discrete population growth between consecutive time periods (Nt and Nt+1) Objective for Today’s Class: Derive an equation to forecast population growth (still discrete growth) Objective for Next Class: Derive continuous population growth equation Text (optional reading): Chapter 1

3. Muskox Case Study • 1936: First introduction • Fig 1.3 in text Nunivak Island

4. Recap from Previous Class • Nt+1 = Nt + B – D • Nt+1 = Nt + b’Nt – d’Nt • Rearrange to get: • B = b’ Nt • D = d’ Nt • Nt+1 = Nt (1 + b’ – d’) • (b’is per capita birth rate) • (d’is per capita death rate) • We defined a new parameter, r’ • r’ = b’ – d’ (r’ is net population change) • and plugged r’into equation: • Nt+1 = Nt (1 + r’) • We defined another new parameter, λ(lambda) λ = 1 + r’ (λ is finite population growth rate) • and plugged λinto equation: • Nt+1 = Ntλ

5. Today’s Goal • Nt+1 = Ntλ • Simple model of multiplicative(geometric) population growth (discrete type) • Multiplicative means the population increases in proportion to its size • i.e., population size increases (or decreases) by a constant fraction per year • (rather than adding, e.g. 50 individuals, per year) • Equation allows us to predict this year from last year, or next year from this year • Today: • We will derive an equation to predict population size over multiple time steps • e.g., 10, 20, or more years from now • We’ll assume our time step is equal to one year for today

6. Deriving Equation to Forecast Growth • Nt+1 = Ntλ • Assume λ is constant across time • (i.e., population grows at a constant rate each year) • Let’s plug specific time steps into the equation: • N1 = N0λ • N1is in both equations… • N2 = N0 λ2 • N2 = (N0 λ)λ • … we can substitute N1= N0 λ • in second equation • N2 = N1 λ • N3 = N0 λ3 • N3 = (N0 λ2)λ • N3 = N2λ • Similarly, • Nt = N0 λt • Pattern continues; eventually we arrive at:

7. Forecasting Population Growth • Nt = N0 λt • General equation for forecasting population size • Can also write this equation in terms of the components of λ: • Nt = N0 (1 + b’ – d’)t • Let’s look at growth of a real population… • Goal: Determine if we can apply our new equation to muskox population growth • If so, then use the model to forecast growth

8. Muskox Case Study • Nt = N0 λt • Our assumption when deriving • was that λwas constant across time (i.e., population grew at a constant rate) • Step 1: Determine if this assumption holds for muskox • Determined by: λ = Nt+1/Nt λ • Fig 1.3 in text • Fig 1.4 in text • Conclusion: λfluctuates, • but shows no trend over time

9. Muskox Case Study • Step 2: Determine if population actually exhibits geometric growth • Last class I said: • Plot curves upward: • Suggestive of “multiplicative growth” [geometric], but not diagnostic λ • Fig 1.3 in text • Fig 1.4 in text • If a population growth is geometric, then population size • should appear linear when expressed on a log scale

10. Muskox Case Study • Step 2: Determine if population actually exhibits geometric growth • Log scale* • *Could also have been plotted as log (Nt) • with “normal” axis • Fig 1.5 in text • Growth looks linear – population is exhibiting geometric growth

11. Muskox Case Study • Step 3: Determine λ(essentially, the average λthrough time) • to use in model: • Nt = N0 λt • Two methods: • Calculate geometric mean (book uses this way) • Use linear regression (book does not address this approach)

12. λDetermination - Geometric Mean • Method 1: Calculate geometric mean • Two types of means: • Arithmetic and geometric • MATH REVIEW • Arithmetic mean: Use when averaging sums: • 20, 22, and 24 people in 3 sections of a course • Total = 66 • Arithmetic mean: (20 + 22 + 24) / 3 = 22 • Checking the calculation: • Total number = 22 + 22 + 22 = 3*22 = 66 • Arithmetic mean works!

13. λDetermination - Geometric Mean • Method 1: Calculate geometric mean • Two types of means: • Arithmetic and geometric • MATH REVIEW • Geometric mean: Use when averaging a multiplying factor: • Example: λ for population growth • Animal Population Size: • Year 0: 1000 animals • Year 1: 1200 animals • Year 2: 1200 animals • Year 3: 1320 animals • Population Growth Rate: • λYear 0-to-1 = 1200/1000 = 1.2 • λYear 1-to-2 = 1200/1200 = 1.0 • λYear 2-to-3 = 1320/1200 = 1.1 λ = Nt+1 / Nt • Geometric meanis cube root of product of λs: • Mean λ=(1.2 * 1.0 * 1.1) 1/3= (1.32) 1/3 = 1.097 • Checking our calculation: • Increase from Year 0 to Year 3 is: • 1000 * 1.097 * 1.097 * 1.097 = 1320 animals • Nt = N0 λt • N3 = N0 λ3

14. λDetermination - Geometric Mean • Method 1: Calculate geometric mean • Two types of means: • Arithmetic and geometric • MATH REVIEW • Geometric mean: Use when averaging a multiplying factor: • Example: λ for population growth • Animal Population Size: • Year 0: 1000 animals • Year 1: 1200 animals • Year 2: 1200 animals • Year 3: 1320 animals • Population Growth Rate: • λYear 0-to-1 = 1200/1000 = 1.2 • λYear 1-to-2 = 1200/1200 = 1.0 • λYear 2-to-3 = 1320/1200 = 1.1 λ = Nt+1 / Nt • Arithmetic mean gives wrong answer • Exercise: Do the calculation to show that arithmetic mean does not work • (i.e., calculate arithmetic mean and plug into forecasting equation)

15. λDetermination – Linear Regression • Method 2: Linear Regression • Nt = N0 λt • Let’s start with our original equation: • Now let’s take the log of both sides (can also do ln): • log (Nt) = log (N0 λt) • log (Nt) = log (N0) + log (λt) • log Nt = log N0 + t log λ Intercept Slope This is a linear relationship between log Nt and t, with slope = log λ, intercept = log (N0) • Linear regression (i.e., plot) of log Ntvs. time (t) can provide slope, and therefore an estimate of λ(specifically, log λ)

16. λDetermination – Linear Regression • Method 2: Linear Regression • Nt = N0 λt • Let’s start with our original equation: • Now let’s take the log of both sides (can also do ln): • log (Nt) = log (N0 λt) • log (Nt) = log (N0) + log (λt) • log Nt = log N0 + t log λ Intercept Slope • Advantage of linear regression: • Can obtain statistical output that gives goodness of fit (R2), • which gives an estimate of uncertainty for λ

17. λDetermination – Linear Regression • See Excel file on website: • “Muskox Linear Regression.xlsx” • Method 2: Linear Regression • Equation and R2 obtained by adding a trendline • Slope = 0.062 = log λ • λ= 10 0.062 = 1.15 • log base 10: log10 • R2 indicates good fit

18. λDetermination – Linear Regression • Method 2: Linear Regression • Can now use our estimate of λ (1.15) and an estimate of population size to forecast future population size, • assuming that the population growth rate does not change • Nt = N0 λt • Exercise: • Given 511 muskox in 1964 and our estimate of λ as 1.15, • what would the population be in: • 1974 ? • 1984 ? • 1994 ?

19. Doubling Time • How long will it take for a population to double in size given its growth rate?  A common question in population analysis Key to answering this question is to recognize that the doubling of a population can be expressed as: Nt= 2N0 or Nt/N0= 2

20. Doubling Time Can develop a general doubling time equation using: Nt/N0= 2 Need to use this relationship with our population forecasting equation • Nt = N0 λt • Nt / N0 = λt • 2 = λt • log (2) • t = • log (2) = log (λt) • log (2) = t log (λ) • log (λ) • 0.301 We can calculate doubling time just knowing ! For muskox, = 1.15, tdoubling = 4.96 years • t = • log (λ) Can also use natural log (as in text): • ln (2) • 0.693 • t = • t = • ln (λ) • ln (λ)

21. Looking Ahead Next Class: Derive continuous population growth equation …and more!