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

Outline for Today Goal for Today: Last Class: 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

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

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

Today’s Goal Nt+1 = Nt λ 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

Deriving Equation to Forecast Growth Nt+1 = Nt λ 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 = N0 λ N1 is in both equations… N2 = (N0 λ) λ N2 = N0 λ2 … we can substitute N1= N0 λ in second equation N2 = N1 λ Similarly, N3 = N2 λ N3 = (N0 λ2) λ N3 = N0 λ3 Pattern continues; eventually we arrive at: Nt = N0 λt

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

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

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

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

Muskox Case Study Nt = N0 λt Step 3: Determine λ (essentially, the average λ through 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)

λ Determination - 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!

λ Determination - 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: λ Year 0-to-1 = 1200/1000 = 1.2 λ Year 1-to-2 = 1200/1200 = 1.0 λ Year 2-to-3 = 1320/1200 = 1.1 λ = Nt+1 / Nt Geometric mean is 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

λ Determination - 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: λ Year 0-to-1 = 1200/1000 = 1.2 λ Year 1-to-2 = 1200/1200 = 1.0 λ Year 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)

λ Determination – Linear Regression Method 2: Linear Regression Let’s start with our original equation: Nt = N0 λt 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 Nt vs. time (t) can provide slope, and therefore an estimate of λ (specifically, log λ)

λ Determination – Linear Regression Method 2: Linear Regression Let’s start with our original equation: Nt = N0 λt 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

λ Determination – Linear Regression Method 2: Linear Regression See Excel file on website: “Muskox Linear Regression.xlsx” Year Nt log Nt 1948 1 57 1.756 1949 2 65 1.813 1950 3 61 1.786 1951 4 76 1.883 1952 5 84 1.924 1953 6 98 1.992 1954 7 109 2.038 1955 8 127 2.102 1956 9 138 2.140 1957 10 157 2.197 1958 11 200 2.300 1959 12 228 2.357 1960 13 282 2.451 1961 14 322 2.508 1962 15 386 2.587 1963 16 444 2.647 1964 17 511 2.708 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

λ Determination – 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 ?

 A common question in population analysis 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

Doubling Time Nt = N0 λt Nt / N0 = λt 2 = λt t = log (2) log (λ) 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 t = log (2) log (λ) log (2) = log (λt) log (2) = t log (λ) t = 0.301 log (λ) We can calculate doubling time just knowing ! For muskox,  = 1.15, tdoubling = 4.96 years Can also use natural log (as in text): t = ln (2) ln (λ) t = 0.693 ln (λ)

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