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

2. 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

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

4. 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)

5. 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

6. 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

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 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

9. 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

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 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)

12. 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: ( ) / 3 = 22
Checking the calculation:
Total number = = 3*22 = 66
Arithmetic mean works!

13. 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 * * * = 1320 animals
Nt = N0 λt
N3 = N0 λ3

14. 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)

15. 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)

16. 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

17. 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 = = log λ
λ = = 1.15
log base 10: log10
R2 indicates good fit

18. 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 ?

19.  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

20. 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)

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