Question

B. Analyzing Data 11.1: Answer questions #1-4 found on page 258 and below. Compute the missing values of A in the Table below, rounding each estimate to two decimal places. Check that the average (the arithmetic mean, defined below) of the seven values of 2 equals 1.03. If it does not, redo your calculations. Year () Population size Yearly growth rate () 0.80 1.26 1,000 800 1008 796 1,018 763 992 4 Use Equation 10.2 to calculate how large a population with a fixed growth rate of2- 1.03 and an initial size of 1,000 (No -1,000) will be after t-7 years. Compare your answer with the value shown in the table for year 7. How has year-to-year variation in affected the growth of the population? For multiplicative processes such as population growth, an altermative is to use the geometric mean (defined below and described more fully in Web Extension 11.1) instead of the arithmetic mean. Calculate the geometric mean of the 7 year-to-year values of A in the table. Use the geometric mean that you determined in Question 2 to calculate how large a population with an initial size of 1,000 will be after 7 years. Compare your answer with the data in the table and with your result in Question 1 (which was based on the arithmetic mean). To describe the growth of a population in a variable environment, would it be better to use the arithmetic or geometric mean of year-to-year values of 2? Explain.

Answer 1

We use the following formula for the calculation of the yearly growth rate ($\lambda$)

N_t+1= N_t x $\lambda$_t , t = (0,1,2,3,4,5,6)

So,

$\lambda$₀ = 0.80

$\lambda$₁ = 1.26

$\lambda$₂ = N₃/N₂ = 796/1008 = 0.79

$\lambda$₃ = N₄/N₃ = 1018/796 = 1.28

$\lambda$₄ = N₅/N₄ = 763/1018 = 0.75

$\lambda$₅ = N₆/N₅ = 992/763 = 1.30

$\lambda$₆ = N₇/N₆ = 1022/992 = 1.03

Average = (0.80 + 1.26 + 0.79 + 1.28 + 0.75 + 1.30 + 1.03)/7 = 1.03

Since Eqn 10.2 is not mentioned in the question, I'll use the equation =

N_t = N₀ x ($\lambda$ )^t

therefore, for t = 7 years and $\lambda$ = 1.03, we get

N₇ = N₀ x ($\lambda$ )⁷ = 1000 x (1.03)⁷ = 1229

As we can see the answer is more than the actual value. This is because of varying population sizes during the course of 7 years. For a lower population size, a growth rate will result is smaller growth as compared to larger population size, and since the population size is never going down in case of a fixed population growth rate, the growth at the end of 7 years is larger than the actual growth.

Geometric mean = ($\lambda$₀ x$\lambda$₁ x$\lambda$₂ x$\lambda$₃ x$\lambda$₄ x$\lambda$₅ x$\lambda$₆ )^1/7

⁼ (0.80 x 1.26 x 0.79 x 1.28 x 0.75 x 1.30 x1.03)^1/7 = 1.003

Using equation, N_t = N₀ x ($\lambda$ )^t

For t = 7 years and $\lambda$ = 1.003, we get

N₇ = N₀ x ($\lambda$ )⁷ = 1000 x (1.003)⁷ = 1021

As we can see, this value is closer to the actual value. Hence we can conclude that population growth follows geometric growth rather than arithmetic growth.

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