A population of lynx was studied in northern Ontario. There was an initial population of 120 lynx during the next 12 months 10 lynx were born into this population and 40 adults died. A) what is the growth rate of this population? B) if the growth rate remains consistent, what would be the population in five years?
The correct answer and explanation is:
A) What is the growth rate of this population?
To calculate the growth rate:
- Initial population = 120 lynx
- Number of births = 10 lynx
- Number of deaths = 40 lynx
- Net change = Births – Deaths = 10 – 40 = -30 lynx
- Growth rate = (Net change / Initial population) × 100
= (-30 / 120) × 100 = -25%
Answer to A: -25%
B) If the growth rate remains consistent, what would be the population in five years?
This is a compound decrease problem. Each year the population decreases by 25%, meaning it retains 75% of its size year over year.
Use the formula for exponential decay:
P = P₀ × (1 + r)ⁿ
Where:
- P₀ = initial population = 120
- r = growth rate per year = -0.25 (so 1 + r = 0.75)
- n = number of years = 5
P = 120 × (0.75)^5
P ≈ 120 × 0.2373 ≈ 28.48
Rounded to the nearest whole number: 28 lynx
Answer to B: 28 lynx
300-Word Explanation:
To determine the growth rate of a population, the basic formula involves comparing the change in population over a period to the initial size. In this case, the lynx population started at 120. Over 12 months, 10 new lynx were born, and 40 adult lynx died. That leads to a net change of -30 lynx.
Dividing this net change by the original population (−30 ÷ 120) and multiplying by 100 gives a growth rate of −25%. This negative value shows the population is shrinking, not growing.
To predict what will happen over a longer time, like five years, we apply exponential decay. This is because each year, the population does not just lose 25 lynx; instead, it loses 25% of whatever the population is at that point. After one year, it’s 75% of the original, then 75% of that amount the next year, and so on.
Using the formula for exponential decay: P = P₀ × (1 + r)ⁿ, where the rate r is −0.25 (or +0.75 as a multiplier), we plug in the values: 120 lynx initially, and n = 5 years.
The result is about 28 lynx remaining after five years. This demonstrates how population dynamics can decline quickly under high mortality or low birth rates and why conservation efforts often focus on balancing these factors.