A population of spotted hyenas in a South African National Park currently has 87 individuals. The carrying capacity for spotted hyenas in the park is estimated to be 125 hyenas & the intrinsic growth rate ($r$) is 0.25 hyenas/year/hyena. What is the instantaneous rate of population growth of the spotted hyena population?

• A. 22 hyenas/year.
• B. 7 hyenas/year.
• C. 15 hyenas/year.
• D. 13 hyenas/year.

Answer: (B)

Verified step by step guidance

1. Understand the concept of population growth: The instantaneous rate of population growth can be calculated using the logistic growth model, which considers the current population size, carrying capacity, and intrinsic growth rate.
2. Identify the formula for logistic growth: The formula to calculate the instantaneous rate of population growth is \( \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \), where \( N \) is the current population size, \( r \) is the intrinsic growth rate, and \( K \) is the carrying capacity.
3. Substitute the given values into the formula: Use \( N = 87 \), \( r = 0.25 \), and \( K = 125 \) in the formula \( \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) \).
4. Calculate the term \( \left(1 - \frac{N}{K}\right) \): This represents the proportion of the carrying capacity that is still available for growth. Substitute \( N = 87 \) and \( K = 125 \) to find this value.
5. Multiply the intrinsic growth rate \( r \), the current population size \( N \), and the calculated proportion \( \left(1 - \frac{N}{K}\right) \) to find the instantaneous rate of population growth.