Logistic Growth Rate Constant Calculation

This solution explains how to find the growth rate constant ($r$) for a population exhibiting logistic growth, given initial population size ($N_0$), carrying capacity ($K$), and the initial per capita growth rate.

Given Parameters

• Initial population size, $N_0 = 80$
• Carrying capacity, $K = 240$
• Initial per capita growth rate = $0.12$

Logistic Growth Model

The logistic growth model describes population growth that slows down as it approaches the carrying capacity ($K$). The rate of population change is given by:

Where:

• $\frac{dN}{dt}$ is the rate of population change
• $r$ is the intrinsic rate of increase (the growth rate constant)
• $N$ is the population size
• $K$ is the carrying capacity

The per capita growth rate is represented by $\frac{1}{N} \frac{dN}{dt}$. From the logistic equation, this is:

Calculating the Growth Rate Constant ($r$)

We are given that the initial growth rate (per capita) is $0.12$ when $N = N_0 = 80$. We can substitute the known values into the per capita growth rate equation:

1. Set up the equation: Use the initial values in the per capita growth rate formula. $ 0.12 = r \left( 1 - \frac{N_0}{K} \right) $
2. Substitute values: Plug in $N_0 = 80$ and $K = 240$. $ 0.12 = r \left( 1 - \frac{80}{240} \right) $
3. Simplify the fraction: $ 0.12 = r \left( 1 - \frac{1}{3} \right) $
4. Simplify the term in parentheses: $ 0.12 = r \left( \frac{2}{3} \right) $
5. Solve for $r$: Rearrange the equation to isolate $r$. $ r = 0.12 \times \frac{3}{2} $ $ r = 0.12 \times 1.5 $ $ r = 0.18 $

Conclusion

The growth rate constant ($r$) for the population is $0.18$. This corresponds to Option C.