A population of glasswing butterflies exhibits logistic growth. The carrying capacity of the population is 200 butterflies, and rmax, the maximum per capita growth rate, of the population is 0.10 butterflies/(butterflies*month). Calculate the maximum population growth rate for the population if the maximum growth occurs when N=k/2.
To calculate the maximum population growth rate, we need to use the logistic growth equation. The equation is:
dN/dt = rmax * N * (1 - (N / K))
Where:
- dN/dt represents the rate at which the population changes over time
- rmax is the maximum per capita growth rate
- N is the size of the population
- K is the carrying capacity of the environment
In this case, we are given that the carrying capacity (K) is 200 butterflies, and the maximum per capita growth rate (rmax) is 0.10 butterflies/(butterflies*month).
To find the maximum growth rate when N = K/2, we substitute N = 100 into the logistic growth equation and solve for dN/dt:
dN/dt = rmax * 100 * (1 - (100 / 200))
Simplifying the equation:
dN/dt = 0.10 * 100 * (1 - 0.5)
dN/dt = 0.10 * 100 * 0.5
dN/dt = 5 butterflies/month
Therefore, the maximum population growth rate for this population of glasswing butterflies occurs when the growth rate is 5 butterflies per month.