Question

Suppose that a population develops according to the logistic equation(dP/dt)=0.05P−0.00005P2 where t is measured in weeks.(a) What is the carriying capacity?(b) Is the solution increasing or decreasing when P is between 0 and the carriying capacity?(c) Is the solution increasing or decreasing when P is greater than the carriying capacity?

Answer 1

a.

$\dfrac{\mathrm dP}{\mathrm dt}=0.05P-0.00005P^2=0.00005P(1000-P)$

so that $\frac{\mathrm dP}{\mathrm dt}=0$ for $P=0$ or $P=1000$. The derivative being equal to 0 gives you some information about the long-term stability of the population $P$; namely, that the carrying capacity is 1000.

b. If $0$, then $0.00005P>0$ and $1000-P>0$, so that $\frac{\mathrm dP}{\mathrm dt}>0$, which means the population would be increasing.

c. If $P>1000$, then $0.00005P>0$ but $1000-P$, so $\frac{\mathrm dP}{\mathrm dt}$ and the population would be decreasing.