Chapter 1.5, Problem 9P

Consider the population model

$\frac{dP}{dt}=r\left(P-T\right)P$, $P\left(0\right)=P_0$, $\left(\mathrm{1.5.7}\right)$

where $r$, $T$, and $P_0$ are positive constants.

(a) Perform a qualitative analysis of the differential equation in the initial-value problem $\left(\mathrm{1.5.7}\right)$ following the steps used in the text for the logistic equation. Identify the equilibrium solutions, the isoclines, and the behavior of the slope and concavity of the solution curves.

(b) Using the information obtained in (a), sketch the slope field for the differential equation and include representative solution curves.

(c) What predictions can you make regarding the behavior of the population? Consider the cases $P_0<T$ and $P_0>T$. The constant $T$ is called the threshold level. Based on your predictions, why is this an appropriate term to use for $T$?