As a modification to the population model considered in the previous two problems, suppo...

As a modification to the population model considered in the previous two problems, suppose that P(t) satisfies the initial-value problem

where r, C, T , P₀ are positive constants, and 0 < T < C. Perform a qualitative analysis of this model. Sketch the slope field and some representative solution curves in the three cases 0 < P₀ < T, T < P₀ < C, and P₀ > C. Describe the behavior of the corresponding solutions.

The next two problems consider the Gompertz population model, which is governed by the initial-value problem where r, C, and P₀ are positive constants.