The logistic equation is the standard nonlinear foil to linear equations. In particular, it is the test bed for more complex graphical analysis (section 3.2, Direction Fields and Phase Lines, p. 108-118), multiple steady states and linearized stability analysis (section 3.3, Steady States, Stability, and Linearization, p. 118-126), the limitations of analytic methods (e.g., separation of variables does not find the solution P = a/s, example 19, p. 152), and the significance of uniqueness (example 46, p. 190).
The emigration model, P' = kP - E(t) with E defined appropriately, motivates the study of the Laplace transform of both the unit step function and the Dirac delta function in sections 10.4, Ramps and Jumps, p. 546-553, and 10.5, The Unit Impulse Function, p. 556-559.