The right side of the differential equation can be written as the quadratic polynomial $ \ -\frac{r}{K} \ N^2 + rN - H $ $ = \ -\frac{r}{K} · \left(N^2 - KN + \frac{KH}{r} \right) \ , $ the zeroes (if any) of which give the equilibrium values of $ \ N \ $ for the system,
$$ N \ = \ \frac{K \pm \sqrt{K^2 - \frac{4KH}{r}}}{2} \ \ . $$
Alternatively, we may "complete the square" in the polynomial to produce
$$ \frac{dN}{dt} \ = \ -\frac{r}{K} \ \left(N - \frac{K}{2} \right)^2 \ - \ \left(H \ - \ \frac{rK}{4} \right) \ \ . $$
These expressions indicate that there are three cases to consider, depending on the "size" of the harvesting term. The parabola alluded to by Lutz Lehmann appears in the graph of $ \ \frac{dN}{dt} \ $ versus $ \ N \ \ . $

From either of the descriptions given above, we can define a "critical" harvesting rate $ \ H_c = \ \frac{rK}{4} \ . $ For $ \ H < H_c \ \ , $ there are two equilibria, which we might call $ \ N_{+} \ $ and $ \ N_{-} \ $ ; since $ \ \frac{dN}{dt} > 0 \ $ for $ \ N_{-} < N < N_{+} \ $ and $ \ \frac{dN}{dt} < 0 \ $ otherwise, the population "runs asymptotically" to $ \ N_{+} \ $ (the stable equilibrium) and away from $ \ N_{-} \ $ (the unstable equilibrium); if the initial population is less than $ \ N_{-} \ $ , then the population drops to zero (actually, to "negative infinity"). If $ \ H > H_c \ $ ("over-harvesting"), any initial population ultimately declines to zero. The threshold rate, $ \ H = H_c \ $ ("critical harvesting"), will have an initial population $ \ N(0) > \frac{K}{2} \ $ tend to $ \ \frac{K}{2} \ $, but an initial population $ \ N(0) < \frac{K}{2} \ $ will fall to zero; this peculiar "one-sided" equilibrium is a merger of the two equilibria of the sub-critical harvesting case and so is sometimes called "semi-stable".
The parabola is useful for characterizing this dynamical system with a single dependent variable, but it is not the "phase-portrait" itself. As Lutz Lehmann remarks, rather than a "portrait" (which is the term used for systems of more than one variable), we produce what is variously called a "slope-", "direction-" or "flow-field", in which we plot curves for $ \ N \ $ as a function of $ \ t \ $ with the tick marks representing values of $ \ \frac{dN}{dt} \ \ , $ as is also done in phase-portraits. (Since $ \ \frac{dN}{dt} \ $ is independent of time, all of the tick-marks at any level of $ \ N \ $ all have the same slope.) Direction fields are displayed below for the three harvesting cases discussed, with curves for various initial populations included.
$ \quad \quad \quad \quad \large \text{sub-critical harvesting} $
$ \quad \quad \quad \quad \large \text{critical harvesting} $
$ \quad \quad \quad \quad \large \text{over-harvesting} $