When `P` is small relative to `M`, the rate of growth `(dP)/(dt)` is approximately `kPM`, i.e., a constant times `P`. This means that `P` grows almost exponentially, i.e., `P~~P_0e^(kMt)`.
When `P` is close to `M`, the factor `M-P` is close to zero and `(dP)/(dt)~~0.` This means that `P` is approximately constant, which corresponds to the leveling off of the curve in Figure 1.
If `P` is greater than `M`, then the factor `M-P` is negative, so the population decreases. As `P` approaches `M`, `(dP)/(dt)` approaches `0,` and the curve levels off to the steady-state of `M`.