Consider two population functions P₁(t) and P₂(t), both of which satisfy the logistic equation with the same limiting population M but with different values k₁ and k₂ of the constant k in Eq. (3). Assume that k₁2. Which population approaches M the most rapidly? You can reason geometrically by examining slope fields (especially if appropriate software is available), symbolically by analyzing the solution given in Eq. (7), or numerically by substituting successive values of t .

dP dt = kP(M-P), (3)