Logistic Differential Equation
The logistic function is the solution of the simple first-order non-linear differential equation
with boundary condition P(0) = 1/2. This equation is the continuous version of the logistic map.
The qualitative behavior is easily understood in terms of the phase line: the derivative is 0 at P = 0 or 1 and the derivative is positive for P between 0 and 1, and negative for P above 1 or less than 0 (though negative populations do not generally accord with a physical model). This yields an unstable equilibrium at 0, and a stable equilibrium at 1, and thus for any value of P greater than 0 and less than 1, P grows to 1.
One may readily find the (symbolic) solution to be
Choosing the constant of integration ec = 1 gives the other well-known form of the definition of the logistic curve
More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative t, which slows to linear growth of slope 1/4 near t = 0, then approaches y = 1 with an exponentially decaying gap.
The logistic function is the inverse of the natural logit function and so can be used to convert the logarithm of odds into a probability; the conversion from the log-likelihood ratio of two alternatives also takes the form of a logistic curve.
The logistic sigmoid function is related to the hyperbolic tangent, A.p. by
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