The exponential growth equation (dN/dt = rN) assumes that the population growth rate (r) is constant, which, over long periods of time, is not realistic. Here we drop that assumption and assume instead that r declines in a straight line as density (N) increases.
Mathematically, the equation for a straight line is y = mx + b, where m is the slope of the line and b is the y intercept. In our case, we replace y with r and x with N to draw a straight line in which the population growth rate r decreases as a function of the density N.
As shown in Figure A, the y intercept is rmax and slope of the line is –rmax/K. Thus, the equation for the decline of r with density is
where N is the population density, rmax is the maximum growth rate, and K is the density at which the population growth rate (r) equals zero.
Next, we replace the constant value of r found in exponential growth with the relation we have just derived. This substitution changes the equation dN/dt = rN into the equation
Finally, if we rearrange the new equation and make it less cumbersome by relabeling rmax as r, we have the logistic equation
