The Moran Equation

Solving the Moran Equation

For some values of r_i the Moran equation can be solved exactly, but often we must solve it numerically. We illustrate both methods for the equation

0.5^d + 0.5^d + 0.5^d + 0.25^d = 1

Numerical solution Most graphing calculators and computer algebra systems have routines for numerical solution of equations. For example, in Mathematica the command is

FindRoot[.5^d + .5^d + .5^d + .25^d == 1,{d,1}]

The {d,1} specifies that d is the variable, and provides a starting guess of 1 for the solution. This command uses Newton's method to find the approximate solution.

For this example, the FindRoot command gives the solution 1.72368.

Mathematica is installed in all of the AM&T clusters. A list is available at http://www.yale.edu/acs/cluster/.

Exact solution This instance of the Moran equation can be solved analytically when we note 1/4 = (1/2)². Then the Moran equation

3(1/2)^d + (1/4)^d = 1

can be written as

3(1/2)^d + ((1/2)²)^d = 1

Interchanging the exponents of the second term of the left side we obtain

3(1/2)^d + ((1/2)^d)² = 1

Writing

(1/2)^d = x,

the Moran equation becomes the quadratic equation

3x + x² = 1.

The quadratic equation gives

x = (-3 ± sqrt(13))/2.

Recalling x = (1/2)^d, which is positive, we have

x = (-3 + sqrt(13))/2

and so

d = Log((-3 + sqrt(13))/2)/Log(1/2).

This approach can be adopted to any situation in which all the scaling factors are (integer) powers of one of the scaling factors.

Return to the Moran equation.