# The Moran Equation

## Solving the Moran Equation

 For some values of ri the Moran equation can be solved exactly, but often we must solve it numerically. We illustrate both methods for the equation 0.5d + 0.5d + 0.5d + 0.25d = 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. For this example, the FindRoot command gives the solution 1.72368. 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 formula gives x = (-3 ± √(13))/2. Recalling x = (1/2)d, which is positive, we have x = (-3 + √(13))/2 and so d = Log((-3 + √(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.