Quadratic solution of the Migration / Selection equilibrium

To solve q_i  = tq_i² - (m + t)q_i + mq_M

Recall the quadratic formula: 0 = [-b (b² - 4ac)] / 2a
                                                    
        Note that   (1 x)     ( 1  (x/2))      [ approximation ]
                      because (1 + x/2)²  =  1 + x + (x/4)²     (1 + x)    iff x << 1,
                 and  q_M = 1 (a constant) in standard model

Then, set terms of quadratic as   a = t    b = - (m + t) = - m - t   c = mq_M
                               
    = [ (m + t) [(-m - t)² - 4tmq_M] ] / 2t                quadratic form

        =  [ (m + t) [m² + t² + 2mt - 4tmq_M] ] / 2t       expand (-(m + t))²
 
          [ (m + t) [t² + 2mt - 4tmq_M] ] / 2t                 if  m < t , then m² << t² 

          [ (m + t) [t² + (t²)(2m/t - 4mq_M / t)] ] / 2t      create common t² term

          [ (m + t)  (t) [1 + 2m/t - 4mq_M / t] ] / 2t         factor out t² term as t

          [ (m + t)  (t)(1 + m/t - 2mq_M / t) ] / 2t           apply approximation
                                                                                            x = (2m/t - 4mq_M / t) / 2

For which the negative root, if  t > 0, is

      (m + t) - [t + m - 2mq_m] / 2t                        t & m terms cancel [ pretty !]

      2mq_m / 2t  =  (m / t) (q_M)

                       =  (m / t)                                    if q_m = 1 as in standard model

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Text material © 2021 by Steven M. Carr

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