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

Then, set terms of quadratic as   a = t    b = - (m + t)   c = mq_m
                               
    = [ (m + t) [( (-1)²(m + t)² - 4tmq_m] ] / 2t          quadratic form

        =  [ (m + t) [m² + t² + 2mt - 4mt] ] / 2t              q_m = 1; expand ( - (m + t) )²
 
          [ (m + t) [t² + 2mt - 4mt] ] / 2t                      if  m < t , then m² << t² 

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

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

          [ (m + t)  (t)(1 + m/t - 2m/t) ] / 2t                   apply approximation
                                                                                        (1 x)     ( 1  (x/2))
                                                                                        because (1 + x/2)²  =  1 + x + (x/4)²
                                                                                                                         (1 + x)    iff x << 1
                                                                                                                       that is (- m / t) << 1

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

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

      2m / 2t  =  (m / t)                                if q_m = 1 as in standard model, & m < t

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Text material © 2025 by Steven M. Carr

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