Evolution in a nutshell

an alternative outline on evoution

and some consequences concerning valuations

by

Gregor Kjellström

            References          

Theorem 6.2.2. The theorem of Gaussian  adaptation

1. The gradient of the mean fitness of a normal p. d. f. with respect to m is equal to

   grad_mP(m) = P M^-1 ( m^* – m).

The maximizing necessary condition for mean fitness is m^* = m (at selective equilibrium).

2. The gradient of phenotypic disorder (entropy, average information, diversity) with respect to m – assuming P constant - points in  same   

    direction as grad_mP(m).

Proof: P(m) = ò  s(x) N(m – x) dx. Since differentiation is here allowed to the right of the integral sign we get

      ¶P(m)/¶m_j = ò  s(x)  { ¶ N(m – x) /¶m_j  } dx

     = - ò s(x) N(m – x) {  u_j^T M^-1(m_j – x_j) +  (m_j – x_j)^T M^-1 u_j }   / 2 dx,

where the components of the vector u_j are = 0, except for the component number j

which is = 1. Thus we have

     grad_m P(m) =  - ò  s(x)  N(m – x)  M^-1 (m – x)  dx

     =  - P  M^-1 { m - ò  x s(x)  N(m – x) dx / P }

     = P M^-1  ( m^* – m )

where we have introduced the mean of phenotypes of the set of selected parents

     m^* = [m_j^*] = ò  x s(x) N(m–x) dx / ò  s(x) N(m–x) dx =

           = ò  x s(x) N(m–x) dx / P

which proves part 1 of the theorem. For the proof of part 2 see theorem 6.2.3.

In contrast to Fisher's fundamental theorem, this result seems more reliable, because in a state of selective equilibrium, we have m^* = m and consequently no increase in P, i. e. grad_m P(m) = 0, but the phenotypic variance – displayed by M and H - must not be equal to zero. Instead, P has been maximized and - as will be shown by the next theorem - det(M) and H are simultaneously maximized with respect to variations in m, keeping P constant, even though this is only a sub-optimal solution as long as the condition M  proportional to  M^* is not utilized.

A correspondence to Fisher’s increase in mean fitness may now be derived from Gaussian adaptation. In this case the increase is defined from the offspring in one generation to the offspring in the next (it is assumed that M is fairly constant from one generation to the next). We have

   DP = (¶P/¶m₁) Dm₁ + (¶P/¶m₂) Dm₂ + … +  (¶P/¶m_n) Dm_n 

If the Gaussian is moved from m to m^* we get the approximation

   DP  = P (m^* -  m)^T M^-1 (m^* - m);