Genetic Drift is stochastic q [unpredictable, random]
                      (cf. deterministic q [predictable, due to selection, mutation, and (or) migration]

      Sewall Wright (1889 - 1988): "Evolution and the Genetics of Populations" (1931)

      Stochastic q > deterministic q in small populations:
            allele frequencies "drift" more in 'smaller' than 'larger' populations

      Drift most noticeable if s ≈ 0, and (or) N small (< 10), on the order of  N ≈ 1/s
            Wright - Fisher Model for haploid organisms,
                                                  finite size N (or 2N ~ diploid) over discrete generations

       q drifts between generations (variation decreases within populations over time);
                  eventually, any allele lost (q = 0) or fixed (q = 1) [all alleles lost, except one]

           Ex: Matlab model of Cepaea nemoralis snails

       q drifts among populations (variance increases among populations over time)
                  eventually, all fixed for one or another allele

      ** Variation 'fixed' or 'lost'  &  populations diverge due to chance **

      Evolutionary significance
            "Gambler's Dilemma" : if you play long enough, you win or lose everything
            All populations finite: many very small, somewhere or sometime
            Evolution occurs on vast time scales: "one in a million chance" a rate (or certainty)
            Reproductive success of individuals in variable: 0, 1, 2 ... / parent

       What happens in really long run?

      Effective Population Size (N_e)
            size of 'ideal' population N_exp with same evolutionary behavior as 'real' population with N_obs
                    same genetic variation (measured as H_obs)
                    same inbreeding coefficient F [see derivation]
                    same 'genetic drift' (measured as genetic variance)
                            in any single generation σ² = (pq)/2N

${\displaystyle V_{t}\approx pq\left(1-\exp \left(-{\frac {t}{2N_{e}}}\right)\right)}$over time t: 

              approximately,  number of breeding individuals in local population 

    Consider four special cases where N_e << N_obs  [observed 'count' of individuals]:

      (1) Unequal sex ratio

     N_e = (4)(N_m)(N_f) / (N_m + N_f)    [see derivation]
 
                 where N_m & N_f = counts of breeding males & females, respectively 

           "harem" structures in mammals (N_m << N_f)
            Ex.: if N_m  = 1 "Alpha Male" and  N_f = 200
                    then     N_e = (4)(1)(200)/(1 + 200) ≈ 4
 
            Single "Alpha" male elephant seal (Mirounga) does almost all breeding [NS 02-04]
                    [Elephant seals have very low genetic variation]

           eusocial (colonial) insects like ants & bees (N_f << N_m)
                     Ex.: if N_f = 1 "Queen" and  N_m= 100 drones
                                   then     N_e = (4)(1)(100)/(1 + 100) ≈ 4
                    Hives of 10,000s act like very small families

      (2) Unequal reproductive success
             In stable population, N_{offspring / parent} = 1
           "Random" reproduction follows Poisson distribution (N = 1  1):
                  Characteristics: = σ² ; if = 1,  σ² = σ
                  ~ 1/3 of (female) parents have 0, ~1/3 have 1, ~1/3 have 2 or more
 

        Conservation strategy: All animals bred equally, gene pool conserved
        K-strategy: keep N constant near carrying capacity (K) by investing parental care
         r-strategy: maximize success with high reproductive rate (r), little parental investment    

      (3) Population size variation over time

           N_e = harmonic mean of N  = inverse of arithmetic mean of inverses
                                                          [much closer to smallest value in series]
                           n
           N_e = n /  [ Σ(1/N_i) ]  where N_i = pop size in i th generation 
                           i=1

       Populations exist in fluctuating environments:
                Population numbers unstable over (very) long periods of time:
                  10^1~2 forest fire , 10^2~3 flood , 10^-4 ice age

           Ex.: if typical N = 1,000,000  & every 100th generation 'bottlenecks' to N = 10 :
                         then  N_e = (100) / [(99)(10^-6) + (1)(1/10)]  100 / [10^-4 + 10^-1] 100 / 0.1 = 1,000 [HOMEWORK]

       Founder Effect & Bottlenecks:
                Populations started by (very) small number of individuals,
                    or undergo dramatic reduction in size [NS 02-03]

                Ex.: Founder origin of Newfoundland moose (Alces):
                            2 bulls + 2 cows introduced at Howley in 1904
                            [1 bull + 1 cow at Gander in 1878 didn't succeed].

                Ex.: Bottlenecks in population cycles: Hudson Bay Co. trapping records (Elton 1925)
                        Population densities of arctic lynx, hare, muskrat cycle over several orders of magnitude
                        Does Lynx cycle "chase" hare cycle ?

    (4) Non-uniform population dispersion

            N_e = (4π)ẟσ²

                Ex.: In old-field Deer Mice (Peromyscus maniculatus),
                            ẟ = 6 / 10⁴ m^-2, σ = 114 m, so σ² = 1.3 x 10⁴ m²
                       N_e = 98 / wild "neighborhood"
                            [cf: Canadian Rules football field = 6 x 10³ m²: ẟ = 3.4 mice / field ] 
              
                 cf. Mus mice in barns at 4X density & 1/10 dispersion: N_e = 4

Effect of drift on genetic variation in populations

          Larger populations more variable (higher H) than smaller
                  if s = 0:  Heterozygosity (H) reflects balance between
                                loss of alleles by drift and replacement by mutation

           The Neutral Equation: H = (4N_eµ) / (4N_eµ + 1)
                    Ex.: if µ = 10^-6   & N_e = 10⁶   then   N_eµ = 1   and H_exp = (4)/(4 + 1) = 0.80
                    But for protein electrophoretic data, typical H_obs 0.20  which suggests  N_e 10⁵
                    => Natural populations have much smaller N_e than observed N_c count

            Stochastic effects may often be more important than deterministic processes in evolution.