Background – A re-look at natural selection using ABMs

Simple Selection from Martin Nowaks book Evolutionary Dynamics, or ‘Survival of the fittest’ when total population is held constant

We define $x_i$ to be the size of the population of the $i$ -th type(phenotype).

We define $f_i$ to be the fertility of the $i$ -th type. A very simplistic model where the population size is not held constant, would then be:

$\dot{x}_i = f_i x_i$

We now define the population size across all types to be constant; $\sum\dot{x}_i=0$ . To account for this extra definition, we extend the simplistic model to:

$\dot{x}_i=x_i ( f_i - \phi )$

Then $\phi\sum x_i=\sum f_i x_i$ . If $x_i$ is the respective type frequency in the population, we can make a further simplification that $\sum x_i=1$ . Which gives the standard formula for selection:

$\phi=\sum f_i x_i$

The variable $\phi$ can be thought of as an average fitness. Then it is easily seen that the type with the largest $f_i$ always wins; or in a phrase ‘survival of the fittest’. Criticism of phrase here please.