Extension to background – A re-look at natural selection using ABMs

Extension of ‘Survival of the fittest’ to varying population size

If $x_i$ is the frequency of type $i$ within the population, with $P$ being the population size, then $X_i$ is the size of the population of type $i$ .

$X_i=P x_i$

$\dot{X}_i=\dot{P}x_i+P\dot{x}_i$

We then assume that the dynamical equation for $X_i$ where $\psi$ is an unknown is:

$\dot{X}_i=X_i(f_i-\psi)$

Converting into an equation in $x_i$ :

$\dot{P}x_i + P\dot{x}_i=P x_i (f_i-\psi)$

$\dot{x}_i = x_i \left(f_i- \frac{\dot{P}}{P}-\psi\right)$

To find the unknown $\psi$ we sum across $i$ :

$0=\sum f_i x_i - \frac{\dot{P}}{P}-\psi$

$\psi = \sum f_i x_i - \frac{\dot{P}}{P}$

Substituting back into the dynamical equation for $x_i$ :

$\dot{x}_i=x_i \left( f_i-\sum f_i x_i \right)$

Or,

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

The same dynamical equation for constant population size. So ‘Survival of the fittest’ holds for varying population size as well.

‘Survival of the fittest’ sometimes means ‘Survival of none’

We use the idea that organism and environment co-evolve (see the earlier section on Richard Lewontin explaining ecological ideas) to demonstrate that the fittest type can sometimes drive the population to extinction.

We generate a simple model by abstracting the environment away (whether it is biological or geophysical) leaving the carrying capacity $C$ of the population. Then we can say that the carrying capacity reacts only to the frequencies of the types as below:

$C=\sum p_i x_i$

Where $p_i$ is the contribution towards the carrying capacity from organisms of type $i$ . So in this highly simplified model, only the organisms in the population affect the environment that they experience. Or, the environment is constructed by the organisms in the population.

We choose the ‘fittest’ type to be $i=1$ , and select $p_1<0$ whilst all other $p_i>0$ . We can see immediately that at some time before $x_1(t)=1$ , the population will have gone extinct, driven there by the ‘fittest’ organisms. This possibility we could call ‘Survival of none’, a theoretical model of a race to the bottom.

Another way of putting this, is that types do not help nor hinder the population size, and sometimes don’t even help themselves.

We shall choose to use the linear model of population carrying capacity $C(t)$ as it is the simplest to implement in our ABM studies.