Link between Population Dynamics and Dynamics of Darwinian Evolution

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of similar populations. Similarity implies that the relative dynamics of the populations is slow compared to, and decoupled from, their aggregated dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the singular points of the evolutionary state space. Here “evolutionary branching” may occur. The diversification of life forms thus is demonstrated to be a natural consequence of the Darwinian process.

Figure 1

Course of evolution in the Lotka-Volterra model (19). Horizontal axes represent the strategy interval $[-1,1]$. Left pane: time dependence. Small panes: Instantaneous fitness function (curve, horizontal line represents zero) superimposed on the population distribution (gray). Each small pane corresponds to an instant of time represented by a horizontal line on the left pane. Observe that uphill evolution ends up in arriving at the minimum of the fitness function, where evolutionary branching occurs. After the branching, the two subpopulations evolve away. $K=10000$, $\sigma =0.5$; mutation probability: 0.002. See 18 for the details of the stochastic modeling of the mutations, which are not rare, and for the multidimensional results.