Solution of the Crow-Kimura model with changing population size and Allee effect

The Crow-Kimura model is commonly used in the modeling of genetic evolution in the presence of mutations and associated selection pressures. We consider a modified version of the Crow-Kimura model, in which population sizes are not fixed and Allee saturation effects are present. We demonstrate the evolutionary dynamics in this system through an analytical approach, examining both symmetric and single-peak fitness landscape cases. Especially interesting are the dynamics of the populations near extinction. A special version of the model with saturation and degradation on the single-peak fitness landscape is investigated as a candidate of the Allee effect in evolution, revealing reduction tendencies of excessively large populations, and extinction tendencies for small populations. The analytical solutions for these dynamics are presented with accuracy $\mathcal{O}(1/N)$, where $N$ is the number of nucleotides in the genome.

Figure 1

Plot of $T\left(x\right)$ as computed from Eq. (12), with $x_0=1$ and $N=1000$. The tendency for the population to diverge from $x_0$ as time progresses can be observed, reflecting the effects of mutation on the gene pool. The inverse function $X\left(t\right)$ can also be derived. Numerical calculations coincide with the presented analytics by Eq. (12) with accuracy of $0.2\%$.

Figure 2

Comparison of our analytical solution (solid line), as given by Eqs. (7) and (12), and numerical solutions (dots) of base equations Eqs. (1) and (2). A fitness function $f\left(x\right)=x^2$ was used, with parameters $D=0.3$, $N=1000$, $x_0=1$, and $\kappa =1$.

Figure 3

Plot of steady state population size $\widehat{P}$ against fitness parameter $c$. A fitness function $f\left(x\right)=c{x}^2/2$ was used, with parameters $D=0.2$, $N=1000$, $x_0=1$, and $\kappa =1$. The solid line represents our analytical result; dots represent numerical calculations.

Figure 4

The analytical population dynamics, as given by Eq. (24). The analytical and numerical solutions coincide with accuracy $\mathcal{O}(1/N)$. A single-peak fitness function $r_0=2$ was used, with parameters $D=0.2$, $N=1000$, $x_0=1$, $\widehat{P}\left(0\right)=10$, $\kappa =0.001$, and $a=0$.

Figure 5

The population dynamics, as given by Eqs. (29) (smooth line) and (32) (dashed lines). A single-peak fitness function was used, with parameters $r_0=2$, $D=1.5$, $N=1000$, $x_0=1$, $\kappa =0.001$, and $\alpha =0.02$. The lower line corresponds to $\widehat{P}\left(0\right)=14.8$; the upper line corresponds to $\widehat{P}\left(0\right)=16$. There is a phase transition near the point $\widehat{P}\left(0\right)=15$. The model has a declining tendency for population size at large $\widehat{P}>20000$, and extinction occurs for small initial population size $\widehat{P}\left(0\right)<15$, typical of the Allee effect.