Figure 3. Relationships among chronological distributions’ shape and selection strength measures.

(A) Graphical representation of various fitness and selection strength measures by $K_X^{\prime }(\xi )$-plot. Blue curve represents $K_X^{\prime }(\xi )$. The area between the horizontal axis and $K_X^{\prime }(\xi )$ in the interval $0\le \xi \le 1$ outlined in red corresponds to population growth $\tau \mathrm{\Lambda }$. The gray and hatched regions correspond to ${⟨h(X)⟩}_{\mathrm{cl}}$ and $S_{\mathrm{KL}}^{(1)}[X]$, respectively. The area between $K_X^{\prime }(\xi )$ and $y={⟨h(X)⟩}_{\mathrm{rs}}$ corresponds to $S_{\mathrm{KL}}^{(2)}[X]$. (B-D) Representative shapes of $K_X^{\prime }(\xi )$ depending on ${\kappa }_3^{(X)}$. Assuming that the contributions from fourth or higher-order cumulants are negligible, $K_X^{\prime }(\xi )$ becomes convex downward when ${\displaystyle {\kappa }_3^{(X)}>0}$ (B); a straight line when ${\kappa }_3^{(X)}=0$ (C); and convex upward when ${\displaystyle {\kappa }_3^{(X)}<0}$ (D). (E-G) Relationships between third-order fitness cumulant and skewness of chronological distribution $Q_{\mathrm{cl}}(h)$.