Figure 1.

Illustration of an example of a generalized demographic model as introduced in the first section of Materials and Methods. This model consists of five epochs (starting from the present on the right): (1) faster-than-exponential ($b>1$) growth (forward in time) from $N_{1,\mathrm{f}}$ to $N_{1,\mathrm{i}}$ between $T_0=0$ and $T_1;$ (2) linear decline (a special case of generalized decline when $b=0$) from $N_{2,\mathrm{f}}$ to $N_{2,\mathrm{i}}$ between $T_1$ and $T_2;$ (3) exponential growth (a special case of generalized growth when $b=1$) from $N_{3,\mathrm{f}}$ to $N_{3,\mathrm{i}}$ between $T_2$ and $T_3;$ (4) slower-than-exponential ($b<1$) decline from $N_{4,\mathrm{f}}$ to $N_{4,\mathrm{i}}$ between $T_3$ and $T_4;$ and (5) constant population size (a special case of generalized growth when $r=0$) at $N_{5,\mathrm{i}}=N_{5,\mathrm{f}}$ starting from $T_4$, which lasts indefinitely backward in time ($T_5=\mathrm{\infty }$). The ending population size of the previous epoch is not necessarily the beginning population size of the next epoch (e.g., $N_{2,\mathrm{f}}\ne N_{3,\mathrm{i}}$, $N_{4,\mathrm{f}}\ne N_{5,\mathrm{i}}$), corresponding to an instantaneous population size change at that time.