Effects of genetic drift on population dynamics

So far we have assumed that the population has a constant size - equivalent to assuming a demography with a finite rate of demographic growth equal to one - or that the population fluctuations are linked to exogenous factors, not intrinsic to the population. We can, however, wonder what may happen if the demographics of the population is also linked to intrinsic factors, such as its density. Many data confirm that the demographic rates depend on both the population size and its genetic structure. For example, in the plant Ipomopsis aggregata, typical of Arizona's mountain areas, the success of seed germination significantly depends on the population size (Fig. 21). In fact, smaller populations are characterized by smaller plants which produce smaller seeds that are more subject to environmental stress. However, when the plants are artificially pollinated with pollen from large populations (thus characterized by greater genetic variability) the germination success increases significantly. This proves that the rate of demographic growth of the plant is positively influenced by the genetic diversity of each population.

Figure 21: Percentage of germinating seeds in populations of different size for the plant Ipomopsis aggregata in Arizona. Reworked from Heschel and Paige (1995).

A very simple way to introduce genetic deterioration into a population demographics is to assume that the finite rate of growth is an increasing function of Wright's factor, or in other words the rate of population growth from one generation to the next is lower than optimal if there is a reduction of heterozygosity between generations. For the sake of simplicity we can assume that the finite growth rate is proportional to the Wright factor and equal to the optimal value $\Lambda ({N_t})$ when there is no loss of heterozygosity (i.e., when Wright's factor is equal to 1). If we denote by $\gamma$ the ratio of effective population size to total population size ( $N_e = \gamma N$), we can then write the following simple model of population dynamics

$${N_{t + 1}} = \left( {1 - \frac{1}{{2\gamma {N_t}}}} \right) \Lambda ({N_t}){N_t} .$$ (2.7)

Of course for $\gamma N < \frac {1}{2}$ Wright's factor would be negative and must therefore be set equal to zero. Actually, the effective population sice cannot be smaller than 2, because otherwise one of the two sexes would be absent. In other words genetic deterioration in small populations has such a large effect that the rate of population growth vanishes. It is easy to understand that population dynamics as described by eq. 5 is characterized by critical depensation. For example, Fig. 22 displays a Moran diagram for a Beverton-Holt model in which the Wright factor has been introduced.

Figure: Moran diagram of a Beverton-Holt model ($\lambda =3$, $\alpha =0.06$, curve in grey), modified via the Wright factor (eq. 5). The ratio $\gamma$ of effective population size to total size is equal to 20%. The non-trivial unstable equilibrium is denoted by $I$, while the non-trivial stable equilibrium is labelled $S$.

It is worth remarking that in this case extinction is a stable equilibrium for the population. There also exist two non-trivial equilibria: one with lower abundance is unstable and acts as an extinction threshold, while the one with higher abundance is stable. It is therefore apparent that genetic deterioration is a phenomenon that leads to consequences very similar to those we have identified and discussed in section 3 of this chapter devoted to the Allee effect.