Island and Stepping-stone Models

Migration, Gene Flow, and the Differentiation of Populations

Ecologists and population biologists pay attention to very different aspects of animal migration. From an ecologist's point of view, they are interested in questions of when animals move, how far do they travel, and why do they migrate in the the first place. Ecologists may be interested in the seasonal migrations of birds, for example, or the movements of salmon to their home waters for breeding.

Population geneticists, however, are more interested in how genes change when an organism migrates. They couldn't care less about the distances, but rather how do the genes move from one population to another. In some cases animals might migrate in vast numbers, but outside of their breeding season. If they return to their home spawning ground (salmon), little or no gene migration will take place. In other cases, numerous breeding opportunities may be presented and the populations will exhibit gene flow along with the physical migration of the population.

Many animals have a patchy distribution based on their behavior and/or the distribution of their environment. As an example, there may be a series of ponds in an area, each inhabited by a population of frogs. Most of the breeding probably takes place within a single population (a particular pond), but frogs will move from one pond to another allowing an exchange of genes among the various populations (thus, gene flow can take place). Gene flow among these discrete populations can be modeled using island models and stepping-stone models.

Island Models

If individuals are just as likely to move from one population to another, that situation can be modeled using an island model.

Island Hopping

In this model, m is the immigration rate from one island to another for each generation. The "natives" for a particular island are all the rest of the individuals: (1-m). If the allelic frequency of natives is q_o and the allelic frequencies of the immigrants is q_m, then the frequency of the gene after one generation will be....

q₁=(1-m)q_o + mq_m

Thus, if no other evolutionary forces are affecting the population (mutation, genetic drift, etc.), the change in the gene frequency depends only on the migration rate and the original frequencies of the alleles in the native and migrating populations. Although this is only for migration into a single native population, the model can be easily expanded into multiple populations. Selection pressures can even change for the allele at each population. In the absence of selection the gene frequencies will eventually converge among all of the populations.

The following simulation assumes that there can be differences in the migration rate, but there is no differential selection and the population sizes are the same. It would be easy to make the model more realistic. For example, differing population sizes could be taken into account by simply weighting the allelic frequencies.

The Excel model is here. When the File Download box opens, save the model to your desktop (or in a temp folder). Double-click on the file to open it in Excel. You can then Alt-Tab between the program and the questions.  HINT: Copy the graphs to a Word document so you can make easy comparisons between runs. Be sure to include the values you change as a title for each graph.

1. Only change the starting values for the cells highlighted in yellow. All other cells contain equations. NOTE: the migration rates in cells B3-E3 are the migration rates into that particular population. Initially, the migration rates

2. Change the migration rates in cells B3-E3 (all rates should be <1.0; change only one at first). Make all the rates equal. What effect does migration rate have on the eventual stabilization of the allele among the four populations? Explain.

3. Now change the migration rates in cells B3-E3, but enter different values. What effect does that have on the stabilization of the allele in the model? Explain.

4. Change one of the rates to zero (or near zero. say, 0.001). What happens to that population? Explain.

5. Does it matter if the zero rates are at the beginning, middle, or end of the row? Explain.

6. Now change the starting allelic frequencies in cells B6-E6 (change only one at first). What effect does that have on the stabilization of the allele in the model. Explain.

7. Try entering symmetric values in cells B6-E6 (say, 0.1,0.4, 0.4, 0.1). What effect does that have on the stabilization of the allele in the model. Explain.

Stepping-Stone Model

Linear stepping-stone model

The spatial distribution of many populations does not fit the island model we just covered. In many cases individuals migrate to those populations that are closest to them rather than randomly (as in the island model). The model that fits this situation is a stepping-stone model. Here we explore a linear model. Often stepping-stone behavior is seen in populations that exist along an environmental gradient (such as mountains, etc.).

Individuals can migrate between adjacent populations

q₁ = [1 - (m₂₁ + m₂₃)]q₀ + m₁₂q_m1 + m₃₂q_m3

The above equation shows the allelic frequency at the end of generation 1 (q₁) in population 2. The variable q₀ represents the beginning allelic frequency for population 2 at time zero, while q_m1 and q_m3 are the current allelic frequencies in populations 1 and 3. The m_ij variable is as shown in the above diagram.

The Excel model is here. When the File Download box opens, save the model to your desktop (or in a temp folder).Double-click on the file to open it in Excel. You can then Alt-Tab between the program and the questions.  Only change the starting values for the cells highlighted in yellow. All other cells contain equations. HINT: Copy the graphs to a Word document so you can make easy comparisons between runs. Be sure to include the values you change as a title for each graph.

The allelic frequencies for populations 1 through 9 lie on a straight line since the migration rates are equal among those populations. If, however, you reduce a migration rate between two populations (try 0.5 or 0.1), this creates a barrier to dispersal and there's a sudden shift in allelic frequencies for the adjacent populations. Try several combinations until you're sure you understand the concept.

Question: Use the internet to find examples of real species that follow the stepping stone and island models (two species for each).