(This exercise is based on Karels, T. J. and R. Boonstra. 2000. Concurrent density dependence and independence in populations of Arctic ground squirrels. *Nature* 408: 460–463.)

[*Note:* The reference above links directly to the article on the journal’s website. In order to access the full text of the article, you may need to be on your institution’s network (or logged in remotely) so that you can use your institution’s access privileges.]

In Chapter 9 you learned about two types of factors that influence population growth: density-dependent factors and density-independent factors. Density-independent factors can *determine* population size; that is, they make population sizes go up or down in a way that has no relationship to current population size. Density-dependent factors *regulate* population size; that is, they may make population sizes go up or down, but whether they go up or down depends on the current population size. We will use a recent paper that experimentally assessed the roles of density-dependent and density-independent factors to explore the effects and interactions of these two types of factors.

Read the abstract of the paper by Karels and Boonstra (2000) about the concurrent effects of density-dependent and density-independent factors on populations of Arctic ground squirrels. Along with population growth rate, the authors measured the proportion of females lactating, the proportion of females that weaned a litter, litter size, summer survival rate, and overwinter survival rate. Density dependence was observed in the proportion of females that weaned a litter and in overwinter survival rate. Evidence of density independence was observed in the proportion of females that weaned a litter. The proposed mechanism for this density-independent effect was winter snow depth and its effect on the body condition of overwintering females.

**Question 1**

Now, let’s assess the effect of Karels and Boonstra’s food addition treatment on density-dependent (acting with increasing severity to increase mortality rates or decrease reproductive rates as density increases) population change. Refer to Figure 1 below (based on Figure 2 from the paper), which gives (the natural log of) the ratio of population size at time *t* to that at time *t* + 1 on the *y* axis and population density (on a log scale) on the *x* axis. Thus, the *y* axis can be interpreted as measuring the rate of population change from one year to the next. The line through the green circles is the best fit of the relationship between the rate of population change and density (for the unmanipulated sites in the current study). Note that several of these sites had been manipulated previously, so we are seeing the recovery from experimental to natural population sizes. The blue circles represent three sites that had food additions during the current study, but they are not included in the calculation of the best-fit relationship.

To get an idea of the effect of food addition (a density-dependent factor) on the rate of population change, calculate the difference in population change between the three food addition sites and the three sites with comparable densities without added food. Do this by averaging the value from the *y* axis for the three food addition sites and the three sites with no added food, then take the anti-log of the average value for each group to display the values in normal units. How do the ratios compare between the food and non-food sites?

**Figure 1** Density-dependent rates of population change of arctic ground squirrels after the experimental treatments of the Kluane project were removed in spring 1996. Data are shown for controls and for the former experimental treatments. The negative slope of the dotted line suggests that low density populations show no change in population size from year to year, but high density populations show negative population growth from year to year.

**Question 2**

Now let’s look at the difference in magnitude of effect between density-dependent and density-independent factors. Refer to Figure 2 below (based on Figure 3a in the paper) showing the effect of density on proportion of females that successfully weaned a litter in two different years, with and without food additions. The effect of year (1996 to 1997) is evidence for a density-independent effect. In other words, there is something about 1996 other than density that caused a higher proportion of females to wean a litter.

**Figure 2** Density-dependence plots of the proportion of females weaning a litter. Data are shown for controls and for former experimental treatments of the Kluane project after the treatments were removed during spring 1996.

Calculate the magnitude of this density-independent effect by calculating the average proportion of females that weaned a brood separately for 1996 and 1997. Do *not* include the food addition treatments in this calculation. What is the magnitude of the density-independent effect?

**Question 3**

Now, using the same graph, let’s look at the effect of food addition (a density-dependent effect) on weaning rate in 1996 and 1997.

First, calculate the average difference in weaning rate between the 1996 food additions (green squares) and the two most comparable densities without food addition in 1996. Next, calculate the difference in weaning rate between the 1997 food addition (brown square) and the most comparable density without food addition in 1997. Finally, calculate the difference in weaning rate between 1996 and 1997 in the food addition treatments. Which effect has a greater magnitude, the density-dependent effect (food addition) or the density-independent effect?

© 2011 Sinauer Associates, Inc.