How can we detect density dependence in the field? For a density-independent population, Tanner (1966) proposed that we can simply use the equation for discrete growth, Nt+1 = XNt. After taking natural logs of both sides of the equation we can write:
When we plot ln Nt+1 versus ln Nt, if X is a constant, we should have a straight line with the slope of 1.0 and a y-intercept equal to ln X= r. But if there is density dependence and the growth rate slows with population size, when ln Nt+1 is graphed against ln Nt, a linear regression through the data should yield a slope less than 1.0.
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Figure 3.2 Effect of stochastic "environmental noise" on population growth. The deterministic growth curve has no density-independent effects. In all cases r = 0.25, N0 = 10, and K = 1000.
♦ Deterministic growth
Figure 3.2 Effect of stochastic "environmental noise" on population growth. The deterministic growth curve has no density-independent effects. In all cases r = 0.25, N0 = 10, and K = 1000.
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Figure 3.3 Exponential growth. Initial population size = 10 and r = 0.50.
For example, consider Figs. 3.3 and 3.4. In Fig. 3.3 we have an exponentially growing population with ln X = r = 0.50 and an initial population size of 10. In Fig. 3.4 we have graphed ln Nt+1 versus ln Nt. Since this population is not showing density dependence we get what we expect, a positive slope equal to 1.0. Now contrast this with a hypothetical yeast population showing density-dependent growth (Fig. 3.5, based on Pearl 1927). If we take natural logs and graph the data as we did in Fig. 3.4, we find that the slope of the
line is indeed less than one (Fig. 3.6). The question remains, however, how far less than one should a regression slope be before we consider it significantly different?
Tanner (1966) examined 70 data sets for animal populations and found slopes significantly different from one in 63 of them. However, this method for detecting density dependence is fundamentally flawed. First, a linear regression assumes data points are independent. In this analysis the x-value in one time series becomes the y-value in the next time series. Second, measurement error in the population estimates inevitably leads to a slope of less than one. Therefore a slope of less than one is often just an artifact of measurement error and not evidence for density dependence. Finally, the expected relationship between Nt+1 and Nt is not necessarily linear if there is environmental variability
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Figure 3.6 Density-dependence test for a yeast population. Based on data from Fig. 3.5.
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Figure 3.7 Per capita growth test for density dependence in a population with exponential growth and no carrying capacity. Based on data from Fig. 3.3.
or if the population has such a high growth rate that it approximates chaos (as described in Chapter 2).
A better test for density dependence is to examine the per capita growth rate versus population size (Turchin 1995, Case 2000). In the logistic equation we expect the per capita growth rate to have a negative slope when graphed against population number (Fig. 2.2 from Chapter 2). By contrast, in exponential growth, the per capita rate remains steady. For example, examine Figs 3.7 and 3.8. In Fig. 3.7 (based on exponential-growth data from Fig. 3.3), the slope equals zero, indicating no change in the per capita growth rate with population density. By contrast, for our yeast population graphed in Fig. 3.5 we find
a significant negative slope for the same analysis (Fig. 3.8). Examining plots of per capita growth versus N have many advantages, including the detection of density dependence in populations with environmental noise (Case 2000).
Let us now look at some field data. The following is based on long-term Christmas Bird Counts of waterfowl populations in the Chesapeake Bay area of Maryland and on the Piedmont of Virginia. These data were obtained by Heath (2002) from the United States Fish and Wildlife Service and from the Virginia Society for Ornithology. Christmas Bird Counts (CBC) were initiated in the late 1800s as a replacement for the traditional Christmas hunt, and may be the oldest wildlife census in the world. The CBC, however, depends on volunteers, many of whom are not professional biologists, and the use of CBC data in scientific studies is problematic. Nevertheless, the CBC often represents the only long-term data on waterfowl in regions such as the Virginia Piedmont. In addition, Maryland and Virginia waterfowl populations have been affected by habitat loss, hunting pressure, and environmental degradation in the Chesapeake Bay. On the other hand, due to land-use changes including the creation of new reservoirs and wetlands, waterfowl populations may be increasing on the Virginia Piedmont (Heath 2002).
Indeed, if we examine CBC estimates of Canada geese (Branta canadensis) populations from 1958 to 2001, there is a distinct increase (estimated r = 0.17) in the Piedmont population, while the Coastal Plain population has no distinct trend other than that of increasing oscillations (Fig. 3.9). In Fig. 3.10 we have tried to test for density dependence in the Piedmont population by the first method, graphing ln Nt+1 versus ln Nt. The resultant regression is so close to one that the conclusion is inescapable that the Piedmont goose population is not density-dependent at this time. By contrast, in Fig. 3.11 we see that the regression slope departs radically from one in the Coastal Plain population. In fact the regression is so weak that the slope is not significantly different from zero. How do we interpret this? On the one hand we might conclude that the Coastal Plain population is very density-dependent. Or we might conclude that this is just an unreliable set of data.
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Figure 3.9 Canada goose (Branta canadensis) population on the Coastal Plain of the Chesapeake Bay and on the Virginia Piedmont. Based on Christmas bird counts.
Figure 3.9 Canada goose (Branta canadensis) population on the Coastal Plain of the Chesapeake Bay and on the Virginia Piedmont. Based on Christmas bird counts.
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Figure 3.10 Test of density dependence of the Piedmont Canada goose population. Based on Christmas bird counts.
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Figure 3.11 Test for density dependence of the Coastal Plain Canada goose population. Based on Christmas bird counts.
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Figure 3.12 Per capita growth versus population size in the Piedmont Canada goose population.
Now, let's try the second method for determining density dependence, using per capita growth rates. From Fig. 3.12 we learn that the regression line for the Piedmont population is not significantly different from zero. This means that the Piedmont goose population has shown no decrease in per capita growth through 2001. Therefore, both tests tell us that the Piedmont population is not density-dependent at this time.
On the Coastal Plain (Fig. 3.13), although the slope is very small, it is negative and the regression is significant. Therefore, although the data are rather weak, it does appear that this population shows density dependence.
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