untitled

increasing wildebeest abundance, so one obvious model candidate would be a linear

decline in rwith N. In mathematical terms, this means we postulate the linear model

r(i) =a+b*N(i), where ais the intercept of the straight line and bis the slope

of the relationship between Nand rin any particular census i. These values can be

estimated from any linear regression package. In this case, the intercept =0.18 and

the slope =−0.00016.

We can plot the observed value of rfor each time interval versus population

abundance at the beginning of the interval and then overlay these values with those

predicted by the linear model whose parameters we have just estimated (Fig. 15.1).

The linear model seems to do a reasonable job of predicting the rate of population

growth shown by Serengeti wildebeest. The consistent tendency for deviation below

the regression line at either very low or very high wildebeest densities, coupled with

deviation above the regression line at intermediate densities, suggests that a curve

might fit these data even better. We will address this possibility shortly.

We now consider how we choose between several possible models that could repre-

sent our observations, in this case the trend in wildebeest numbers. We have illus-

trated one model of trend, the straight line, and we observed that a curve may be a

better model. However, there could be several types of curve. Therefore, we need to

decide how “likely” each of several alternative mathematical models might be, based

on their ability to explain the census data obtained for the Serengeti wildebeest. First,

we describe the distribution of residual variability around the postulated relation-

ships (i.e. the models). Before we go any further, however, we are going to convert

our equation for the linear relationship between the exponential growth rate (ri) and

population density (Ni) estimated at the beginning of time interval iinto a form more

familiar to students of wildlife biology, based on ecologically relevant parameters.

The intercept ais usually called rmaxby ecologists (see Chapter 6). It is the maxi-

mum exponential rate of increase of which the population is capable, applying under

optimal growth conditions of low population density and high food availability. The

slope brefers to −rmax/K, where Kis known as the ecological carrying capacity, that

is, the maximum number or density of animals that can be sustained over the long

term in a particular place (Chapter 6). After making these conversions, rmax=0.18

and K=1142. In the case of the Serengeti data, there are 17 estimates of ri(a

256 Chapter 15

0.2

0.1

0

• 0.1


• 0.2
  0 200 400 600 800 1000 1200 1400 1600
  Number of wildebeest (thousands)

r

Fig. 15.1Predicted
(line) and observed
(circles) exponential
rates of increase shown
by Serengeti wildebeest
in relation to population
density.

15.3Measuring the
likelihood of models
in light of the
observed data