untitled

There are a great many ways that we can interpret underlying causes of this

population growth. Ecological interactions among wildebeest, other large mam-

mals, and the rest of the environment could influence the patterns of wildebeest

population growth and natural regulation. We can begin by determining how fast

the population is growing. A useful way to do this is to convert the census data

into estimates of the exponential rate of increase (r) between sequential population

estimates. We recall from Chapter 6 that the growth rate for a population can be

expressed either as the finite growth rate (λ=Nt+ 1 /Nt) or as its exponential equi-

valent (r=loge(Nt+ 1 /Nt)). The exponential growth rate is especially convenient when

population censuses or estimates are timed irregularly, rather than occurring every

year, because it can be readily translated into shorter or longer time intervals by

simple multiplication or division operations. We calculate the natural log of the ratio

of subsequent to initial population abundance for each time interval and divide this

ratio by the number of years between successive population estimates (τ):

rt=loge(Nt+τ/Nt)/τ

We encourage you to check the procedure by calculating the first two or three

estimates of rby hand. Why do we need to divide by τ? In many cases, we will not

have annual data to work from. In these cases, we can handle the irregular timing

between censuses by dividing by the number of years between them, τ. The result

of these calculations for the Serengeti wildebeest is shown in Table 15.1. Once the

values of rhave been calculated, we can readily translate back into values of λby

exponentiation: λ=er.

The next step is to fit a mathematical relationship to the multiple estimates of r

recorded over time. The accepted convention in such matters is to find a mathematical

model whose values best fit the existing data, where “best fit” means to minimize

the sum of squared deviations between the model estimates and the observed data.

The recorded estimates for Serengeti wildebeest certainly seem to decline with

MODEL EVALUATION AND ADAPTIVE MANAGEMENT 255

Year N(thousands) Rate of increase (r)

1958 190.000 0.108835146

1961 263.262 0.130874700

1963 356.124 0.104751424

1965 439.124 0.047919596

1967 483.292 0.090021783

1971 629.777 0.109589002

1972 773.014 0.124420247

1977 1440.000 −0.142352726

1978 1248.934 0.03443494

1980 1337.979 −0.050802822

1982 1208.711 0.050754299

1984 1337.849 −0.077244421

1986 1146.340 0.012747402

1991 1221.783 −0.095578878

1994 917.204 0.070080128

1998 1165.908 0.110475091

1999 1302.096 −0.044661447

Table 15.1The
instantaneous rate of
increase calculated for
the Serengeti wildebeest
population.