untitled

The Gompertz model is based on a logarithmic curvilinear relationship between r

and N, which is steepest at small values of N, becoming progressively less steep at

larger values of N(Fig. 15.5). It has been applied to many invertebrate and small

mammal populations, but is less common in species of larger wildlife. As a first step,

we calculate the natural logarithm of N:

X=loge(N)

We then proceed as before to estimate the slope (β) and intercept (α) of rversus X:

α=0.657

β=−0.093

Residual variance or mean-squared error is calculated as follows:

σ=0.062

The likelihood calculation for the Gompertz model is:

Λ 3 =1.015 ×^1010

Λ 3

0

(^12)
2

1

2 2

exp

[ ( log ( ))]

=

⎡−− +

⎣

⎢

⎤

⎦

⎥

=

−

∏

σ

αβ

i π σ

n

rNiei

σ = MSE

MSE

[ ( log ( ))]

=

−+

=

−

∑

rN

n

iei

i

n αβ 2

0

1

MODEL EVALUATION AND ADAPTIVE MANAGEMENT 261

15.4.3Gompertz
logistic model

• 0.2
  0 500 1000 1500
  N

r

0.8

0.6

0.4

0.2

0

Fig. 15.5Predicted
(line) and observed
(circles) exponential
rates of increase shown
by Serengeti wildebeest
in relation to population
density, based on the
Gompertz model.