From this we obtain the residual standard deviation:

σ=0.04

The likelihood calculation for the theta logistic model is calculated in a similar man-

ner as we did for the Ricker model, except that we modify the expected value and

the residual variance:

Λ 2 =1.681 × 1013

p 2 = 4

Note that we now have four parameters (rmax, K, θ, and the standard deviation of the

residuals around the linear regression line), necessary for the more complex, non-

linear model:

AIC 2 =−49.573

AIC 222

2

22

1

log ( )

=− +

−−

⎛

⎝⎜

⎞

⎠⎟

e p

n

np

Λ

−− −

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

rr

N

i K

i

max^1

2

2

2

θ

σ

Λ 2

0

(^11)
2
= exp

−
∏
i σ π
n
σ = MSE

MSE

(^) max

−−

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

=

−

∑

rr

N

K

n

i

i

i

n

1

2

0

1

θ

260 Chapter 15

0.1

0

• 0.1


• 0.2


• 0.3
  0 500 1000 1500
  N

r

Fig. 15.4Predicted
(line) and observed
(circles) exponential
rates of increase shown
by Serengeti wildebeest
in relation to population
density, based on the
theta logistic model.