untitled

number of parameters are available to model the data. On the other hand, the AIC

approach applies a penalty to each parameter pin the model. This enables us to iden-

tify the most parsimonious model (i.e. the model that explains the data reasonably

well, using a moderate number of parameters). For further details, the interested reader

should consult the comprehensive treatise on AIC by Burnham and Anderson

(1998). Perhaps the easiest way to introduce the procedure here is to apply it to our

data for the Serengeti wildebeest.

The Ricker logistic model that we have fitted to the Serengeti wildebeest data (we

will call it model 1) has three parameters, because we estimated rmax, K, and the

standard deviation of the residuals around the linear regression line.

AIC 1 =−42.346

where p 1 is the number of parameters estimated from the data and nis data sample

size. This AIC value only has meaning relative to those arising from other plausible

models. We now consider three such candidates in the following sections.

The theta logistic model is appropriate for populations that have a threshold curvi-

linear relationship between rand N. According to this model, growth rates change

little with density when Nis modest, but the density-dependent response becomes

much steeper as Nbecomes large. In other words, the intensity of density-dependent

processes becomes disproportionately severe at high population densities. If there is

a threshold effect, the theta logistic might be a more plausible model. In the theta

logistic model, exponential growth rates are the following non-linear function of N,

with the precise pattern dictated by three parameters: rmax, K, and θ:

Most statistical programs have a routine to estimate the parameters for non-linear

relationships such as this (including MATHCADsupplied with this book). In this case,

the parameters are estimated as follows:

rmax=0.105

K=1.241 × 103

θ=5.946

With these values we can plot the theta logistic curve against the data (Fig. 15.4).

Then, we calculate the residual variance or mean-squared error (MSE):

FNr K r

N

K

(, max, , ) θ max^

θ

=−⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1

AIC 1 log ( )

=− +

−−

⎛

⎝⎜

⎞

⎠⎟

22

1

11

1

e p

n

np

Λ

MODEL EVALUATION AND ADAPTIVE MANAGEMENT 259

15.4.1Ricker logistic
model

15.4.2Theta logistic
model