The Preservation of the Northern Spooky Tooky

* Due March 10 *

The Spooky Tooky is a large owl-like creature that tends to inhabit
and mate in Old Growth Forest. The Spooky Tooky was once abundant
in the wilderness but with the coming of the chain saw, the Spooky
Tooky population is starting to dwindle. The Spooky Tooky population
is important to the overall health of the ecosystem since they prey
on the spineless manga toad that likes to suck the sap out of old
growth trees. In addition, the Spooky Tooky perform barrel rolls and
loop to loop on command for the the entertainment of tourists visiting
our National Forests. Thus, there is a keen need to develop effective
policy to preserve the Spooky Tooky. At last count, there were
** 8000 ** individual Spooky Tookies known. They are known to
be long lived creatures and Sparky, the best known of all Spooky Tookies,
is at least 60 years old.

The Spooky Tooky, like most creatures in the forest, mate and produce offspring. Preservation of the Spooky Tooky then requires a habitat which a) lets them mate b) lets them have female offspring and c) lets the female offspring have a reasonable probability of living long enough to mate and have more female offspring. The Spooky Tooky long ago discovered the Viagra tree and even old Sparky can still produce offspring. For a species like the Spooky Tooky, it can be shown that the geometric growth rate of the population can be approximated by the following horrible looking equation:

The terms in the above horrible equation are the following:

- lambda = the growth rate; for a stable species lambda = 1;
values of lambda less than 1 lead to population crashes.
- alpha = the age at which the species starts breeding
- s = the adult annual survival probability
- b = adult average female fecundity - fecundity means the power of a
species to multiply rapidly or its capacity to form reproductive elements.
It is one measure of fitness. In the case of the Spooky Tooky
means the average reproductive rate of female offspring per adult female in the overall population per year.**b** - l
_{alpha}represents the probability that a new born will survive to age alpha (the breeding age).- note that l

_{alpha}in turn = s_{o}* s_{1}* s where s_{o}= survival probability at birth and in fledgling stage of life, s_{1}= sub-adult annual survival probability, and s is as given above. Note that, of course, s, must be correlated at some level with s_{o}and s_{1}.

Given a value of lambda, we can then estimate the characteristic timescale of the population crash as

where N is the number of known individuals in the population (8000) in this case.

Below are three sets of data that have observationally determined the
necessary parameters. Your ** first task ** is to derive estimates of lambda
and tau (the population crash time) from each of these data sets.

Data Set 1 (from university biologists):

- alpha = 3 years
- s = 0.95
- s
_{o}= 0.15 - s
_{1}= 0.72 - b = 0.24

Data Set 2 (from the Government Study)

- alpha = 4 years
- s = 0.85
- s
_{o}= 0.10 - s
_{1}= 0.50 - b = 0.28

- alpha = 2 years
- s = 0.65
- s
_{o}= 0.05 - s
_{1}= 0.35 - b = 0.20

When a group of university biologists went to the Spooky Tooky policy
hearings, they were met by an angry mob of legislatures that a)
complained they were overpaid and b) that they had done their study
wrong by implicitly assuming that the Spooky Tooky, if they survive
as an adult, lives for a very long time. The legislatures whipped out
a formula produced at the Institute for Scientists that Whip
out Formulas in which a new parameter * w * is now introduced.
This parameter represents the maximum age for survival and reproduction.
This whipped out formula is even more horrible that the first one:

The republican legislatures insist that w = 10 years while the democrats
insist that w = 15 years. Using these values of w your ** third
task ** is to use the three data sets to re-estimate values for
lambda and tau and to explain why your results are sensitive to this
new parameter.

Note:

http://homework/pub/class/2004/es399/whipped-out-formula.xls

will down load an excel spread sheet with the above formula coded in.

Lastly there is the important question of equilibrium occupancy in
Old Growth territory. In general, a species will not inhabit all of
the available habitat as that could lead to extinction (as in the
classic overgrazing scenario). A working model is that
habitable and uninhabitable patches of land are randomly dispersed
throughout the forest ecosystem. This model makes an important prediction
for
**p _{o}** which is the equilibrium occupancy of suitable habitat
by females (the model assumes the males will find them ...) and it has
two parameters

For simplicity we will use now use **p** for **p _{o}**
If

Below are three data sets each with N_{p} = 10.

Data Set 1 (from university biologists):

- p = 0.4
- h = 0.38

Data Set 2 (from the Government Study)

- p = 0.7
- h = 0.50

- p = 0.2
- h = 0.25

Your ** last task ** is to use this data to do the following for
each of the three data sets

- derive k
- show the uncertainty in k (e.g. the standard deviation for k)
- assume that before the coming of the chain saw that h was 0.65
and calculate what p would have been.
- Map your determined values of k and its uncertainty into the
critical parameter 1-k for each data set; remember,
this value means that if
**h**falls below it, extinction of the Spooky Tooky will occur. - Discuss the policy implications depending on which of the
3 data sets is adopted as
*the truth*...