Life System and Environmental & Evolutionary Biology II
Lab 8: Population Viability Analysis:
The role of chance in the growth of a population.
PVA's are based on models that relate a dependent variable such as population
size to independent variables that influence it, such as weather, disease,
mortality rates, etc. The relationship between independent and dependent
variables are mediated through the model's parameters, such as survival
rates and reproductive rates of individuals. A PVA uses a model that combines
both the magnitude of the parameters and the amount that they vary. It
generally involves three steps.

A single population projection is made over a specified period of time
(say 10 or 100 yrs.). The population size (N) at any given point in time
(t) is a function of the population size at the previous point in time
(t1) and of values drawn at random. The random values come from statistical
distributions that describe the variability in the model's parameter.

Many projections are made (say 500 or more). Each projection will give
slightly different results.

The proportion of projections in which the population reached a certain
threshold is calculated.
Thus a PVA has three elements: a population threshold (typically 0  the
point of extinction), a probability (0 to 1) that the population will reach
the threshold, and an interval of time for which the prediction pertains.
It gives us not only an average value for the predicted size of the population,
but also a range of possible future values. Therefore we make many population
projections and ask what percentage of those projections fall above or
below a certain population level. For example, the risk of extinction would
be 40% if 400 of 1000 population projections reached a threshold of zero.
1. A model and coin flipping exercise.
Consider the following model of a population in which the total number
of individuals (N) can change over a discrete time interval (from t to
t+1) only because of births and deaths:
N_{t+1} = (NT * S) + (NT * B * S)
Where S = the probability of an individual surviving from t to t+1 and
B = the average number of offspring produced per individual at each time
interval. So (NT * S) represents the survival of adults and (NT * B * S)
represents the production of offspring and their subsequent survival.
If NT = 10, S = .5, and B = 1, than the population will remain stable
at 10 individuals no matter how far we project it into the future.
Or will it? What about the role of chance? Try the following exercise.
Rather than using the deterministic equation above and simply multiplying
population size by survival, we can examine the fate of each individual
through the flip of a coin. Start with two coins (N = 2. Not enough coins?
Keep track of your numbers on paper). Now breed them by adding a new coin
to the group for each of the two coins (so now N = 4). This is analogous
to the birth rate of B = 1. Now flip each coin to determine if it survives.
Heads survives, tails dies. Because heads and tails are equally likely,
S = .5. Repeat this birthdeath process five times, for 5 years.

Record the final population size. The deterministic model would predict
a population of N = 2 every time. Is this what you found? Compare with
the results of your classmates.

Repeat the exercise for 10 years. Record your results and compare to
your classmate's results. What is the importance of the time interval for
our projections of extinction probabilities?
Now repeat the exercise for five years with a starting population of
N = 10 coins. Compare to your classmate's results.

How does starting population size effect the probability of chance extinction?
2. A real world example.
Go to the bandigraph
website and run the program there to conduct a Population Viability
Analysis of the longterm survival of an endangered Bandicoot. For
more information about the LongNosed
Bandicoot, you could go to this page from the Unique Australian Mammals
website. You may teamup with a classmate.
Answer the three questions that are bolded above as well as the following
questions:

Are the removal program and the removal levels prudent?

The data needed to do PVAs such as these require extensive field work.
If we didn't know the survival rate of the youngest age class, could we
still do a PVA on this species?

Which is more important for the rapid rebound of this species: juvenile
(03 month) survival, subadult (36 month) survival, or adult (over 6 months
of age) survival? Why?

PVA's are obviously useful tools for gaining insights about the population.
However, what might be some drawbacks to their use for making management
decisions?