## 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.

1. 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 (t-1) and of values drawn at random. The random values come from statistical distributions that describe the variability in the model's parameter.
2. Many projections are made (say 500 or more). Each projection will give slightly different results.
3. 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:

Nt+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 birth-death 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 long-term survival of an endangered Bandicoot.  For more information about the Long-Nosed Bandicoot, you could go to this page from the Unique Australian Mammals website. You may team-up with a classmate.

Answer the three questions that are bolded above as well as the following questions:

1. Are the removal program and the removal levels prudent?
2. 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?
3. Which is more important for the rapid rebound of this species: juvenile (0-3 month) survival, subadult (3-6 month) survival, or adult (over 6 months of age) survival? Why?
4. 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?