Before addressing the viability of biological populations, let's consider some things and processes for whose viability we might have a better intuitive understanding. Of course, it it often the converse of viability (extinction, ruin, bankruptcy, etc.) for which we actually have the intution. These are really the same thing, the only difference being the mathematical sign or the subtraction of the probability from 1.0. For example, consider:
Most of us have gone to casinos to gamble. And most of us have "gone broke" playing in such casinos, where this means that we have at least lost as much money as were were initially prepared to lose. How to understand this? What are some of the things, the statistics, the probabilities that we might want to know about gambling?
Perhaps you can see that such questions as these have parallels in population extinction.
To be concrete, consider a particular form of gambling, for example, craps. Specifically, consider the basic PASS bet, which is simulated with the following Java applet. One starts with $10, and one repeatedly rolls the dice, and changes in the amount of money follow the resolution of the PASS bet.
The very important basic understanding involves the rules of the game. For the PASS bet in craps, these rules are fairly simple. If the coming out throw is a 7 or an 11, it is a win, if a 2, 3 or 12, it is a loss. Otherwise, the coming out throw becomes the point and throwing continues until the point is again thrown (a win, making the point), or until a 7 is thrown (a loss, 7 out). Thus, a player can understand the rules of the game exactly. These rules are invarient in time; and all craps tables follow the identical rules. The randomness in winning or losing comes in with the actual rolling of the dice.
The applet allows additional explorations beyond the trial by trial role of the dice. One can play till broke and see how long it takes. Press this button ten or twenty times. The number of steps is shown at the upper left. Note that the player always go broke. Note that the time till gamblers ruin can vary considerably.
One can also do 100 simulations of the time-to-ruin problem. From this a distribution of the time to extinction can be displayed as a frequency histogram. A couple of things should be noted from this distribution. The mean is much greater than the mode and the median. The bulk of the extinctions occur early, but some cases take a very long time.
Experiments can be done with the players strategy. In particular, she can bet $5 instead of $1. What does tis do to the time to extinction?
With more initial money, the time to extinction is longer. In fact, no one would worry about Bill Gates going extinct at a $1 craps table.
One might investigate different betting strategies. For example, the player could raise her bet if she got ahead, for example betting $2 if she has > $10 and betting $3 if she has > $20, and so on (the Step Up Strategy). Or, she might start to tip the dealer $1 for a win when she has greater than $20 (this puts a $20 ceiling on her amount of money--the Ceiling at $20 Strategy). Or, the drunken husband might walk by every hundredth trial and give her a dollar if she has less than $10 (this is the rescue or Infusion Strategy). An applet to investigate these strategies is given below. Think about what you expect to happen and then run 100 trials a few times to chech your prediction. Email your results to Maeveen.
One could even change the rules of craps, giving the player an advantage. Surprisingly, the player can still go extinct surprisingly often, due to the low initial amount of money. However, the mean time to extinction now becomes infinite.
There are questions one can ask about groups of players (metapopulations). What is the time to group extinction if 10 players play the same bets at the same table? What if they each play at a different table?
It is important to understant that the foregoing analysis was based on a complete understanding of the dynamics, the rules, of the system. What does one do if one doesn't know the rules? There are only two approaches, very different in conduct, philosophy and on-the-ground approach. They are basically 1) to attempt to learn the rules of the system, build a model and do simulation, such as with the craps system, or 2) to watch the process and to compute summary statistics from the resulting time series. Both approaches can and are terrmed population viability analysis, PVA. However, the fact that they are so very different often causes confusion.
Learn the rules
This is science. It minimally depends on observation and can be carried out more efficiently with manipulative experimentation. (Endangered species status often precludes experimentation). This approach takes time, money, expertise. An example is provided below.
Observe the process and compute statistics
This also takes time, money and expertise. Obviously, one can learn more from observing replicate systems. However, the typical endangered species case has but one population on which to observe. Almost invariably, the population is sampled rather than completely censused The number of samples, their inherent error, and the total duration of the sampling period are all important for accurate trend extrapolation.
Consider the following applet, which computes a trend from two samples at two points in time of a state variable. Player money described the state of the craps game. Population size is a basic state variable for the extinction process. In this simulation game, the magnitude of the per time step population decline can be changed with the radio buttons as -1%, -2% or -3%. That is, you know the actual growth rate. The issues is whether you can accurately detect this growth rate. You can control the fraction of the population that you sample. For example, you might decide on a sampling fraction of 0.1, which might mean that you send out a biologist to count the deer he sees in 10% of the populations range. You then multiply this number by 10 to get the extimate of the total population size. Explore this process.
One of the important aspects of this process is the degree to which you get false positives, that is, your process says that you have a growing population.
You can experiment with the parameters of this process to find methods to reduce the probability of false positives. The two answers are readily apparent: Take the samples further apart in time. And sample a larger fraction of the population. The first approach costs time, the second approach costs money.
This same basic consideration is illustrated with population projection applet below. This is a game. The player is to try to guess whether the population is growing or declining from only a few years of observations. The growth rate is randomly positive or negative, with equal probability of each. The magnitude of the growth rate can be changed by pressing one of three radio buttons. Start a new trial of 10 samples. Then, press the Next button to see the first few samples from a 50-year growth process. Study it and then make your guess by pressing either the '+' or the '-' buttons at the left. Your percent accuracy for 10 trials is computed.
Single Population with Environmental Stochasticity
A local population may be subject to a variable environment due to variation in, say, rainfall, winter temperatures, pray availability, and so forth. Independent of this variation will be limits that produce a carrying capacity. The growth is quantified by its expectation (r) and the variance of this rate over time (var r). The following applet allows an investigation of the sensitivity of extinction to r and var_r.
Fragmented Populations (Metapopulations)
The following applet considers multiple local populations, each of which is subject to environmental stochasticity and to a carrying capacity proportional to their area.
PVA: An example, the Concho River Water Snake
Michael Soule and I did this work in 1987. We had 'excellent' data. We produced a spatially explicit model. This model addressed viability. But, more importantly, it addressed jeopardy. Under Section 7 of the ESA, federal agencies are supposed to consult with the USFWS as to the impact of proposed projects on listed species (the Concho River Water Snake was not quite listed at the time of our analysis). Will the project 'jeopardize' the listed species. Note that the meaning of jeopardy is not given in the act. However, clearly the issue that lies at the heart of jeopardy is viability and predicted changes of viability.
The metapopulation model we produced is given below. It simulates the extinction-recolonization dynamics of local populations (habitat pathes) 300 years into the future. It allows some sensitivity analysis and also the inclusion of a proposed dam at the confluence of the two rivers.