294 Scalar, Stochastic, Discrete Dynamic SystemsConsider modeling a population of sand-hill cranes in year n by the first-order, deterministic re-currence equationy(n) = Ry(n − 1) + u(n)whereR = 1 + r = 1 + b − d .In this expression, the growth rate r is the difference between the birth rate b and the death rate d.The parameters b and d are numbers between zero and one, so that R is between zero and two. Ifthe recurrence above describes something other than a population (say, an amount of money), thennegative values of R are meaningful as well (debit versus credit), and we saw in class that differentvalues of R yield a relatively rich variety of possible responses even when the input u(n) to therecurrence is zero.However, the output y(n) in the sand-hill crane example had to be interpreted in any case asthe average number of birds, rather than the actual number, if nothing else because when R is areal number y(n) is not necessarily an integer number. A deeper reason for thinking of y(n) as anaverage is that the exact number of cranes is hard or impossible to predict. Consider the experimentof observing a population of sand-hill cranes over, say, N = 10 years from an initial population y0.Repeating the experiment would entail reproducing the same environmental conditions, restart thepopulation at y0, and observe its evolution over another ten years. Different experiments are boundto produce different time sequences y(0), y(1), . . ., although y(0) in particular would be the samein all experiments. Discrepancies are caused by uncontrolled variations in the environment, bythe health and genetic makeup of the specific initial population, and by other unpredictable factorssuch as whether a particular alligator was or was not able to capture and kill a particular bird onday 37 of year 4.In summary, a bird population is a stochastic quantity, that is, a quantity whose exact variationsdefy detailed modeling, and are rather described in an aggregate sense. Another example of astochastic quantity is the outcome of the roll of a die. A classical, mechanistic view of physicswould have posited that it is possible in principle to know enough about the circumstances inwhich a die is cast to predict the outcome. In practice this is unrealistic, and in more modern viewsof physics utterly impossible. An aggregate description is more feasible, and would state that theprobability of any of the six possible outcomes is the same for a fair die. In one interpretation, thismeans that if a fair die is rolled K times, then each of the possible outcome values 1 through 6 islikely7to occur about K/6 times, and the approximation improves indefinitely as K increases.A less detailed description would state that the average outcome of the roll of a die is 3.5. Inone interpretation, this means that if a fair die is rolled K times with outcomes o1, . . . , oK, thenthe quantity1KKXk=1ok7The astute reader will have noticed that the expression “is likely” reeks itself of probability. This observation iscorrect, and can be made precise.30 4 SCALAR, STOCHASTIC, DISCRETE DYNAMIC SYSTEMSis likely to become closer and closer to 3.5 as K increases.Similarly, a growth coefficient of R could be interpreted by stating that if the experiment men-tioned above were repeated K times, and the ratios, say, Rk= [y(1)/y(0)]experiment kwere com-puted from empirical observations over all experiments, then one would obtain1KKXk=1Rk≈ Rfor a large enough K.8The average outcome y(n) from a recurrence based on the average growth coefficient R isincomplete information, just as the statement that a roll of a die yields 3.5 on average is incom-plete. Much more detailed information could be obtained if the recurrence under study were toalso model the stochastic variations from experiment to experiment. This greater amount of infor-mation is both a curse and a blessing: It is a curse in that running such a recurrence once through asequence of years would only provide information about that particular sequence, and would there-fore be of limited predictive value. More detailed information is a blessing in that a recurrence canbe run multiple times through a sequence of years, and by doing so one can compute aggregateinformation that includes but is not limited to the average behavior of the system.A recurrence that includes stochastic behavior is called a stochastic dynamic system. Such arecurrence requires some mechanism for generating random outcomes. Even once such a mecha-nism is available, there is a wide choice of options for how to inject randomness into a recurrence.We start from a conceptually straightforward method next, for motivation and intuition. Some pre-liminaries on probability theory follow, and a more quantitative treatment is presented thereafter,together with some alternative options for injecting randomness.Russian RouletteThe simplest and most direct way of thinking about randomness in the sand-hill crane example isto flip a coin every year n and for each of the y(n) birds in turn: head means that the bird survives,tail means that it dies. This only makes sense for a population that does not grow (0 ≤ R ≤ 1). Inother words, we assume a zero birth rate b for now, and model 1 − d only, rather than 1 + b − d.Births will be handled in the next Section.With a fair coin, each bird has a 50-50 chance of survival, so R = 0.5. For different valuesof R between zero and one, we can think of Russian roulette, in which the revolver’s cylinder isspun anew before each bird is ... visited. By varying the number of chambers and bullets, one canachieve any desired probability p that a bird survives. The resulting recurrence is as follows:y(n) = number of survivals when simulating Russian roulette y(n − 1) times.Although this may not look much like a recurrence, it is, because y(n) is a (stochastic) function ofy(n − 1).8The fact that a similar result would be obtained for ratios y(n)/y(n − 1) for other values of n is a bit of a coin-cidence, and corresponds to the fact that the growth coefficient R is assumed to be the same regardless of populationsize or time (that is, it is independent of both y(n) and n).31In Matlab, we can generate what is called a pseudo-random number between zero and onewith the instructionrandPseudo-random means that the sequence of numbers produced by repeated calls to rand isdeterministic but hard to predict. If you restart Matlab, or, more conveniently, you callrand(’seed’, 0)then