Raup-Chapt 3 Gambler_s Ruin

Raup-Chapt 3 Gambler_s Ruin - EXTINCTION BAD GENES OR BAD...

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Unformatted text preview: EXTINCTION: BAD GENES OR BAD LUCK? Raup, D. M., and S. M. Stanley. 1978. Principles of paieontology. 2d ed. San Francisco: W. H. Freeman. College text. Raup, D. M., andJ. W. Valentine. 1983. Multiple origins of life. Proceedings of the National Academy of Sciences 80:2981—84. Research article exploring the possibility that life originated more than once. Schopf, J. W., ed. 1983. Earth’s earliest biosphere: Its origin and evolution. Princeton: Princeton University Press. Detailed and comprehensive collection of articles on Precambrian environ— ments and early life. Schopf, J. W., and C. Klein, eds. 1991. The proterozoic hiosphere. Cambridge: Cambridge University Press. Forthcoming. An even more comprehensive treatment of Precambrian life. Stanley, S. M. 1973. An ecological theory for the sudden origin of multicellular life in the late Precambrian. Proceedings if the National Academy of Sciences 70:1486—89. Research article out- lining the case for cropping as the cause of the Cambrian Explosion. Stanley, S. M. 1989. Earth and [ye through time. 2d ed. New York: W. H. Freeman. A college text. 44 CHAPTER 3 GAMBLER’S RUIN AND OTHER PROBLEMS GAMBLING Suppose you are a casino gambler fortunate enough to find an even—odds game—you and the house each have a fifty— fifty chance of winning on each play. This might be a rou- lette table with no zeros (green numbers), if such there be. You are playing only red or black on this wheel, the two colors occurring with equal frequency. You have entered the game with ten dollars and bet one of them on red. If the wheel comes up red, you win one dollar and now have eleven; if it comes up black, you lose one dollar and have nine. Continuing in this manner, your stake will fluctuate in steps of one dollar until one of three things occurs: (1) you go broke, (2) the casino goes broke (or asks you to leave 45 EXTINCTION: BAD GENES OR BAD LUCK? because you are winning too much), or (3) you run out of time or otherwise decide to leave. Figure 3—1 shows several possible outcomes of the casino scenario. Though simulated with a random—number genera— tor on a home computer, they could just as easily have been created manually by flipping a coin or drawing randomly from a deck of playing cards, using red and black. In each case, our gambler ultimately went broke, but in one of them (Game #3), he / she did pretty well for a while. The Gambler’s Ruin problem, just described in its sim— plest form, has been used for years by statisticians as a model of kinds of probability. Inevitably, a specialized language has grown up around Gambler’s Ruin. For example, the paths followed by the gambler’s fortunes in Figure 3—1 are called random walks. Once started, the random walk has no tendency to return to a level previously occupied. If the gambler starts with ten dollars, no force induces the path to stay near ten or return to ten. The system has no memory. Every good gambler knows this, of course: a long string of failures does not change the odds for the future. The horizontal base of each graph in Figure 3—1 denotes a zero stake—the level at which the gambler has lost the original stake and is broke. This is called an absorbing bound- ary because if the path reaches this level, there is no return; the game is over. We could change this condition by stipu— lating that when you go broke, the casino gives you a single dollar to keep going. In that case, the bottom of the graph becomes a reflecting boundary— you bounce back with at least one upward step. I know of no casino that does this, except by oHering credit. In Game #3 of Figure 3—1, the gambler’s stake dropped from 10 to 1 early in the play, then rose to 14, and ulti— 46 GAMBLER,S RUIN AND OTHER PROBLEMS Gambler’s Rum Game # 1 Game #2 M O ~A O Gambler’s stake U] Gambler’s stake U] D Time Time Game #3 Gambler’s stake absorbing bwnda”?! _l 0 40 so 120 160 Time (number of plays) FIGURE 3-1. Simulated gambling results in an even—odds game (fifty-fifty chance of winning on each play). The gambler’s initial stake is $10, and each bet is $1. Thus, the stake fluctuates up and down randomly, in steps of $1. Gambler’s Ruin occurs when the absorbing boundary—zero—is reached. Each game is like the fate of a genus that starts with ten species. The number of species goes up when a species branches to form another species and goes down when a species goes extinct. 47 EXTINCTION: BAD GENES OR BAD LUCK? mately dropped to O (the absorbing boundary). Suppose the gambler had started with nine instead of ten dollars and the play was otherwise identical. Zero would have been hit in that initial drop, and the gambler would never have enjoyed the later success—and the opportunity to leave with a small profit. This emphasizes the importance of the size of the initial stake: the higher it is, the farther it is from the absorb— ing boundary and the more likely that the gambler will remain in the game for a long time. Theoretically, casinos could make a profit by offering an even—odds game as long as they put limits on how much players bet. This is because the typical gambler enters the casino with an amount of money much closer to zero than to the assets of the house. The upper boundary in the Gam— bler’s Ruin problem is also an absorbing boundary (bank- ruptcy of the casino), but this boundary is generally so high as to be not relevant for the individual customer. Consider, however, a high roller who enters our even—odds casino with assets equal to precisely half those of the house. If this gambler plays long enough or is allowed large enough bets, there is a fifty—fifty chance of his breaking the house. In the extinction context, we may think of the gambler’s stake as the number of species in an evolutionary group. Our example’s initial stake of 10 might be a genus with ten species living at some instant in the geologic past. We will use a time scale in millions of years instead of the gambler’s time scale. For every interval of one million years, each species has a fifty—fifty chance of surviving to the start of the next million year interval; if it survives, it has a fifty—fifty chance of speciating to produce-an additional species. What predictions can we make about the fate of the genus? Several interesting predictions are possible. For example, 48 GAMBLER’S RUIN AND OTHER PROBLEMS the number of species (diversity) willfluctuate up and down as zn a random walla. Extinction of species lowers d1vers1ty; speciation increases diversity. As long as the chance of ex— tinction is identical to that of speciation (fifty—fifty), a ran- dom walk will result. Furthermore, eventual extinction of the genus is inevitable. This is somewhat counterintuitive, but it follows from the presence of the single absorbing boundary at zero species. As we have seen, a random walk is free to wander up and down indefinitely. If there is no upper absorbing boundary, the random walk is bound to hit the lower boundary eventually. We could, of course, specify an upper absorbing bound— ary, analogous to the casino’s total assets. In the context of global biology, the upper absorbing boundary would be all the spaces for organisms in the world. An, evolutlonary group, such as a genus, could “break the bank by spec1at1ng so many times that no other genus could ex1st. All spec1es 1n the world would belong to the same genus. Th1s 1s as un— likely as an ordinary gambler’s winning the whole casmo. Thus, for all practical purposes, the ultimate ext1nction of the genus is assured. CONCEPTS or RANDOMNESS What is the meaning of randomness in the natural world? The flip of a coin is said to be random. To most people, this means that the outcome is a matter of pure chance—With- out cause. But coin flipping is surely governed by cause and effect. Whether the coin lands with head— or tail-side up depends on an almost uncountable number of phys1cal fac— tors, including the side facing up at the start and the strength 49 EXTINCTION: BAD GENES OR BAD LUCK? of the flip—and therefore the number of times the coin turns in the air. Also relevant must be the state of air cur— rents (including wind) and perhaps also barometric pressure, to say nothing of the condition of the coin. (In the some— what similar case of horseshoe tossing, by the way, good players become expert at controlling the number of turns of the shoe in the air.) Coin flipping is so complex that we cannot or choose not to monitor all determining factors. Instead, we choose to assume that the complex of causes will combine to make the coin behave as if the process were random. Having made this choice, we can ignore all the wind currents and other details and adopt the statistical pre— sumption that heads and tails are equally likely. This, in turn, makes available an arsenal of mathematical procedures for dealing with random events, enabling us to answer ques— tions like, How often should tails be expected to come up twenty times in a row? The assumption of randomness is a clever dodge. By choosing to ignore cause and effect and all the complexity that goes along with it, we have a tractable phenomenon about which we can make interesting and use— ful predictions. Most scientists and philosophers now agree that nothing is truly random in the natural world. The motion of mole— cules in a gas, the advance of a glacier, the formation of a hurricane, the occurrence of an earthquake, and the spread of an epidemic all have causes. In some cases, it is possible and worthwhile to investigate causes. This is certainly true for earthquakes and disease. But in other cases, we cannot or choose not to learn all there is to know. For example, by assuming randomness in the movement of molecules in a gas, we can derive the gas laws (Boyle’s law and the like), which are critical in countless engineering applications. 50 GAMBLER’S RUIN AND OTHER PROBLEMS Perhaps the best working definition of randomness in natural systems is the following: Random events are events that are unpredictable except in terms ofprobabz'lzty. The 70 percent rain forecast is a case in point. Using the Gambler’s Ruin approach to extinction, we deal with the same sort of probabilities. We assume that there is some number of reasons, probably large, for a spe— cies to go extinct. We observe patterns of extinction in the fossil record that are indistinguishable, at some scale, from patterns that could have been produced by a purely random process. Regarding the fate of a species, this is tantamount to saying that if one stress or calamity doesn t get you, another will. Once we have made the assumption of random behav— ior, we are free to work with the patterns. This approach makes possible generalizations that would be out of reach of the more conventional case—by—case approach. GAMBLING FOR SURVIVAL In the evolutionary history of a genus or any other group of related species, there is a certain amount of randomness in the sense just defined. A complex of physical and biological factors determines how long each species in a genus Will survive and whether species will branch to form new spe— cies. Species extinction weakens the future of the genus, and speciation protects it. It has been said that genera do not struggle for existence—they gamble. By good luck alone, a genus (or any other group) may thrive for a considerable time, just as a gambler may have some dramatic successes. Furthermore, if a run of good luck has produced a lot of species—equivalent to a Winning 5r EXTINCTION: BAD GENES OR BAD LUCK? streak at the casino——the chance that the genus will go extinct in the next few million years is lessened. A large number of species thus provides the group with temporary protection from extinction. There are more species of rodents today (about seventeen hundred) than of any other order of living mammals. Next mhdiversity are the bats, some nine hundred species strong T u — . _ _ . . s, almost two thirds of all livmg mammal spec1es are rodents or bats. Could this be just luck? Did these two extinction early in their history? Or are they simply good at surziving and / or speciating for some definite biological rea— son. One problem in deciding between these alternatives is that random processes have a wide range of outcomes; the position in that range is largely unpredictable. The question for the rodents or bats is as follows: Is the evolutionary history of the group within the expected limits of an even— odds game, or is the surplus of speciations beyond reason— able statistical expectation? If the latter is true, they must must be doing something right. Will Cuppy, in How to Become Extinct, thought he knew the answer. He wrote, “Bats are going to flop, too, and everybody knows it except the Bats themselves.” DIFFERING EXTINCTION AND SPECIATION RATES . In all of this, I have been assuming an equality of specia— tion and species extinction rates. Birth and death of species 52 GAMBLER’S RUIN AND OTHER PROBLEMS have been assigned precisely the same likelihood. How could this be realistic, especially since species birth and spe— cies death are such different phenomena? There are two responses. First, the random—walk logic works perfectly well with un equal probabilities-«only the mathematics changes slightly. To return to the casino anal— ogy, the house normally establishes an imbalance in the odds so that its profit is favored. The random walk still works but with a bias slightly against the customer. Therefore, we can easily handle cases in evolution where the likelihoods of speciation and extinction are different. We may suspect, for example, that the rodents and bats have lived in situations where speciation is more likely than extinction. We must be wary, as was noted earlier, that the rodents and bats were merely lucky enough to beat the odds. Some casino gam— blers do win and win consistently. The second response to the question about the assumption of equality of rates is to note that the total number of speciation events in the history of life is approximately the same as the total number of extinctions. This follows from the 1,00021 ratio between extinct and living species discussed in the first chapter. If 40,000,000,000 species were formed in the past and if 40,000,000 species are living today (keeping in mind that we do not know the exact numbers), then there have been 40,000,000,000 speciations and 39,960,000,000 species extinctions. Thus, the long—term average rates of speciation and species extinction have been about the same. Regardless of the reasons for this near—equality of rates, the numbers indicate that the even—odds model is not unreason— able. 53 EXTINCTION: BAD GENES OR BAD LUCK? SKEWED HISTOGRAMS Paleontologists have done many computer simulations in order to gain a feel for the range of outcomes of random walks in biodiversity, with both equal and unequal rates of speciation and extinction. Some simulated groups expand to the point of swamping the computer’s memory; other groups go extinct rather soon. Extinction of a group is most common when the group starts out small, just as a casino gambler is most likely to go broke quickly if he starts with a small stake, close to the absorbing boundary. In evolution, a group such as a genus or family must, by definition, start with a single species. For the fledgling group to survive, the founding species must speciate before it goes extinct. Because new evolutionary groups start small, they usually don’t last long. This, in turn, yields an impor- tant facet of the history of life: most groups of species have life spans shorter than the average of all groups. Figure 3—2 shows a histogram of life spans of fossil genera. It has a skewed shape, with many short durations and only a few long ones. The skewed (asymmetrical) shape of variation is typical of important biological properties germane to the extinc- tion question. These include . numbers of species in a genus . life spans of species . numbers of individuals in a species . geographic ranges of species 54 Zr g Number of genera GAMBLER,S RUIN AND OTHER PROBLEMS 8000 Fossil Genera 6000 4000 2000 0 50 100 Life spa/n. (millions of years) FIGURE 3-2. Histogram showing the distribution of geologic life spans of fossil genera. The mean duration of fossil genera is about twenty million years. The graph is highly skewedf—many more genera have life spans less than the mean than have life spans greater than the mean. (Based on time ranges of 17,505 genera; tabulated by J. J. Sepkoski, Jr.) In each category, the small “thing” is most abundant. Let me give another example. There are about 4,000 liVing species of mammals, grouped into about 1,000 .genera. About half these genera have only a single spec1es, and about 15 percent have only two species. The numbers drop off smoothly (see Figure 3—3), so there are only a few genera with more than 25 species. The most spec1ose 55 150 200 EXTINCTION! BAD GENES OR BAD LUCK? 500 Living Mammals Number of geneTa 10 20 30 Number of species per genus FIGURE 3—3. Skewed distribution of genus size among living mammals. About half the genera have only one species. Genera with as many as ten species are rare. (Based on data from Ander— son and Jones, 1967.) living mammal genus (a small insectivore) has about 160 species. The overall average is 4 species per genus (4,000/1,000), but because of the asymmetry, fully three— quarters of the genera have 1, 2, or 3 species and are thus below average. Let me summarize a few of the foregoing points. For reasons that develop from both theory and observation—— some depending on the Gambler’s Ruin problem—we can make the following generalizations: 1. Most species and genera are short—lived (compared with the averages). 2. Most species have few individuals. 56 _fi 160 GAMBLER’S RUIN AND OTHER PROBLEMS 3. Most genera have few species. 4. Most species occupy small geographic areas. Skewed variation is extremely common in nature. Strangely, however, most of us have been trained to believe that variation in natural phenomena is bell shaped, having just as many items above as below the average—whether we are talking about heights or weights of people, weather, or baseball averages. Nothing could be further from the truth. Classic examples of skewed variation include incubation times of infectious diseases and life expectancies of cancer patients. In both, the majority of cases fall below the aver— age, because the average is constructed by summing many short time intervals and a few long ones and then divid— ing by‘the total number of cases. It would be far better to use the median time—~that time exceeded by half the in— dividuals. Of course, bell—shaped curves (called normal or Gaussian distributions by mathematicians) do sometimes occur in nature. It is only that other shapes are more common. Stat— isticians wrestle with this problem because many of the best statistical tests are designed for the bell—shaped curve. Often they avoid the problem by transforming the raw data—that is, distorting the scale of measurement so that they can treat the results as If they had a bell—shaped distri— bution. One such transformation that sometimes works is to convert all the measurements to their logarithms (or even square roots). If the transformed numbers have a bell—shaped distribution, the analyst can proceed with tests that assume this shape. 57 EXTINCTION: BAD GENES OR BAD LUCK? OTHER MODELS The Gambler’s Ruin problem has led us to generaliza- tions about species that are relevant to the extinction prob— lem. However, many of the patterns, especially the skewed distributions, can also be approximated by processes having nothing to do with gambling or biology. Suppose you take a stick that is 100 inches long and break it at 25 random points—not favoring the middle or any other part. When you are done, you will have 26 short sticks. Now, measure and count the short sticks and con- struct a histogram. The shape of variation in stick length will look very much like t...
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