Connections

Escape from a domain of attraction, or more generally the transition between two deterministic steady states driven by fluctuations, has wide applicability in biology, chemistry, economics, engineering, and physics (Klein 1952, Brinkman 1956, Kubo et a/. 1973, Arnold 1974, van Kampen 1977, Schuss 1980, Ricciardi and Sato 1990). Many of the ideas go back to Hans Kramers, a master of modern physics (ter Haar 1998) who modeled chemical reactions as Brownian motion in a field of force (Kramers 1940). Other introductions to the problem from the perspective of physics or chemistry can be found in van Kampen (1981b), Gardiner (1983), Gillespie (1992) and Keizer (1987). In biology, the classic paper of Ludwig (1975) brings to bear many of the tools that we have discussed. The more mathematical side of the question is interesting and challenging because the problems involve large deviations (Bucklew 1990). There are ways to use the method of thinking along sample paths to understand the general problem (Freidlin and Wentzell 1984), but the mathematical difficulty rises rapidly.

The Theory of Is/and B/ogeography spawned an industry (a good place to start is Goel and Richter-Dyn (1974)). Indeed, the late 1960s were heady times for theoretical biology. In the remarkable period of the late 1960s, optimal foraging theory (MacArthur and Pianka 1966, Emlen 1966), island biogeography (MacArthur and Wilson 1967), and metapopulation ecology (Levins 1969) developed. The theories of optimal foraging and island biogeography developed rapidly and led to experiments, and the development of new fields such as behavioral ecology. On the other hand, metapopulation theory languished for quite a while before a phase of development, and the subsequent development in the 1980s was mainly theoretical (see, for example, Hanski (1989)). The rapid success of island biogeography and optimal foraging theory relative to metapopulation ecology teaches us two things. First, the developers of optimal foraging theory and island biogeography provided a prescription: (i) measure a certain set of empirically clear parameters, and (ii) given these parameters, compute a quantity of interest. Levins did not do this as explicitly. For example, in classical rate-maximizing optimal foraging theory as we discussed at the start of the book, one measures handling times of, energy gain from, and encounter rates with, food items, and then is able to predict the diet breadth of a foraging organism. In classical island biogeography, one measures per capita birth and death rates and carrying capacity of an island and is able to predict the mean persistence time. On the other hand, it is not exactly clear what to measure in metapopulation theory or how to apply it. Indeed, authors still revisit the original Levins model trying to operationalize it (Hanski 1999). Second, Levins published his seminal paper in an entomology journal and on biological control. In the heady times of the late 1960s, such ''applied'' biology was scorned by many colleagues. A very interesting discussion of the role of theory in conservation biology is found in Caughley (1994), which caused an equally interesting rejoinder (Hedrick et a/. 1996).

Catastrophic changes in population size can occur for many reasons, and in the past decade or so there has been increasing recognition of the role of catastrophes in regulating populations. Connections to the literature can be found in Mangel and Tier (1993, 1994), Young (1994), Root (1998), and Wilcox and Elderd (2003).

Mangel and Tier (1993) show that the second moment of the persistence time satisfies S(n) = — 2M—11T(n) from which the variance and coefficient of variation of the persistence time can be calculated. By using that calculation, they conclude that persistence times are approximately exponentially distributed. Ricciardi and Sato (1990) provide a more general discussion of first-passage times.

In population biology, the most general formulation of the diffusion approximation (Halley and Iwasa 1998, Diserud and Engen 2000, Hakoyama and Iwasa 2000, Lande et a/. 2003) takes the form dX = b(X)dt + v/0eXd We + VadXdWd, where ae and ad are the environmental and demographic components of stochasticity and dWe and dWd are independent increments in the Brownian motion process, the former interpreted according to Stratonovich calculus (to account for autocorrelation in environmental fluctuation) and the latter according to Ito calculus (to account for demographic stochasticity). Engen et a/. (2001) show results similar to those in Figure 8.5, for the decline of the barn swallow, using a model that has both environmental and demographic fluctuations. The main message, however, is the same (see their

Figure 4). It is worthwhile to wonder when the diffusion approximation gives valid conclusions for life histories that do not meet the assumptions of the model (Wilcox and Possingham 2002).

In general, we will need to estimate extinction risk and mean time to extinction from time series that may often be short and sparse. This presents new challenges, both conceptually and technically (Ludwig 1999, Hakoyama and Iwasa 2000, Fieberg and Ellner 2000, Iwasa et a/. 2000).

As with some of the other topics in this book, there are probably 1000 papers or more on punctuated equilibrium, what it means, and what it does not mean (Gould and Eldredge 1993). A recent issue of Genet/ca (112-113 (2001)) was entirely dedicated to the rate, pattern and process of microevolution (see Hendry and Kinnison (2001) for the introduction of the issue). Pigliucci and Murren (2003) have recently wondered if the rate of macroevolution (the escape from a domain of attraction) can be so fast as to pass us by. West-Eberhard (2003) is a grand source of ideas for models (but not of models) in this area. The calculation by Lande (1985) using a very similar approach to the one that we did, with a quantitative genetic framework, warranted a news piece in Science (Lewin 1986). Jim Kirchner (Kirchner and Weil 1998, 2000; Kirchner 2001, 2002) has written a series of interesting and excellent papers on the nature of rates in the fossil record.

Demography - generally understood today as a social science - is the statistical study of human populations, especially with respect to size and density, distribution and vital statistics. The goal is to describe patterns, understand pattern and process, and predict the consequences of change on those patterns. The foundations of demography are the life table, the Gompertz mortality model (Gompertz 1825), and the stable age distribution that arises as the solution of the Euler-Lotka renewal equation. Biodemography seeks to merge demography with evolutionary thinking (Gavrilov and Gavrilova 1991, Wachter and Finch 1997, Carey 2001, Carey and Judge 2001, Carey 2003). The result, for example, will be to use the comparative method to explore similarities and differences of patterns across species and to understand the patterns and mechanisms of vital statistics as the result of evolution by natural (and sometimes artificial) selection. The Gavrilovs (Gavrilov and Gavrilova 1991) note that Raymond Pearl actually understood the importance of doing this - and wanted to do it - but lacked the tools. For example, Pearl and Miner (1935) wrote ''For it appears clear that there is no one universal 'law' of mortality ... different species may differ in the age distribution of their dying just as characteristically as they differ in their morphology'' and that ''But what is wanted is a measure of the individual's total activities of all sorts, over its whole life; and also a numerical expression that will serve as a measure of net integrated effectiveness of all the environmental forces that have acted upon the individual throughout its life''; the methods of life history analysis that we have discussed in other chapters allow exactly this kind of calculation. The papers of Pearl are still wonderful reads, and most are easily accessible through JSTOR; I encourage you to take a look at them (Pearl 1928, Alpatov and Pearl 1929; Pearl and Parker 1921, 1922a, b, c, d, 1924a, b; Pearl et al. 1923, 1927, 1941). As I write the final draft (April 2005) one of the most interesting issues in biodemography, with enormous importance for aging modern societies, is that of mortality plateaus. The Gompertz model can be summarized as dN

That is, the population declines exponentially and the coefficient of mortality characterizing the decline grows exponentially. However, in the past twenty years many studies of the oldest members of populations (see, for example, Vaupel et al. (1998)) have shown that mortality rates may not grow exponentially in the oldest individuals, but may plateau or even decline. Why this is so is not understood and is an area of active and intense research (Mueller and Rose 1996, Pletcher and Curtsinger 1998, Kirkwood 1999, Wachter 1999, Demetrius 2001, Mangel 2001a, Weitz and Fraser 2001, de Grey 2003a, b).

Financial engineering: a different way of thinking

Louis Bachelier developed much of the theory of Brownian motion in the same manner as Einstein did, but five years before Einstein in his (Bachelier's) doctoral thesis ''Theory of Speculation''. This thesis is translated from the French and published by Cootner (1964). Thus, many of the tools that we have discussed in the previous and this chapter apply to economic problems; this area of research is now called financial engineering (Wilmott 1998) and some readers may decide that this is indeed an attractive career for them. I was very tempted to include a detailed section on these methods, but both one of the referees of the proposal and my wife thought that financial engineering did not belong as an application of this tool kit, so I leave it to Connections. The basic ideas behind the pricing of stock options, due to Merton (1971) and to Black and Scholes (1972), employ stochastic differential equations but in a somewhat different manner than we have used. There are three key components. The first is a stock whose price S(t) follows a log-normal model for dynamics where we interpret p as a measure of the mean rate of return of the stock and g as a measure of the volatility of the price of the stock. The second component is a riskless investment such as a bank account or bond paying interest rate r, in the sense that if B(t) is the price of the bond then dB/dt = rB. Looking backwards from a time T at which we know the value of the bond, B(T), we conclude that the appropriate price at time t is B(t) = B(T)e—r(T— z). The third component is the option, which is a right to buy or sell a stock at a fixed price (called the exercise or strike price k) up to a fixed time T (called the expiration or maturity date). With an American option one can exercise at any time prior to T, with a European option only at time T. A put option is exercised by selling the stock; a call option is exercised by buying the stock. An option does not have to be exercised, but a future has to be exercised (so that a future might better be called a Must). With these definitions, we can compute the values of call and put options. For example, a call option will be exercised only if the price of the stock on day T exceeds the exercise price; for a put option, the reverse is true. Ultimately, the goal is to find the value of the option on days prior to T, so we define for which we have the end condition for a call option W(s, T, T) = max{s — k, 0}. Unlike evolutionary problems, in which one maximizes a fitness function, option pricing is based on the concepts of hedging and no arbitrage. Hedging consists of buying an amount A of actual stock (in addition to the right to buy stock at a later date) in such a manner that whether the price of the stock rises (making the option more valuable) or falls (making it less valuable), the net value of the portfolio, n(t) = W(s, t, T) — As consisting of the option minus the amount spent on stock, stays the same. The condition of no arbitrage (arbitrage is the general process of profiting from price discrepancies) means that

W (s, t, T ) = EfEuropean option on day T |S(t) = sg (8.96)

the portfolio grows at the same rate as the riskless investments, so that dn = rn dt. These conditions are sufficient to allow one to derive the diffusion equation for the price of the option. See Wilmott (1998) for an excellent introduction to such matters. Another terrific book on these topics, and which will seem familiar to you technically if not scientifically, is Dixit and Pindyck (1994).

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