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Let us summarize the analysis. The deterministic return time from L to a vicinity of the origin scales as log(L//e), the mean time for all stochastic trajectories to escape from [— L, L] scales as exp(L2/e) and the mean time to escape without ever returning to 0 scales as log(L/v/e). These are vastly different times - indeed many orders of magnitude when L is moderate and " is small. The mean time to escape, conditioned on not returning to the origin, is much, much smaller than the average escape time. Thus, the mean time to escape, conditioned on not returning to the origin appears as a punctuated trajectory. Gavrilets (2003) refers to those trajectories that escape as "lucky" ones and notes that they do it quickly.

That was a lot of hard work. And to some extent, the payoff is in a deeper understanding of the problem, rather than in the details of the mathematical analysis. Indeed, in retrospect, our discussion of the gambler's ruin can shed light on this problem. Recall that, in the gambler's ruin, we decided that in general one is very rarely going to be able to break the bank, but that if it is going to happen, it will happen quickly (with a run of extreme good luck). And the same holds in this case: it is rare for a trajectory starting at X(0) = x to escape without returning to the origin, but when a trajectory does escape, the escape happens quickly.

I feel obligated to end this section with a discussion of punctuated equilibrium. In 1971, Stephen J. Gould and Niles Eldredge (then youngsters aiming to become the Waylon and Willy - the outlaws - of evolutionary biology (see http://en.wikipedia.org/wiki/Outlaw_country if you do not understand the context of this metaphor) coined the phrase "punctuated equilibrium" and offered punctuated equilibria as an alternative to the gradualism of Darwinian theory as it was then understood (Gould and Eldredge 1977; Gould 2002, p. 745 ff.) Writing about it thirty years later, Gould said ''First of all, the theory of punctuated equilibrium treats a particular level of structural analysis tied to a particular temporal frame ... Punctuated equilibrium is not a theory about all forms of rapidity, at any scale or level, in biology. Punctuated equilibrium addresses the origin and deployment of species in geological time" (Gould 2002, pp. 765-766). The two key concepts in this theory are stasis and punctuation, which I have illustrated schematically in Figure 8.8; Lande (1985) describes the situation in this manner "species maintain a constant phenotype during most of their existence and that new species originate suddenly in small localized populations'' (p. 7641). The question can be put like this: since the geological record

Figure 8.8. The key concepts in Eldredge and Gould's challenge to Darwinian gradualism are stasis and punctuation. These are illustrated here for a hypothetical trajectory of three species characterized by a generic trait. For the first 2000 time units (call one time unit a thousand years if you like) or so, the trait fluctuates around the value 1 (stasis) but then around time 2000 there is a rapid transition (punctuation) to trait value equal to 3, which persists for about another 5000 time units (more stasis) after which another punctuation event occurs. During the periods of stasis, there are fluctuations around trait values. The question, and challenge, is whether this picture is consistent with the notion of gradual modification. Our answer is yes.

Trait does look like the schematic in Figure 8.8, what challenge is posed to the Darwinian notion of gradualism? Indeed, some authors (Margulis and Sagan 2002) have argued that the entire mathematical and technical machinery associated with the gradualist Darwinian paradigm falls apart because of punctuated equilibrium. In her recent and wonderful book, West-Eberhard (2003) emphasizes (pp. 474-475) that ''punctuated equilibrium is a hypothesis regarding rates of phenotypic evolution and does not challenge gradualism. The patterns and causes of change in evolutionary rates are at issue, not the relative importance of selection versus development." Gould (2002) makes this clear on p. 756: ''Rather, punctuated equilibrium refutes the third and most general meaning of Darwinian gradualism, designated in Chapter 2 (see pp. 152-155) as 'slowness and smoothness (but not constancy) of rate'.'' Put more simply, we could ask: can a single mechanism account for the pattern shown in Figure 8.8 or does one require multiple mechanisms and processes? There is a flip answer to the question, as there always is a flip answer to any question. In this case, it is that stasis corresponds to a relatively constant environment and microfluctuations around an adaptive peak of fitness and punctuation corresponds to an environmental change in which the current adaptive peak becomes non-adaptive, another peak arises and there is strong selection from the formerly adaptive peak to the new one. But this answer is somewhat dissatisfying since it is flip and it makes key assumptions about the link between the environment and the fitness peaks that are not present in the underlying Darwinian framework. At first it would seem that we have answered the question in this section and to some extent, we have in that we now understand how the pattern of stasis and punctuation might be consistent with gradualism. However, Gavrilets (2003) emphasizes that this kind of analysis is not the full story, which can be found in his paper.

Anderson's theory of vitality and the biodemography of survival

We now turn to the application of diffusion processes to understanding survival. To begin, we will review life tables, the Euler-Lotka equation of population demography and methods for solving it. We will then see how a diffusion model of vitality can be used to characterize survival.

In a life table (Kot 2001, Preston et a/. 2001), we specify the schedule of survival to age a {/(a), a = 0, 1, 2, ..., amax} and the expected reproduction at age a {m(a), a = 0, 1, 2, ..., amax}, where amax is the maximum age, for individuals in a population. To characterize the growth of the population, we compute the number of births B(t) at time t. These have two sources: individuals who were born at a time t — a and who are now of age a, and individuals who were present at time 0 and are still contributing to the population. If we denote the births due to the latter individuals by Q(t), we can write that amax

Equation (8.78) is called Lotka's renewal equation for population growth. If t ^ amax we assume that Q(t) = 0 since none of the individuals present at time 0 will have survived to produce offspring at time t. Let us do that, for convenience drop the upper limit in the summation, and assume that B(t) = Cert, where the constant C (which actually becomes immaterial) and the population growth rate r are to be determined. Setting Q(t) = 0 and substituting into Eq. (8.78) we have

from which we conclude that

Equation (8.80) is called the Euler-Lotka equation. In the literature, it is usually treated as an equation for r, depending upon {/(a), m(a), a = 0, 1,2, ..., amax}. Dobzhansky and Fisher recognized that if there are no density-dependent effects, then r is also a measure of fitness for a genotype with schedule of births and survival given by {m(a), /(a), a = 0, 1,2, ..., amax}. Note also that since e—ra/(a)m(a) sums to 1, we can think of it as the probability density function for the fraction of the population at age a (Demetrius 2001).

According to Eq. (8.80), the solution r is a function r(/(a), m(a)). We may ask: how does r change with a change in the schedule of fecundity or survival (Charlesworth 1994)? For example, let us implicitly differentiate Eq. (8.80) with respect to m(y):

from which we conclude that

The denominator of Eq. (8.82) has units of time and in light of our interpretation of e-ra7(a)m(a) as a probability density, we conclude that the denominator is a mean age; indeed it is generally viewed as the mean generation time. Note that the right hand side of Eq. (8.82) declines with age as long as r > 0; this observation is one of the foundations of W. D. Hamilton's theory of senescence (Hamilton 1966, 1995).

We can ask the same question about the dependence of r on the schedule of survival. This is slightly more complicated.

Set l(a) — n"-o s( y) so that s(y) has the interpretation of the probability of surviving from age y to age y + 1. Show that

o-raa a0

Now, to actually employ Eqs. (8.82) or (8.83), we need to know r. In my experience, Newton's method, which I now explain, has always worked to find a solution for the Euler-Lotka equation. Think of Eq. (8.80) as an equation for r, which we write as H(r) — 0, where

Now suppose that rT is the solution of this equation (the subscript T standing for True), so that H(rT) — 0. If we Taylor expand H(rT) around r to first order in rT — r, we have H(rT) ~ H(r) + Hr(rT — r), where the derivative is evaluated at r. Now the left hand side of this equation is 0 and if we solve the right hand side for rT we obtain rT « r — (H(r)/Hr), which suggests an iterative procedure by which we might find the true value of r. Choose an initial value r0 and then iteratively define rn by r» = rn—1 — PTA (8.85)

Under very general conditions, rn will converge to the true value. A good starting value is often r0 = 0 or, to be a bit more elaborate, one might write that the expected lifetime reproduction of an individual R0 = Sa=o l(a)m(a) as exp(rTg), where Tg is the average generation time in a population that is not growing, given by the denominator of Eq. (8.82) when r = 0. In that case, a starting value could be r0 = log(R0)/Tg.

If the preceding material is new to you, or you feel kind of rusty and would like more familiarity, I suggest that you try the following exercise.

Exercise 8.14 (E)

Waser et al. (1995) published the following information on the life history of mongoose in the Serengeti. Some of it is shown in the table below.

Age (a)

l(a)

m(a)

0 0

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