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Figure 2.6. (a) The function r = log(X) is concave. This implies that fluctuating environments will have lower growth rates than the growth rate associated with the average value of X. (b) The two color morphs of desert snow Linanthus parryae are maintained by fitness differences in fluctuating environments. (c) An example of why this plant is called desert snow. Photos courtesy of Paulette Bierzychudek.

Figure 2.6. (a) The function r = log(X) is concave. This implies that fluctuating environments will have lower growth rates than the growth rate associated with the average value of X. (b) The two color morphs of desert snow Linanthus parryae are maintained by fitness differences in fluctuating environments. (c) An example of why this plant is called desert snow. Photos courtesy of Paulette Bierzychudek.

to the per capita growth rates or the arithmetic mean to the logarithm of per capita growth rates.

What about measuring the growth rate of an actual population? Data in a situation such as this one would be population sizes over time N(0), N(1), ... N(t) from which we could compute the per capita growth rate as the ratio of population size at two successive years. We would then replace the frequency average by a time average and estimate the growth rate according to r « 1 [log(X(0)) + log(X(1)) + ••• + log(X(t — 1))] (2.22)

with the understanding that t is large. Since the sum of logarithms is the logarithm of the product, the term in square brackets in Eq. (2.22) is the same as log(I(0)I(1)I(2) ... I(t — 1)). But I(s) = N(s + 1)/N(s), so that when we evaluate the product of the per capita growth rates, the product is log(I(0)I(1)I(2)... I(t — 1))

= log{(N(1)/N(0))(N(2)/N(1))... (N(t)/N(t — 1))} = logfN(t)/N(0)}

However, in a fluctuating environment, the sequence of per capita rates (and thus population sizes) is itself random. Thus, Eq. (2.22) provides the value of r for a specific sequence of population sizes. To allow for others, we take the arithmetic average of Eq. (2.22) and write

This formula is useful when dealing with data and when using simulation models (for a nice example, see Easterling and Ellner (2000)). A wonderful application of all of these ideas is found in Turelli et al. (2001), which deals with the maintenance of color polymorphism in desert snow Linanthus parryae, a plant (Figure 2.6b, c) that plays an important role in the history of evolutionary biology (Schemske and Bierzychudek 2001). If you stop reading this book now, and choose to read the papers, you will also encounter the ''diffusion approximation.'' We will briefly discuss diffusion approximations in this chapter and then go into them in great detail in the later chapters on stochastic population theory.

Before leaving this section, I want to do one more calculation. It involves a little bit of probability modeling, so you may want to hold off until you've been through the next chapter. Suppose that we do not know the probability distribution of the per capita growth rate, but we do know the mean and variance of I, which I shall denote by I and Var(I). We begin by a Taylor expansion of r = log(I) around its mean value, keeping up to second order terms:

log(I) = log(I)+- (I — I) — - (I — I)2 (2.24)

and we now take the expectation of the right hand side. The first term is a constant, so does not change, the second term vanishes because E{I} = I and the expectation of the quantity in round brackets in the last term is the variance of the per capita growth rate. We thus conclude r ~ log(I) —1 Var(I) (2.25)

This is a very useful expression for fitness or growth rate in a fluctuating environment. The method is often called Seber's delta method, for G. A. F. Seber who popularized the idea in ecology (Seber 1982). I first learned about it while working in the Operations Evaluation Group of the Center for Naval Analyses (Mangel 1982), so I tend to call it the ''method of Navy math.'' Whatever you call it, the method is handy.

The logistic equation and the discrete logistic map - on the edge of chaos

It is likely true that every reader of this book - and especially any reader who has reached this point - has encountered the logistic equation previously. Even so, by returning to an old friend, we have a good starting point for new kinds of explorations. As in the previous section, we will begin with relatively simple material but end with remarkably sophisticated stuff.

The logistic equation

We allow N(t) to represent population size at time t and assume that it changes according to the dynamics

In this equation, r and K are parameters; K is the population size at which the growth rate of the population is 0. It is commonly called the carrying capacity of the population. When the growth rate is 0, births and deaths are still occurring, but they are exactly balancing each other. The right hand side of Eq. (2.26) is a parabola, with zeros at N = 0 and N = K and maximum value rK 4 when N= K 2, which is called the population size that provides maximum net productitivity (MNP); see Figure 2.7a.

In order to understand the parameter r, it is easiest to consider the per capita growth rate of the population

Inspection of the right hand side of Eq. (2.27) shows that it is a decreasing function of population size and that its maximum value is r, occurring when N= 0. Of course, if N= 0, this is biologically meaningless - there won't be any reproduction if the population size is 0. What we mean, more precisely, is that in the limit of small population size, the per capita growth rate approaches r - so that r is the maximum per capita growth rate.

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