A B

where A and B are constants that you determine by creating the common denominator and simplifying.

The discrete logistic map and the edge of chaos

We now come to what must be one of the most remarkable stories of good luck and good sleuthing in science. To begin this story,

I encourage you to stop reading just now, go to a computer and plot the trajectories for N(t) given by the formula for N(t) in the previous exercise, for a variety of values of r - let r range from 0.4 to about 3.5. After that return to this reading.

Now let us poke around a bit with the logistic equation by recognizing the definition of the derivative as a limiting process. Thus, we could rewrite the logistic equation in the following form:

This equation, of course, is no different from our starting point. But now let us ignore the limiting process in Eq. (2.28) and simply set dt = 1. If we do that Eq. (2.28) becomes a difference equation, which we can write in the form

This equation is called the logistic map, because it ''maps'' population size at one time to population size at another time. You may also see it written in the form which makes it harder to connect to the original differential equation. Note, of course, that Eq. (2.29) is a perfectly good starting point, if we think that the biology operates in discrete time (e.g. insect populations with non-overlapping generations across seasons, or many species of fish in temperate or colder waters).

Although Eq. (2.29) looks like the logistic differential equation, it has a number of properties that are sufficiently different to make us wonder about it. To begin, note that if N(t) > K then the growth term is negative and if r is sufficiently large, not only could N(t + 1) be less than N(t), but it could be negative! One way around this is to use a slightly different form called the Ricker map

This equation is commonly used in fishery science for populations with non-overlapping generations (e.g. salmonids) and misused for other kinds of populations. It has a nice intuitive derivation, which goes like this (and to which we will return in Chapter 6). Suppose that maximum per capita reproduction is A, so that in the absence of density dependence N(t + 1) = AN(t), and that density dependence acts in the sense that a focal offspring has probability f of surviving when there is just

one adult present. If there are N adults present, the probability that the focal offspring will survive is /N. Combining these, we obtain N(t + 1) = AN(t)/N(t), which surely suggests a good exercise.

Often we set/^ = e~bN, so that the Ricker map becomes N(t + 1) = AN(t)e~bN(t). First, explain the connection between / and B and the relationship between the parameters A, b and r, K. Second, explain why the Ricker map does not have the nasty property that N(t) can be less than 0. Third, use the Taylor expansion of the exponential function to show how the Ricker and discrete logistic maps are connected.

But now let us return to Eq. (2.29) and explore it. To do this, we begin by simply looking at trajectories. I am going to set K = 100, N(0) = 20 and show N(t) for a number of different values of r (Figure 2.8). When r is moderate, things behave as we expect: starting at N(0) = 20, the population rises gradually towards K = 100. However, when r = 2.0 (Figure 2.8c), something funny appears to be happening. Instead of settling down nicely at K = 100, the population exhibits small oscillations around that value. For r slightly larger (r = 2.3, panel d) the oscillations become more pronounced, but still seem to be flipping back and forth across K = 100. The behavior becomes even more complicated when r gets larger - now there are multiple population sizes that are consistently visited (Figure 2.8e). When r gets even larger, there appears to be no pattern, just wild and erratic behavior. This behavior is called deterministic chaos. It was discovered more or less accidentally in a number of different ways in the 1960s and 1970s (see Connections).

Before explaining what is happening, I want to present the results in a different way, obtained using the following procedure. I fixed r. However, instead of fixing N(0), I picked it randomly and uniformly (all values equally likely) between 1 and K. I then ran the population dynamics for 500 time steps and plotted the point (r, N(500)). I repeated this, with r still fixed, for 50 different starting values, then changed r and began the process over again. The results, called a bifurcation (for branching) diagram, are shown in Figure 2.9. When r is small, there is only one place for N(500) to be - at carrying capacity K = 100. However, once we enter the oscillatory regime, N(500) is never K - it is either larger or smaller than K. And as r increases, we see that we jump from 2 values of N(500) to 4 values, then on to 8, 16, 32 and so forth (with the transition regions becoming closer and closer). As r continues to increase, virtually all values can be taken by N(500). You may want to stop reading now, go to your computer and create a spreadsheet that does this same set of calculations.

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