## Four examples and a metaphor

Robert Peters (Peters 1991) - who (like Robert MacArthur) tragically died much too young - told us that theory is going beyond the data. I thoroughly subscribe to this definition, and it shades my perspective on theoretical biology (Figure 1.1). That is, theoretical biology begins with the natural world, which we want to understand. By thinking about observations of the world, we conceive an idea about how it works. This is theory, and may already lead to predictions, which can then flow back into our observations of the world. Theory can be formalized using mathematical models that describe appropriate variables and processes. The analysis of such models then provides another level of predictions which we take back to the world (from which new observations may flow). In some cases, analysis may be insufficient and we implement the models using computers through programming (software engineering). These programs may then provide another level of prediction, which can flow back to the models or to the natural world. Thus, in biology there can be many kinds of theory. Indeed, without a doubt the greatest theoretician of biology was Charles Darwin, who went beyond the data by amassing an enormous amount of information on artificial selection and then using it to make inferences about natural selection. (Second place could be disputed, but I vote for Francis Crick.) Does one have to be a great naturalist to be a theoretical biologist? No, but the more you know about nature - broadly defined (my friend Tim Moerland at Florida State University talks with his students about the ecology of the cell (Moerland 1995)) - the better off you'll be. (There are some people who will say that the converse is true, and I expect that they won't like this book.) The same is true, of course, for being able to

Figure 1.1. Theoretical biology begins with the natural world, which we want to understand. By thinking about observations of the world, we begin to conceive an idea about how it works. This is theory, and may already lead to predictions, which can then flow back into our observations of the world. The idea about how the world works can also be formalized using mathematical models that describe appropriate variables and processes. The analysis of such models then provides another level of predictions which we can take back to the world (from which new observations may flow). In some cases, analysis may be insufficient and we choose to implement our models using computers through programming (software engineering). These programs then provide another level of prediction, which can also flow back to the models or to the natural world.

develop models and implementing them on the computer (although, I will tell you flat out right now that I am not a very good programmer -just sufficient to get the job done). This book is about the middle of those three boxes in Figure 1.1 and the objective here is to get you to be good at converting an idea to a model and analyzing the model (we will discuss below what it means to be good at this, in the same way as what it means to be good at opera).

On January 15, 2003, just as I started to write this book, I attended a celebration in honor of the 80th birthday of Professor Joseph B. Keller. Keller is one of the premier applied mathematicians of the twentieth century. I first met him in the early 1970s, when I was a graduate student. At that time, among other things, he was working on mathematics applied to sports (see, for example, Keller (1974)). Joe is fond of saying that when mathematics interacts with science, the interaction is fruitful if mathematics gives something to science and the science gives something to mathematics in return. In the case of sports, he said that what mathematics gained was the concept of the warm-up. As with athletics, before embarking on sustained and difficult mathematical exercise, it is wise to warm-up with easier things. Most of this chapter is warm-up. We shall consider four examples, arising in behavioral and evolutionary ecology, that use algebra, plane geometry, calculus, and a tiny bit of advanced calculus. After that, we will turn to two metaphors about this material, and how it can be learned and used.

Foraging in patchy environments

Some classic results in behavioral ecology (Stephens and Krebs 1986, Mangel and Clark 1988, Clark and Mangel 2000) are obtained in the

Figure 1.2. Two stars of foraging experiments are (a) the great tit, Parus major, and (b) the common starling Sturnus vulgaris (compliments of Alex Kacelnik, University of Oxford). (c) Foraging seabirds on New Brighton Beach, California, face diet choice and patch leaving problems.

study of organisms foraging for food in a patchy environment (Figure 1.2). In one extreme, the food might be distributed as individual items (e.g. worms or nuts) spread over the foraging habitat. In another, the food might be concentrated in patches, with no food between the patches. We begin with the former case.

The two prey diet choice problem (algebra)

We begin by assuming that there are only two kinds of prey items (as you will see, the ideas are easily generalized), which are indexed by i = 1, 2. These prey are characterized by the net energy gain Ei from consuming a single prey item of type i, the time hi that it takes to handle (capture and consume) a single prey item of type i, and the rate 1i at which prey items of type i are encountered. The profitability of a single prey item is Ei/hi since it measures the rate at which energy is accumulated when a single prey item is consumed; we will assume that prey type 1 is more profitable than prey type 2. Consider a long period of time T in which the only thing that the forager does is look for prey items. We ask: what is the best way to consume prey? Since I know the answer that is coming, we will consider only two cases (but you might want to think about alternatives as you read along). Either the forager eats whatever it encounters (is said to generalize) or it only eats prey type 1, rejecting prey type 2 whenever this type is encountered (is said to specialize). Since the flow of energy to organisms is a fundamental biological consideration, we will assume that the overall rate of energy acquisition is a proxy for Darwinian fitness (i.e. a proxy for the long term number of descendants).

In such a case, the total time period can be divided into time spent searching, S, and time spent handling prey, H. We begin by calculating the rate of energy acquisition when the forager specializes. In search time S, the number of prey items encountered will be 21S and the time required to handle these prey items is H = h1(21S). According to our assumption, the only things that the forager does is search and handle prey items, so that T = S + H or

We now solve this equation for the time spent searching, as a fraction of the total time available and obtain

Since the number of prey items encountered is 21S and each item provides net energy E1, the total energy from specializing is E121S, and the rate of acquisition of energy will be the total accumulated energy divided by T. Thus, the rate of gain of energy from specializing is

An aside: the importance of exercises

Consistent with the notion of mathematics in sport, you are developing a set of skills by reading this book. The only way to get better at skills is by practice. Throughout the book, I give exercises - these are basically steps of analysis that I leave for you to do, rather than doing them here. You should do them. As you will see when reading this book, there is hardly ever a case in which I write ''it can be shown'' - the point of this material is to learn how to show it. So, take the exercises as they come -in general they should require no more than a few sheets of paper - and really make an effort to do them. To give you an idea of the difficulty of exercises, I parenthetically indicate whether they are easy (E), of medium difficulty (M), or hard (H).

Repeat the process that we followed above, for the case in which the forager generalizes and thus eats either prey item upon encounter. Show that the rate of flow of energy when generalizing is

We are now in a position to predict the best option: the forager is predicted to specialize when the flow of energy from specializing is greater than the flow of energy from generalizing. This will occur when Rs > Rg.

Show that Rs > Rg implies that

Equation (1.5) defines a "switching value'' for the encounter rate with the more profitable prey item, since as l1 increases from below to above this value, the behavior switches from generalizing to specializing. Equation (1.5) has two important implications. First, we predict that the foraging behavior is ''knife-edge'' - that there will be no partial preferences. (To some extent, this is a result of the assumptions. So if you are uncomfortable with this conclusion, repeat the analysis thus far in which the forager chooses prey type 2 a certain fraction of the time, p, upon encounter and compute the rate Rp associated with this assumption.) Second, the behavior is determined solely by the encounter rate with the more profitable prey item since the encounter rate with the less profitable prey item does not appear in the expression for the switching value.

Neither of these could have been predicted a priori.

Over the years, there have been many tests of this model, and much disagreement about what these tests mean (more on that below). My opinion is that the model is an excellent starting point, given the simple assumptions (more on these below, too).

The marginal value theorem (plane geometry)

We now turn to the second foraging model, in which the world is assumed to consist of a large number of identical and exhaustible patches containing only one kind of food with the same travel time between them

Travel time

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