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How can the future number of deer, agricultural pests, or cod be calculated based on the present number of individuals and their age distribution? How long will it take for a viral outbreak in a particular city to reach another city five hundred miles away? In addressing such basic questions, ecologists today are as likely to turn to complicated differential equations as to life histories—a dramatic change from thirty years ago. Population ecology is the mathematical backbone of ecology. Here, two leading experts provide the underlying quantitative concepts that all modern-day ecologists need.
John Vandermeer and Deborah Goldberg show that populations are more than simply collections of individuals. Complex variables such as the size distribution of individuals and allotted territory for expanding groups come into play when mathematical models are applied. The authors build these models from the ground up, from first principles, using a much broader range of empirical examples—from plants to animals, from viruses to humans—than do standard texts. And they address several complicating issues such as age-structured populations, spatially distributed populations, and metapopulations.
Beginning with a review of elementary principles, the book goes on to consider theoretical issues involving life histories, complications in the application of the core principles, statistical descriptions of spatial aggregation of individuals and populations as well as population dynamic models incorporating spatial information, and introductions to two-species interactions.
Complemented by superb illustrations that further clarify the links between the mathematical models and biology, Population Ecology is the most straightforward and authoritative overview of the field to date. It will have broad appeal among undergraduates, graduate students, and practicing ecologists.
"Amidst the recent plethora of undergraduate books on population ecology emerges this superbly crafted volume. . . . What distinguishes this from most of its predecessors is an uncommon breadth of subject matter . . . a fine balance between patronizing the knowledgeable reader and overwhelming the interested novice, and a highly commendable means by which the information is communicated. Following an impressive range of topics in the opening chapter is one of the best-written introductions to the life history theory that this reviewer has encountered in 20 years of research on the topic."—Choice
"What distinguishes this book from others in the field is the diverse array of topics covered that are rarely or only cursorily treated in other books. . . . What I enjoyed most about this book were the frequent discussions on the ecological interpretation of the mathematical results and the corresponding caveats. . . . Vandermeer and Goldberg do an admirable job of explaining the ecological meaning and assumptions behind all of the mathematical results presented. They include many figures that illustrate their points clearly and these are accompanied with detailed verbal explanations."—Helen M. Regan, Ecology
In many contexts it is important to understand the characteristics of single populations of organisms. A wildlife manager, for example, needs to predict what the density of a population of deer or cod would be under different management plans. Or an agronomist may wish to know the yield of a population of maize plants when planted at a particular density. In more theoretical applications, we are interested in knowing, for example, the rate at which a population changes its density in response to selection pressure. These topics are typical of the field called population ecology.
The unit of analysis is, not surprisingly, the population, a concept that is at once simple and complicated. The simple idea is that a population is a collection of individuals. But, as most ecologists intuitively know, the idea of a population is considerably more complex when one deals with the sort of real-life examples mentioned above. To know what size limits one should place on catch for a fish species, one must know not only how many fish are in the population but also the size distribution of that population and how that distribution is related to the population's overall reproduction. To decide when to take action on the emergence of pest species in forests or farms, one must know the distribution of individuals within life stages. In the determination of whether a species is threatened with extinction, its distribution in space and the amount of movement among subpopulations (i.e., metapopulation dynamics) are far more important than simply its numerical abundance. And, to cite the most cited example, the absolute abundance of the human population has little to do with anything of interest compared with the activities undertaken by the members of that population.
Thus, the subject of population ecology can be very complicated. But, as we do in any science, we begin by assuming that it is simple. We eliminate the complications, make simplifying assumptions, and try to develop general principles that might form a skeleton upon which the flesh of real-world complications might meaningfully be attached. This chapter covers the first two essential ideas of that skeleton: density independence and density dependence.
Density Independence: The Exponential Equation
It is surprising how quickly a self-reproducing phenomenon becomes big. The classic story goes like this: Suppose you have a lake with some lily pads in it and suppose each lily pad replicates itself once a week. If it takes a year for half the lake to become covered with lily pads, how long will it take for the entire lake to become covered? If one does not think too long or too deeply about the question, the quick answer seems to be about another year. But a moment's reflection retrieves the correct answer, only one more week.
This simple example has many parallels in real-world ecosystems. A pest building up in a field may not seem to be a problem until it is too late. A disease may seem much less problematical than it really is. The simple problem of computing the action threshold (the density a pest population must reach before you have to spray pesticide) requires the ability to predict a population's size on the basis of its previous behavior. If half the plants have been attacked within 3 months, how long will it be before they are all attacked?
To understand even the extremely simple example of the lily pads, one constructs a mathematical model, usually quite informally in one's head. If all the lily pads on the pond replicate themselves once a week, then, in a pond half-filled with lily pads, each one of those lily pads will replicate in the next week and thus the pond will be completely filled up. To make the solution to the problem general we say the same thing, but instead of labeling the entities lily pads, we call them something general, say organisms. If organisms replicate once a week and the environment is half full, it will take only one week for it to become completely full. Implicitly, the person who makes such a statement is saying out loud the following equation:
[N.sub.t+1] = 2[N.sub.t] (1)
N is the number of organisms, in this case lily pads. Instead of t(time), say this week, and instead of t + 1, say next week, and equation 1 is simply "the number of lily pads next week is equal to twice the number this week."
Of course, writing down equation 1 is no different than making any of the statements that were made previously about it. But by making it explicitly a mathematical expression, we bring to our potential use all the machinery of formal mathematics. And that is actually good, even though beginning students sometimes do not think so.
Using equation 1, we can develop a series of numbers that reflect the changes of population numbers over time. For example, consider a population of herbivorous insects: if each individual produces a single offspring once a week, and those offspring mature and also produce an offspring within a week, we can apply equation 1 to see exactly how many individuals will be in the population at any point in time. Beginning with a single individual we have, in subsequent weeks, 2, 4, 8, 16, 32, 64, 128, and so on. If we change the conditions such that the species replicates itself twice a week, equation 1 becomes
[N.sub.t+1] = 3[N.sub.t] (2)
(with a 3 instead of a 2, because before we had the individual and the single offspring it produced, now we have the individual and the two offspring it produced). Now, beginning with a single individual, we have, in subsequent weeks, 3, 9, 27, 81, 243, and so on.
We can use this model in a more general sense to describe the growth of a population for any number of offspring at all (not just 2 and 3 as above). That is, write,
[N.sub.t+1] + R[N.sub.t] (3)
where R can take on any value at all. R is frequently called the finite rate of population growth (or the discrete rate).
It may have escaped notice in the above examples, but either of the series of numbers could have been written with a much simpler mathematical notation. For example, the series 2, 4, 8, 16, 32, is a actually [2.sup.1], [2.sup.2], [2.sup.3], [2.sup.4], [2.sub.5], and the series 3, 9, 27, 81, 243, is actually [3.sup.1], [3.sup.2], [3.sup.3], [3.sup.4], [3.sup.5]. So we could write,
[N.sub.t] = [R.sup.t] (4)
which is just another way of representing the facts as described by equation 3. (Remember, we began with a single individual, so [N.sub.0] = 1.0.)
We now wish to represent the constant R (of equation 4) in a different fashion, to make further exposition easier. It is a general rule that any number can be written in many ways. For example, the number 4 could be written as 8/2, or 9 - 5, or [2.sup.2], or many other ways. In a similar vein, an abstract number, say R, could be written in any number of ways: R = 2b, where b is equal to R/2, or R = [2.sup.b], in which case b = ln(R/2) (where ln stands for natural logarithm). If we represent R as [2.7183.sup.r], a powerful set of mathematical tools becomes immediately available. The number 2.7183 is Euler's constant, usually symbolized as e (actually 2.7183 is rounded off and thus only approximate). It has the important mathematical property that its natural logarithm is equal to 1.0.
So, rewrite equation 4 as,
[N.sub.t] = [e.sup.rt] (5)
which is the classical form of the exponential equation (where R has been replaced with [e.sup.r]). One more piece of mathematical manipulation is necessary to complete the toolbag necessary to model simple population growth. Another seemingly complicated but really rather simple relationship that is always learned (but frequently forgotten) in elementary calculus is that the rate of change of the log of any variable is equal to the derivative of that variable divided by the value of the variable. This rule is more compactly stated as,
D(ln N)/dt = 1/N dN/dt (6)
So, if we rewrite equation 5 as,
ln ([N.sub.t]) = rt
we can differentiate with respect to t to obtain,
d (ln N)/dt = r (7)
and we can use equation 6 to substitute for the left-hand side of 7 to obtain,
dN/Ndt = r
and after multiplying both sides by N, we obtain,
dN/dt = rN (8)
Equations 5 and 8 are the basic equations that formally describe an exponential process. Equation 8 is the differentiated form of equation 5, and equation 5 is the integrated form of equation 8. They are thus basically the same equation (and indeed are quite equivalent to the discrete form-equation 3). Depending on the use to which they are to be put, any of the above forms may be used, and in the ecological literature one finds all of them. Their basic graphical form is illustrated in figure 1.1.
In the examples of exponential growth introduced above, the parameter (r, or R) was introduced as a birth process only. The tacit assumption was made that there were no deaths in the population. In fact, all natural populations face mortality, and the parameter of the exponential equation is really a combination of birth and death rates. More precisely, if b is the birth rate (number of births per individual per time unit), and d is the death rate (number of deaths per individual per time unit), the parameter of the exponential equation is
r = b - d (9)
where the parameter r is usually referred to as the intrinsic rate of natural increase.
One other simplification was incorporated into all of the above examples. We presumed always that the population in question was initiated with a single individual, which almost never happens in the real world. But the basic integrated form of the exponential equation is easily modified to relax this simplifying assumption. That is,
[N.sub.t] = [N.sub.0][e.sup.rt] (10)
which is the most common form of writing the exponential equation. Thus, there are effectively two parameters in the exponential equation: the initial number of individuals, [N.sub.0], and the intrinsic rate of natural increase, r.
Putting the exponential equation to use requires estimation of the two parameters. Consider, for example, the data presented in table 1.1. Here we have a series of observations over a 5-week period of the average number of aphids on a corn plant in an imaginary corn field.
As a first approximate assumption, let us assume that this population originates from an initial cohort that arrived in the milpa on March 18 (one week before the initial sampling). We can apply equation 10 to these data most easily by taking logarithms of both sides, thus obtaining,
ln([N.sub.t]) = ln([N.sub.0]) + rt (11)
which gives us a linear equation of the natural logarithm of the number of aphids versus time (where we code March 18 as time = 0, March 25 as time = 1, April 1 as time = 2, etc....). Figure 1.2 is a graph of this line along with the original data to which it was fit, and figure 1.3 is a graph of the original data along with the fitted curve on arithmetic axes.
From these data we estimate 1.547 aphids per aphid per week added to the population (i.e., the intrinsic rate of natural increase, r, is 1.547, which is the slope of the line in figure 1.2). The intercept of the regression is -4.626, which indicates that the initial population was 0.0098 (that is, the anti ln of -4.626 is 0.0098), which is an average of about one aphid per 10 plants. Now, if we presume that once the plants become infected with more than 40 aphids per plant the farmer must take some action to try and control them, we can use this model to predict when, approximately, this time will arrive. The regression equation is,
ln(N of aphids) = -4.626 + 1.547t
which can be rearranged as,
t = [ln(N of aphids) + 4.626]/1.547
The natural log of 40 is 3.69, so we have,
t = (3.69 + 4.626)/1.547 = 5.375
Translating this number into the actual date (April 22 was time = 5), we see that the critical number will arrive about April 24 (actually at 3:00 P.M. on April 24, theoretically).
Naturally, the natural world contains many complicating factors, and the exact quantitative predictions made by the model could be quite inaccurate. As we discuss in later chapters, including some of the complicating factors will increase the precision of the predictions. On the other hand, April 24 really does represent the best prediction we have, based on available data. It may not be a very good prediction, but it is in fact the only one available. It may seem quite counterintuitive that, having taken five full weeks to arrive at only 14 aphids per plant, in only 2 more days the critical figure of 40 aphids per plant will be reached, but such is the nature of exponential processes. A simple model like this could help the farmer plan pest control strategies.
Density Dependence: Intraspecific Competition
In the above section, we showed that any population reproducing at a constant per capita rate will grow according to the exponential law. Indeed, that is the very essence of the exponential law; each individual reproduces at a constant rate. However, the air we breath and the water we drink are not completely packed with bacteria or fungi or insects; as they would be if populations grew exponentially forever. Something else must happen. That something else is usually referred to as intraspecific competition, which means that the performance of the individuals in the population depends on how many individuals are in it; this concept is commonly known as density dependence. Density dependence is a complicated issue, one that has inspired much debate and acrimony in the past and one that still forms an important base for more modern developments in ecology.
The idea of density dependence was originally associated with the human population and was brought to public attention as early as the eighteenth century by Sir Thomas Malthus (1830). Verhulst (1838) first formulated it mathematically as the "true law of population," better known today as the logistic equation (see below). Later, Pearl and Reed (1920), in attempting to project the human population size of the United States, independently derived the same equation.
Excerpted from Population Ecology by John H. Vandermeer Deborah E. Goldberg Copyright © 2003 by Princeton University Press. Excerpted by permission.
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|List of Figures|
|List of Tables|
|Ch. 1||Elementary Population Dynamics||1|
|Density Independence: The Exponential Equation||3|
|Density Dependence: Intraspecific Competition||10|
|Ch. 2||Life History Analysis||35|
|Investment in Survivorship versus Reproduction: The r, K Continuum||37|
|Cost of Reproduction||41|
|Optimal Reproductive Schedules||44|
|Ch. 3||Projection Matrices: Structured Models||51|
|Elementary Population Projection Matrices||52|
|Non-age Structure: Stage Projection Matrices||62|
|Eigenvectors, Reproductive Value, Sensitivity, and Elasticity||69|
|Applications of Population Projection Matrices||73|
|Density Dependence in Structured Populations||80|
|App. A||Basic Matrix Manipulations||94|
|Ch. 4||A Closer Look at the "Dynamics" in Population Dynamics||101|
|Intuitive Ideas of Equilibrium and Stability||103|
|Eigenvalues: A Key Concept in Dynamic Analysis||114|
|Basic Concepts of Equilibrium and Stability in One-Dimensional Maps||120|
|Ch. 5||Patterns in Space and Metapopulations||155|
|The Poisson Distribution||158|
|The Question of Scale||163|
|Ch. 6||Predator-Prey (Consumer-Resource) Interactions||177|
|Predator-Prey Interactions: First Principles||179|
|Functional Response and Density Dependence Together||193|
|Paradoxes in Applications of Predator-Prey Theory||195|
|Predator-Prey Dynamics: A Graphical Approach||198|
|Predator-Prey Interactions in Discrete Time||205|
|Direct Disease Transmission||210|
|Indirect Disease Transmission||217|
|Ch. 8||Competition and a Little Bit of Mutualism||221|
|Competition: First Principles||222|
|The Competitive Production Principle: Applications of Competition Theory to Agriculture||234|
|Competition: The Details||240|
|Ch. 9||What This Book Was About||255|
"Population ecology is rapidly maturing as a theoretical science. One sign of this maturity is the ongoing synthesis between sophisticated mathematical theory and innovative experimental approaches. Yet the traditional education in biology does not equip students with the tools they need to fully appreciate these new theoretical developments. Here is where this admirable book by Vandermeer and Goldberg comes in. A particularly enjoyable aspect of Population Ecology: First Principles is the ability of the authors to relate the complex tapestry of ecological theory to a few fundamental quantitative principles."—Peter Turchin, University of Connecticut
"Vandermeer and Goldberg have written an outstanding book that synthesizes and summarizes the fundamental concepts and principles of population ecology. Its highly approachable treatment of models should give students deep and intuitive insights into population ecology. Because the mathematical techniques presented in the book represent the core toolbox of the discipline, this book is essential reading for anyone going into population and community ecology."—David Tilman, University of Minnesota
"This is an excellent book that I look forward to using in the classroom. It is one of the most understandable in the field. The authors present, in more tractable fashion than do some similar books, fundamental material that all ecologists need. Anyone in the life sciences should immediately recognize the importance of the material. Scholars in economics and the social sciences should also see that the book is very relevant to their disciplines."—David J. Moriarty, California State PolytechnicUniversity
"One of this book's greatest strengths is the way it emphasizes the processes underlying standard ecological models. Rather than relying on plausibility arguments, the authors start from simple mechanistic models—for instance, deriving the Lotka-Volterra competition equations from a model of resource competition. They also present a nice range of practical examples."—Ben Bolker, University of Florida