CHAPTER 1

Elementary Population Dynamics

In 1960 the famous cyberneticist Heinz von Foerster and colleagues devisedan equation predicting that the human population would become effectivelyinfinite on Friday the 13th of November, 2026, meaning that at that pointin time all humans would perish because the next individual to be born wouldcrush everyone else—mass death due to squashation! In fact von Foerster andhis colleagues were making a tongue-in-cheek argument to call attention to anissue they thought quite important. This was one, perhaps humorous, exampleof the application of simple quantitative principles of population dynamicsto problems considered important.

Indeed there are many contexts in which it is important to understand thequantitative characteristics of single populations of organisms. In fisheries management,for example, the manager is interested in being able to predict thedensity of a fish population in the future under different management plans. Anagronomist may wish to know the yield of a population of maize plants whenplanted at a particular density; an epidemiologist will want to know the densityof disease-infected humans next month. Many other examples could be citedwith clear practical importance. Of perhaps even more importance are theoreticalapplications that give us a more detailed understanding of more complexecological systems. We might be interested in knowing the rate at which a populationchanges its density in response to selection pressure as part of a generalprogram of understanding the consequences of natural selection under somehypothetical or real constraints. These topics, both applied and theoretical, aretypical of the field called population ecology, and they all start with some basicideas of what single populations of organisms do.

The unit of analysis is, not surprisingly, the "population," a concept that isat once simple and complicated. The simple idea is that a population is simplya collection of individuals. But, as most ecologists intuitively know, the idea ofa population is considerably more complex when one deals with it in any ofthe applied or theoretical contexts alluded to above. To know what size limitsone should place on a fish species one must know not only the number of fishin the population but the size distribution of that population and how thatdistribution relates to the population's reproductive effort. To decide whento take action on the emergence of pest species in an agricultural or forestrycontext, the distribution of individuals in various life stages must be known.In deciding whether a species is threatened with extinction, its distribution inspace and movement among subpopulations (i.e., metapopulation dynamics)is far more important than simply its numerical abundance. And, to use themost frequently cited example, given the huge variation in the consumptionof resources per person around the globe, the absolute numbers of the humanpopulation may be much less important than the activities undertaken by themembers of that population—doomsday could be at hand well before 2026.

Thus the subject of population ecology can be a very complicated oneindeed. But, as in the case of any science, we begin by assuming that it israther simple. We eliminate the complications, make simplifying assumptions,and try, as much as possible, to develop general principles that might forma skeleton onto which the flesh of real-world complications might meaningfullybe attached. In 2013 it seems that, unlike some other fields of biology,population ecology has a certain core subject matter that has come to be the"conventional wisdom." That is, wherever a university course in populationecology is taught, pretty much the same material is covered, at least at thebeginning of the course. This text is our attempt to present that core materialprecisely, and this chapter covers the first two essential ideas—the densityindependence and density dependence of population growth.

Density Independence: The Exponential Equation

It is often surprising how quickly a self-reproducing event becomes a bigevent. The classic story is this: suppose you have a pond with some lily padsin it, and suppose each lily pad replicates itself once per week. If it takes ayear for half the pond to become covered with lily pads, how long will it takefor the entire pond to become covered? If one does not think too long or toodeeply about the question, the quick answer seems to be about another year.But with a moment's reflection one can retrieve the correct answer—only onemore week.

This simple example has many parallels in real-world ecosystems. A pestbuilding up in a field may not seem to be a problem until it is too late. A diseasemay seem much less problematical than it really is. The simple problemof computing the "action threshold" (the population density a pest mustreach before you have to spray pesticide) requires the ability to predict howlarge a population will be based on its prior behavior. If half the plants in thefield are attacked within three months, how long will it be before they are allattacked?

To understand even the extremely simple example of the lily pads, oneconstructs a mathematical model, usually quite informally, in one's head. Ifall the lily pads on a pond replicate themselves once per week, in a pond half-filledwith lily pads each one of those lily pads will replicate itself in the nextweek and thus the pond will be completely filled up. To make the solutionto the problem general, we simply say the same thing, but instead of labelingthe entities lily pads, we call them something general, say organisms. Iforganisms replicate themselves once per week, by the time...