A population is a set of organisms of the same species living in a particular place and time. This simple definition begs the question of how to define "species," since the traditional criterion that it be composed of "interbreeding" organisms is difficult or impossible to apply in many cases. Nevertheless, the definition works for most purposes. The key idea, to which we will return below, is that populations are composed of interacting individuals. An operational definition of the concept of ecological community is more elusive, however. One anonymous, but cynical, wag defined it as a set of populations about which it is interesting to speak. There is a frighteningly important element of truth in this definition. And we could accept it, provided community ecologists were never boring. This not always being the case, we content ourselves with the more typical definition: the set of co-occurring and interacting populations in a place. In practice, the set of populations and relations studied is often confined to specific taxa and ecological processes.
In this chapter, we describe some of the elementary models of populations and communities. In so doing, we will again return to the principles developed in Part I. In particular, we examine more complex nullcline analysis using mechanistic models of competing species. We introduce the concept of individual-based models and revisit stochastic models in the form of demographic stochasticity and time to extinction. Finally, we encounter again the problem of model validation in testing simple, alternative predator-prey models with laboratory experiments; we will also use bioenergetic models to predict and test size distributions of fish in lakes.
The central questions that these models address include: (1) Can population dynamics be predicted from the bioenergetics of individuals? (2) What is the simplest model needed to describe accurately predator-prey dynamics in simple aquatic microcosms? (3) How does predator learning affect predator-prey cycles? (4) Can pesticides effectively control insect pest outbreaks?
We have already introduced, through examples in Part /, density-independent and density-dependent population growth. We will not repeat that now, but rather will give a simple, phenomenological generalization of the models. We wish to formulate an hypothesis of population growth based on the effects that the entire population has on the reproduction of an average individual. (By average, we mean average in all respects: sex, weight, age, and so on.) In density-independent models, the relation is a straight line with zero slope; in the density-dependent logistic model, it is a straight line with negative slope (Sec. 2.3). To generalize the biological hypothesis that increased population size always decreases per capita birth rate, we could use a nonlinear relation such as Richard's equation as illustrated in Chapter 5.
A more dramatic departure is a phenomenon called the Allee effect in which two processes are operating: decreases in per capita birth rate due to competition, and increases in per capita birth rate with increases in population numbers due to increased chances of encountering mates at low population density. If our aim is simply to describe this relation, we can use any functional form that possesses a maximum and that can be scaled to biologically realistic numbers. Two candidates from Chapter 4 are the maximum function and the Blumberg function. The former, being a product of two separate subfunctions, has the advantage that each subfunction and its parameters can be associated with the two biological processes (mate location and competition).
Here is one possible phenomenological model of population growth using the Allee effect (Wilson and Bossert 1971):
where M is a lower threshold below which the per capita rate is negative. Above the threshold the per capita rate increases to a maximum then decreases to 0 at N - K. The importance of the Allee effect will become apparent in Section 17.8.6, where we discuss chaos.
These models, because of their nonlinear structure, can fit many data sets (Berryman 1991), but being general, they do not satisfy our desire for more mechanistic explanations. One point in which they fail to capture basic biological mechanisms is their assumption that all individuals are equal. All individuals, of course, are not equal and everyone eventually grows old and dies. Individuals differ because of their age and other physiological and ecological variables often correlated with age (e.g., the effect of age or size on energy demands, foraging efficiency, running speed, etc.). The simplest model of an age structured population is one analogous to the density-independent finite difference model. As a simple example, assume the population has four age classes, only the oldest reproduces, and at each time step the fate of an individual is either to die or to live and become one time step older (i.e., advance to the
next age class). So, the fate of age class i is
Nilt+1 = Nu - diNu - (Si)Nu + i.-ity-u = Si-iNi-u, where d is the fraction dying, and s is the fraction surviving and aging 1 time interval (so that d = 1 - s). In addition to survival, the youngest age class increases by the addition of individuals through reproduction. This is modeled as J\, the average birth rate per female of age i. Since all individuals within an age class are considered equivalent, the net number of newborn individuals from females of age i is /¡Nt. The total number of newborn individuals is the sum of the contributions of all reproductive age classes. With this, the complete set of equations for all age classes is
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