Chapter 5. Carrying Capacity

We applied the Law of Tolerance to describe the distribution of population density for a species across varying levels of environmental factors as a bell-shaped curve. Although we also observed that the tolerance range can shift under certain circumstances, we shall return to this in a later chapter. For now, we want (again in the sense of “I want to, and thus intend to…”) to turn our attention to the implications of the Law of Tolerance, as I developed it, for a theory of Plant Community Ecology. As usual, this requires that we look first at the capacity of populations to increase in the absence of limiting factors as a basis for determining what population densities ought to be on a particular site. The rationale for this is that we need to know how populations behave on specific sites before we can approach the broader questions of the organization of communities across the Vegetation (or Landscape).

Growth Potential:

As observed by Malthus in 1798, the growth potential of living organisms is quite large. For example, imagine that a pair of house mice (Mus musculus) immigrated to a Human house on Wednesday, September 22, 2010 (the autumnal equinox). These creatures have litters averaging 5 or 6. The gestation period is approximately 3 weeks, and the offspring reach sexual maturity by the age of 5 to 7 weeks. Thus, the generation time (the time from conception to reproduction by the offspring) for these mice is about 10 weeks. So… by December 1, 2010, there could easily be 3 reproducing pairs of mice in the house, plus the immigrant pair. If we assume non-overlapping generations (each generation reproduces only once, then dies after the young are able to survive on their own), and equal numbers of males and females (3 each) in the litter, we can project the growth of the population [without complicated math], as illustrated in the table. After one year, on the autumal equinox in 2011, there could be 486 immature mice; and after two years, on the autumnal equinox of 2012, the population could be 118,098 mice which will soon be old enough to reproduce!
    If we allow overlapping generations (the parental generation (P-1) reproduces more than once, and the filial (offspring) generation(F-1) reaches reproductive maturity, and reproduces before the P-1 generation dies) so the individual mice continue to live and reproduce for about 7 months, which is more realistic for House Mouse populations in the ‘wild,’ the initial immigrant pair would produce a population of 199,734 mice by the end of one year. It should be obvious to just about everyone that almost 200 thousand mice are not likely to be found in any house, so something must happen to slow the growth of the population. In controlled experiments, population growth not only slows, but seems to approach a stable Population density (the number of individuals of the species population per unit area).
    Darwin (1859) suggested that “A struggle for existence inevitably follows from the high rate at which all organic beings tend to increase. … Hence, as more individuals are produced than can possibly survive, there must be in every case a struggle for existence, either one individual with another of the same species, or with the individuals of another species, or with the physical conditions of life.” (quoted from Charles Darwin, 1958. The Origin of Species, (a Mentor Book) The New American Library of World Literature, Inc, New York NY, first printing). The underlying concepts have not changed, but we now refer to the struggle for existence as competition, in which more than one individual is using the same limited resource (food, nest site or germination site, etc.), so we anticipate that not all competing individuals will gain enough of the resource to support survival. When the competition is between individuals of the same species, it is said to be “intraspecific competition;” and when it involves individuals of distinct species, it is called “interspecific competition.” In either event, competition is considered to be a density-dependent mechanism, suggesting the intensity of the competition depends on the population density. The effects observable in the field include the following: reduced growth rate (which may lead to longer generation times), fecundity (the number of offspring produced during reproduction), self-thinning (a consequence of incresed death rates), and dispersal (an increase in emigration). Some papers published in the mid 20^th Century suggested that there are also density-independent factors which cause population growth rates to slow, but it is not clear that these would be effective. Examples of density-independent mechanisms include periodic environmental events (such as unusually severe winters, droughts, floods, etc) and disease outbreaks (which may be density-dependent) plus some reasonably constant factors (such as predation, in some cases, but more often predation should be density-dependent due to coupling of predator-prey cycles).

Carrying Capacity:

Early in the efforts (of the 20^th Century) to develop Ecology as a mathematically sound Science, attempts were made to describe the decline in population growth and approach to a stable population density [Lotka (1925) & Voterra (1926), working independently]. The result was the “logistic growth curve” which used differential calculus:
dN/dt = rN_(t-1)*(K - N_t)/K
N is population density, r is the reproductive potential growth rate, t is time. dN/dt (a calculus term) is the change in population density (N) with respect to time (t) [not dN (the difference in N from time t to time t+1) divided by dt (the difference in time from time t to time t+1)]. This equation approaches, as a limit, the value K which is the derived stable population density. It was called Carrying Capacity defined as the capacity of the environment to support a population of a species, stated in terms of population size or density.
    Sibly, et al. (Science 309 (22 July 2005), p. 607-610) point out that the logistic growth model assumes a linear relationship between growth rate and population size, but based on data from 1,780 studies, the relationship can be linear, concave, or convex. In the linear relationship, the population growth rate changes at constant rate. In the concave relationship, the population growth rate changes quickly at first, then slows as K is approached; and in the convex relationship, the population growth rate changes slowly at first, increasing as K is approached. An examination of the growth curves predicted by the logistic growth model reveals that the model actually assumes that the relationship between growth rate and population size is concave, not linear as suggested by Sibley. However, in my opinion, the major problem with the logistic growth curve, and other calculus-based Ecological models, is that the underlying mathematic assumptions are, at best, poorly met by the ecological systems being modelled. Calculus is inherently deterministic, and predicts an absolute solution (K in this case) which works quite well in Physics, but ecological systems exhibit a strong stochastic (probabilistic) component.

    An alternative to the above view of carrying capacity would be to look at carrying capacity as a characteristic of the population rather than as a characteristic of the environment. The revised definition (definition 2 in the glossary) of carrying capacity would be “the fitness of the species in the environment, stated as the theoretical maximum population size or density, given the values of the environmental factors.” Thus, if we knew the values for the environmental factors at the site, we should expect the population density to be equal to, or less than, the carrying capacity. As will be shown in a later chapter (Chapter 10), this use of the concept of carrying capacity will allow us to develop a Theory of Plant Community Ecology.

“Boom & Crash” cycles:

Some species do not ever reach stable densities, but fluctuate between very high and very low abundances or densities. Such population cycles are described as Boom & Crash cycles (where “Boom and Crash” are ‘borrowed’ from economists' descriptions of stock market cycles). During the Boom phase, the population grows to excessively high densities, resulting in degradation of the environment. This triggers the Crash phase, in which the population undergoes a rapid decline to excessively low densities near local extinction, sometimes actually becoming locally extinct, perhaps followed by re-invasion. If a small population remains, or if re-invasion establishes a small population, another boom phase begins as soon as the environment recovers (which seems to be normal for the degraded environment). A good example of a Boom & Crash population cycle occurs in the White-tailed deer (Odocoileus virginianus) populations of the eastern United States where the natural predators of the species have been effectively eradicated by Human hunting. There have been reports of sightings (a few confirmed) of a return of the Mountain lion (Puma concolor) and the Gray wolf (Canis lupus), the major natural predators on the White-tail deer, to many sites in their former ranges.

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