and where N is the number of individuals in the population, t is time, and dN/dt is the change in N over time. The intrinsic growth rate of the population (r) is the value p approaches when resources are not limiting growth and there is no intraspecific competition. The effects of environmental conditions other than resources, for instance temperature, are modeled by changes in r. The number of individuals that the resources of a habitat can support (K) is referred to as the carrying capacity, and models intraspecific competition with a constant level of resource supply. Other impacts of the population on the environment, such as the accumulation of waste, are also modeled by K. As a population grows, the resources must be shared among many individuals, decreasing the reproduction rate and increasing the rate of death. In the equation, N approaches K, causing p to approach 0. If the population is above the carrying capacity, it cannot be supported by the resources present, p becomes negative, and the population declines. The relationship in which population growth rate is sensitive to population size is known as density-dependent population regulation.

The logistic growth equation is used by population ecologists mainly as a conceptual tool, rather than to forecast changes in population size, because the mechanisms of population regulation are not explicitly modeled in terms of reproduction and death rates. It is also rare in plant and animal populations for individuals of different ages to have the same rate of death or equal probabilities of reproduction. Intraspecific competition often affects certain classes of individuals more than others. The common method of forecasting population sizes is currently the use of structured demographic models that consider the effects of age, size, or developmental stage on probabilities of reproduction and death. Populations of each class of organism and the transition of individuals between classes are tracked separately in empirical models, using matrix algebraic methods. While the assumptions of the logistic growth equation are thought to be more appropriate for microorganisms such as bacteria and yeasts, structured demographic models have also been used for these organisms to take into account a dormant stage or vegetative (nonreproductive) stage.

Lifetime patterns of growth and reproduction, including timing of reproductive and dormant stages are known as a species' "life history." Life history strategies have important consequences for population dynamics. The logistic growth equation led to MacArthur and Wilson's (1967) model of r and K selection, which, although controversial, is still used as a method of generalizing about species life histories. K-selected species (with high K values and low r values) are selected for traits that favor the persistence of individuals under conditions of scarce resources and high intraspecific competition. These conditions occur when populations remain near their carrying capacity (K). In contrast, r-selected species have the opposite characteristics, with relatively high efficiency in converting resources to offspring. The K-selected strategy is an adaptation to environments in which conditions are relatively stable, resulting in density-dependent mechanisms of population regulation, while the r-selected strategy is an adaptation to a variable environment with high levels of resources. Hence, environments can be classified as r- or K-selecting.

Pianka (1970) made several other predictions about the correlates of r/K selection. An r-selected species was predicted to have more variable population size (typically below carrying capacity), weak intra- and interspecific competitive interactions, rapid maturation, early reproduction, small body size, semelparity (one reproductive event per lifetime), short life span, and high productivity. Ecologists soon realized that selection pressure and reproductive value at different ages can lead to traits opposite to those predicted by the correlates of the r/K-selection model (Reznick et al., 2002). Like the logistic growth equation, the r/K-selection model is now used mainly as a conceptual tool to explain life history and has been supplanted by structured demographic models that can be used to test specific hypotheses about life history evolution.

The classification of organisms as r- or K-selected is common in soil microbiology. Typically, colonies are classified based on the amount of time it takes for the colony to appear in laboratory isolation medium. These designations must be made with reference to a particular environment. Laboratory isolation conditions represent a small range of the conditions encountered in the environment. The environment of a batch culture changes continuously as nutrients are not replenished and wastes are not removed. Hence r-selected species can be positively identified (with respect to the isolation conditions), but K-selected species cannot. Soil microbiologists have created other classifications similar to the r/K-selection dichotomy, but are focused more on the species' preferred resources than on intraspecific competition. In 1925, S. Winogradsky used the term autochthonous to describe organisms that grow steadily on organic matter with a constant presence in the environment and zymogenous for organisms that proliferate on fresh organic matter (Panikov, 1995). In another scheme, oligotrophs grow only at low nutrient levels, while copiotrophs grow quickly at high nutrient levels.

An early model that linked population growth and environmental resources was proposed by Jacques Monod in 1957 (Panikov, 1995). The Monod model was developed for microbial growth in chemostats, but is now used in a wide array of situations for macro- as well as microorganisms. It has been modified to account for a variety of situations, such as colimitation by multiple resources, growth inhibition, and migration. Maintaining the same notation as above, one version is dN dt where

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