824. Environmental Limits to Population GrowthLearning Objectives
Although life histories describe the way many characteristics of a population (such as their age structure) change over time in a general way, population ecologists make use of a variety of methods to model population dynamics mathematically. These more precise models can then be used to accurately describe changes occurring in a population and better predict future changes. Certain models that have been accepted for decades are now being modified or even abandoned due to their lack of predictive ability, and scholars strive to create effective new models. Exponential GrowthCharles Darwin, in his theory of natural selection, was greatly influenced by the English clergyman Thomas Malthus. Malthus published a book in 1798 stating that populations with unlimited natural resources grow very rapidly, and then population growth decreases as resources become depleted. This accelerating pattern of increasing population size is called The best example of exponential growth is seen in bacteria. Bacteria are prokaryotes that reproduce by prokaryotic fission. This division takes about an hour for many bacterial species. If 1000 bacteria are placed in a large flask with an unlimited supply of nutrients (so the nutrients will not become depleted), after an hour, there is one round of division and each organism divides, resulting in 2000 organisms—an increase of 1000. In another hour, each of the 2000 organisms will double, producing 4000, an increase of 2000 organisms. After the third hour, there should be 8000 bacteria in the flask, an increase of 4000 organisms. The important concept of exponential growth is that the The bacteria example is not representative of the real world where resources are limited. Furthermore, some bacteria will die during the experiment and thus not reproduce, lowering the growth rate. Therefore, when calculating the growth rate of a population, the [ΔN (change in number) / (ΔT (change in time)] = B (birth rate)  D (death rate) The birth rate is usually expressed on a per capita (for each individual) basis. Thus, B (birth rate) = bN (the per capita birth rate “b” multiplied by the number of individuals “N”) and D (death rate) =dN (the per capita death rate “d” multiplied by the number of individuals “N”). Additionally, ecologists are interested in the population at a particular point in time, an infinitely small time interval. For this reason, the terminology of differential calculus is used to obtain the “instantaneous” growth rate, replacing the change in number and time with an instantspecific measurement of number and time. dN/dT = bN  dN = (b  d)N Notice that the “d” associated with the first term refers to the derivative (as the term is used in calculus) and is different from the death rate, also called “d.” The difference between birth and death rates is further simplified by substituting the term “r” (intrinsic rate of increase) for the relationship between birth and death rates: dN/dT = rN The value “r” can be positive, meaning the population is increasing in size; or negative, meaning the population is decreasing in size; or zero, where the population’s size is unchanging, a condition known as dN/dT = r_{max}N Logistic GrowthExponential growth is possible only when infinite natural resources are available; this is not the case in the real world. Charles Darwin recognized this fact in his description of the “struggle for existence,” which states that individuals will compete (with members of their own or other species) for limited resources. The successful ones will survive to pass on their own characteristics and traits (which we know now are transferred by genes) to the next generation at a greater rate (natural selection). To model the reality of limited resources, population ecologists developed the Carrying Capacity and the Logistic ModelIn the real world, with its limited resources, exponential growth cannot continue indefinitely. Exponential growth may occur in environments where there are few individuals and plentiful resources, but when the number of individuals gets large enough, resources will be depleted, slowing the growth rate. Eventually, the growth rate will plateau or level off (Figure 1). This population size, which represents the maximum population size that a particular environment can support, is called the The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression “K – N” is indicative of how many individuals may be added to a population at a given stage, and “K – N” divided by “K” is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation: dN/dT = r_{max}dN/dT = r_{max}N(K  N)/K Notice that when N is very small, (KN)/K becomes close to K/K or 1, and the right side of the equation reduces to r_{max}N, which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is large, (KN)/K come close to zero, which means that population growth will be slowed greatly or even stopped. Thus, population growth is greatly slowed in large populations by the carrying capacity K. This model also allows for the population of a negative population growth, or a population decline. This occurs when the number of individuals in the population exceeds the carrying capacity (because the value of (KN)/K is negative). A graph of this equation yields an Role of Intraspecific CompetitionThe logistic model assumes that every individual within a population will have equal access to resources and, thus, an equal chance for survival. For plants, the amount of water, sunlight, nutrients, and the space to grow are the important resources, whereas in animals, important resources include food, water, shelter, nesting space, and mates. In the real world, phenotypic variation among individuals within a population means that some individuals will be better adapted to their environment than others. The resulting competition between population members of the same species for resources is termed Examples of Logistic GrowthYeast, a microscopic fungus used to make bread and alcoholic beverages, exhibits the classical Sshaped curve when grown in a test tube (Figure 2a). Its growth levels off as the population depletes the nutrients that are necessary for its growth. In the real world, however, there are variations to this idealized curve. Examples in wild populations include sheep and harbor seals (Figure 2b). In both examples, the population size exceeds the carrying capacity for short periods of time and then falls below the carrying capacity afterwards. This fluctuation in population size continues to occur as the population oscillates around its carrying capacity. Still, even with this oscillation, the logistic model is confirmed. Art ConnectionIf the major food source of the seals declines due to pollution or overfishing, which of the following would likely occur?
Section SummaryPopulations with unlimited resources grow exponentially, with an accelerating growth rate. When resources become limiting, populations follow a logistic growth curve. The population of a species will level off at the carrying capacity of its environment. Art ConnectionsExercise 1Figure 2b. If the major food source of the seals declines due to pollution or overfishing, which of the following would likely occur?
Show/Hide Solution Figure 2b. A Review QuestionsExercise 2Species with limited resources usually exhibit a(n) ________ growth curve.
Show/Hide Solution A Exercise 3The maximum rate of increased characteristic of a species is called its ________.
Show/Hide Solution C Exercise 4The population size of a species capable of being supported by the environment is called its ________.
Show/Hide Solution B Free ResponseExercise 5Describe the rate of population growth that would be expected at various parts of the Sshaped curve of logistic growth. Show/Hide Solution In the first part of the curve, when few individuals of the species are present and resources are plentiful, growth is exponential, similar to a Jshaped curve. Later, growth slows due to the species using up resources. Finally, the population levels off at the carrying capacity of the environment, and it is relatively stable over time. Glossarybiotic potential (r_{max}) birth rate (B) carrying capacity (K) death rate (D) exponential growth intraspecific competition Jshaped growth curve logistic growth population growth rate Sshaped growth curve zero population growth
