Learning Goals
By the end of this reading you should be able to:
- Describe the conditions in which exponential growth might occur
- Calculate the intrinsic and maximum growth rate of a population
- Explain the relationship between logistic growth and carrying capacity
- Describe how intraspecific competition can impact population growth
- Predict the effect of changing environmental conditions on population growth
Introduction
Life histories describe the way many characteristics of a population (such as their age structure) change over time in a general way. However, there are other factors, including environmental conditions that impact the rate that a population can grow. When resources are unlimited (or in many cases just readily available) populations can often grow at very fast rates and in an exponential manner. When resources becoming limiting the rate of growth usually begins to slow and in some cases the population may actually begin to decline. Scientists have developed some models that help to predict the growth of populations based on the life histories of organisms and the environmental conditions. However, these models are only predictors and newer more complex models are constantly being built to take into account new or changing factors.
Exponential Growth
Charles 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 exponential growth.
The best example of exponential growth is seen in bacteria. Bacteria are prokaryotes that reproduce by binary fission. In some species of bacteria the time between bouts of binary fission can be as short as 60 minutes, and in some species even as quickly as 20 minutes. Consider this, if 1000 bacteria can divide every 60 minutes are placed in a large flask with an unlimited supply of nutrients (so the nutrients will not become depleted), after an hour, assuming each individual divides, there will be 2000 organisms—an increase of 1000. In another hour, each of the 2000 organisms will divide, 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 population growth rate—the number of organisms added in each reproductive generation—is accelerating; that is, it is increasing at a greater and greater rate. After 1 day and 24 of these cycles, the population would have increased from 1000 to more than 16 billion. When the population size, N, is plotted over time, a J-shaped growth curve is produced.
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 death rate (D) (number of organisms that die during a particular time interval) is subtracted from the birth rate (B) (number of organisms that are born during that interval). This is shown in the following formula:
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 instant-specific measurement of number and time.
dNdT = 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:
dNdT = 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 is known as zero population growth. A further refinement of the formula recognizes that different species have inherent differences in their intrinsic rate of increase (often thought of as the potential for reproduction), even under ideal conditions. Obviously, a bacterium can reproduce more rapidly and have a higher intrinsic rate of growth than a human. The maximal growth rate for a species is its biotic potential, or rmax, thus changing the equation to:
dNdT=rmaxN
Review Question:
r=-345 _______
Logistic Growth
Exponential 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 logistic growth model.
Carrying Capacity and the Logistic Model
In 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 (Fig. 2a). This population size, which represents the maximum population size that a particular environment can support, is called the carrying capacity, or K.
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:
dNdT=rmaxdNdT=rmaxN(K − N)K
Notice that when N is very small, (K-N)/K becomes close to K/K or 1, and the right side of the equation reduces to rmaxN, which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is large, (K-N)/K comes 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 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 (K-N)/K is negative).
A graph of this equation yields an S-shaped curve (Fig. 2b), and it is a more realistic model of population growth than exponential growth. There are three different sections to an S-shaped curve. Initially, growth is exponential because there are few individuals and ample resources available. Then, as resources begin to become limited, the growth rate decreases. Finally, growth levels off at the carrying capacity of the environment, with little change in population size over time.
Role of Intraspecific Competition
The 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 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 intraspecific competition (intra- = “within”; -specific = “species”). Intraspecific competition for resources may not affect populations that are well below their carrying capacity—resources are plentiful and all individuals can obtain what they need. However, as population size increases, this competition intensifies. In addition, the accumulation of waste products can reduce an environment’s carrying capacity.
Examples of Logistic Growth
Review Question:
Summary
Populations 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.
End of Section Review Questions:
REVIEW: Rates of growth
REVIEW: Limits to growth
decreasing
increasing
Not changing
A
C
B
