## Exponential Population Growth

When resources are unlimited, a population can experience exponential growth, where its size increases at a greater and greater rate.

### Learning Objectives

Describe exponential growth of a population size

### Key Takeaways

#### Key Points

- To get an accurate growth rate of a population, the number that died in the time period (death rate) must be removed from the number born during the same time period (birth rate).
- When the birth rate and death rate are expressed in a per capita manner, they must be multiplied by the population to determine the number of births and deaths.
- Ecologists are usually interested in the changes in a population at either a particular point in time or over a small time interval.
- The intrinsic rate of increase is the difference between birth and death rates; it can be positive, indicating a growing population; negative, indicating a shrinking population; or zero, indicting no change in the population.
- Different species have a different intrinsic rate of increase which, when under ideal conditions, represents the biotic potential or maximal growth rate for a species.

#### Key Terms

**fission**: the process by which a bacterium splits to form two daughter cells**per capita**: per person or individual

### Exponential growth

In his theory of natural selection, Charles Darwin 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, after which 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 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 will be one round of division (with each organism dividing once), resulting in 2000 organisms. In another hour, each of the 2000 organisms will double, producing 4000; after the third hour, there should be 8000 bacteria in the flask; and so on. 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; the number organisms that die during a particular time interval) is subtracted from the birth rate (B; the number organisms that are born during that interval). This is shown in the following formula:

[latex]\Delta \text{N} / \Delta \text{T} = \text{B} - \text{D}[/latex]

where [latex]\Delta \text{N}[/latex] = change in number, [latex]\Delta \text{T}[/latex] = change in time, [latex]\text{B}[/latex] = birth rate, and [latex]\text{D}[/latex] = 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.

[latex]\text{dN} / \text{dT} = \text{BN DN} = (\text{B} \text{D})\text{N}[/latex]

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:

[latex]\text{dN} / \text{dT} = \text{rN}[/latex]

The value “r” can be positive, meaning the population is increasing in size; negative, meaning the population is decreasing in size; or zero, where the population’s size is unchanging, a condition 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 r_{max}, thus changing the equation to:

[latex]\text{dN} / \text{dT}=\text{r}_{\text{max}}\text{N}[/latex]

## Logistic Population Growth

Logistic growth of a population size occurs when resources are limited, thereby setting a maximum number an environment can support.

### Learning Objectives

Describe logistic growth of a population size

### Key Takeaways

#### Key Points

- The carrying capacity of a particular environment is the maximum population size that it can support.
- The carrying capacity acts as a moderating force in the growth rate by slowing it when resources become limited and stopping growth once it has been reached.
- As population size increases and resources become more limited, intraspecific competition occurs: individuals within a population who are more or less better adapted for the environment compete for survival.

#### Key Terms

**phenotype**: the appearance of an organism based on a multifactorial combination of genetic traits and environmental factors, especially used in pedigrees**carrying capacity**: the number of individuals of a particular species that an environment can support; indicated by the letter “K”

### 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: a process known as 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 becomes large enough, resources will be depleted, slowing the growth rate. Eventually, the growth rate will plateau or level off. 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:

[latex]\text{dN} / \text{dT} = \text{rmax} * (\text{dN} / \text{dT}) = \text{rmax} * \text{N} * ((\text{K N}) / \text{K})[/latex]

Notice that when N is very small, (K-N)/K becomes close to K/K or 1; 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, (K-N)/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 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; 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 the important resources, whereas in animals, important resources include food, water, shelter, nesting space, and mates.

In the real world, the variation of phenotypes 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 as 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

Yeast, a microscopic fungus used to make bread and alcoholic beverages, exhibits the classical S-shaped curve when grown in a test tube ( a). 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 ( b). 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.

## Density-Dependent and Density-Independent Population Regulation

Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population.

### Learning Objectives

Differentiate between density-dependent and density-independent population regulation.

### Key Takeaways

#### Key Points

- The density of a population can be regulated by various factors, including biotic and abiotic factors and population size.
- Density-dependent regulation can be affected by factors that affect birth and death rates such as competition and predation.
- Density-independent regulation can be affected by factors that affect birth and death rates such as abiotic factors and environmental factors, i.e. severe weather and conditions such as fire.
- New models of life history incorporate ecological concepts that are typically included in r- and K-selection theory in combination with population age structures and mortality factors.

#### Key Terms

**interspecific**: existing or occurring between different species**intraspecific**: occurring among members of the same species**fecundity**: number, rate, or capacity of offspring production

### Density-dependent regulation

In population ecology, density-dependent processes occur when population growth rates are regulated by the density of a population. Most density-dependent factors, which are biological in nature (biotic), include predation, inter- and intraspecific competition, accumulation of waste, and diseases such as those caused by parasites. Usually, the denser a population is, the greater its mortality rate. For example, during intra- and interspecific competition, the reproductive rates of the individuals will usually be lower, reducing their population’s rate of growth. In addition, low prey density increases the mortality of its predator because it has more difficulty locating its food source.

An example of density-dependent regulation is shown with results from a study focusing on the giant intestinal roundworm (*Ascaris lumbricoides*), a parasite of humans and other mammals. The data shows that denser populations of the parasite exhibit lower fecundity: they contained fewer eggs. One possible explanation for this phenomenon was that females would be smaller in more dense populations due to limited resources so they would have fewer eggs. This hypothesis was tested and disproved in a 2009 study which showed that female weight had no influence. The actual cause of the density-dependence of fecundity in this organism is still unclear and awaiting further investigation.

### Density-independent regulation and interaction with density-dependent factors

Many factors, typically physical or chemical in nature (abiotic), influence the mortality of a population regardless of its density. They include weather, natural disasters, and pollution. An individual deer may be killed in a forest fire regardless of how many deer happen to be in that area. Its chances of survival are the same whether the population density is high or low.

In real-life situations, population regulation is very complicated and density-dependent and independent factors can interact. A dense population that is reduced in a density-independent manner by some environmental factor(s) will be able to recover differently than would a sparse population. For example, a population of deer affected by a harsh winter will recover faster if there are more deer remaining to reproduce.