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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 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:

ΔN/ΔT=B−DΔN/ΔT=B−D

where ΔNΔN = change in number, ΔTΔT = change in time, BB = birth rate, and DD = 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.

dN/dT=BN DN=(BD)NdN/dT=BN DN=(BD)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=rNdN/dT=rN

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:

dN/dT=rmaxN

## 45.5 Human Population Growth

By the end of this section, you will be able to do the following:

- Discuss exponential human population growth
- Explain how humans have expanded the carrying capacity of their habitat
- Relate population growth and age structure to the level of economic development in different countries
- Discuss the long-term implications of unchecked human population growth

Population dynamics can be applied to human population growth. Earth’s human population is growing rapidly, to the extent that some worry about the ability of the earth’s environment to sustain this population. Long-term exponential growth carries the potential risks of famine, disease, and large-scale death.

Although humans have increased the carrying capacity of their environment, the technologies used to achieve this transformation have caused unprecedented changes to Earth’s environment, altering ecosystems to the point where some may be in danger of collapse. The depletion of the ozone layer, erosion due to acid rain, and damage from global climate change are caused by human activities. The ultimate effect of these changes on our carrying capacity is unknown. As some point out, it is likely that the negative effects of increasing carrying capacity will outweigh the positive ones—the world’s carrying capacity for human beings might actually decrease.

The human population is currently experiencing exponential growth even though human reproduction is far below its biotic potential (Figure 45.14). To reach its biotic potential, all females would have to become pregnant every nine months or so during their reproductive years. Also, resources would have to be such that the environment would support such growth. Neither of these two conditions exists. In spite of this fact, human population is still growing exponentially.

A consequence of exponential human population growth is a reduction in time that it takes to add a particular number of humans to the Earth. Figure 45.15 shows that 123 years were necessary to add 1 billion humans in 1930, but it only took 24 years to add two billion people between 1975 and 1999. As already discussed, our ability to increase our carrying capacity indefinitely my be limited. Without new technological advances, the human growth rate has been predicted to slow in the coming decades. However, the population will still be increasing and the threat of overpopulation remains.

### Link to Learning

Click through this video of how human populations have changed over time.

### Overcoming Density-Dependent Regulation

Humans are unique in their ability to alter their environment with the conscious purpose of increasing carrying capacity. This ability is a major factor responsible for human population growth and a way of overcoming density-dependent growth regulation. Much of this ability is related to human intelligence, society, and communication. Humans can construct shelter to protect them from the elements and have developed agriculture and domesticated animals to increase their food supplies. In addition, humans use language to communicate this technology to new generations, allowing them to improve upon previous accomplishments.

Other factors in human population growth are migration and public health. Humans originated in Africa, but have since migrated to nearly all inhabitable land on the Earth. Public health, sanitation, and the use of antibiotics and vaccines have decreased the ability of infectious disease to limit human population growth. In the past, diseases such as the bubonic plaque of the fourteenth century killed between 30 and 60 percent of Europe’s population and reduced the overall world population by as many as 100 million people. Today, the threat of infectious disease, while not gone, is certainly less severe. According to the Institute for Health Metrics and Evaluation (IHME) in Seattle, global death from infectious disease declined from 15.4 million in 1990 to 10.4 million in 2017. To compare to some of the epidemics of the past, the percentage of the world's population killed between 1993 and 2002 decreased from 0.30 percent of the world's population to 0.14 percent. Thus, infectious disease influence on human population growth is becoming less significant.

### Age Structure, Population Growth, and Economic Development

The age structure of a population is an important factor in population dynamics. Age structure is the proportion of a population at different age ranges. Age structure allows better prediction of population growth, plus the ability to associate this growth with the level of economic development in the region. Countries with rapid growth have a pyramidal shape in their age structure diagrams, showing a preponderance of younger individuals, many of whom are of reproductive age or will be soon (Figure 45.16). This pattern is most often observed in underdeveloped countries where individuals do not live to old age because of less-than-optimal living conditions. Age structures of areas with slow growth, including developed countries such as the United States, still have a pyramidal structure, but with many fewer young and reproductive-aged individuals and a greater proportion of older individuals. Other developed countries, such as Italy, have zero population growth. The age structure of these populations is more conical, with an even greater percentage of middle-aged and older individuals. The actual growth rates in different countries are shown in Figure 45.17, with the highest rates tending to be in the less economically developed countries of Africa and Asia.

### Visual Connection

Age structure diagrams for rapidly growing, slow growing, and stable populations are shown in stages 1 through 3. What type of population change do you think stage 4 represents?

### Long-Term Consequences of Exponential Human Population Growth

Many dire predictions have been made about the world’s population leading to a major crisis called the “population explosion.” In the 1968 book *The Population Bomb*, biologist Dr. Paul R. Ehrlich wrote, “The battle to feed all of humanity is over. In the 1970s hundreds of millions of people will starve to death in spite of any crash programs embarked upon now. At this late date nothing can prevent a substantial increase in the world death rate.” 8 While many experts view this statement as incorrect based on evidence, the laws of exponential population growth are still in effect, and unchecked human population growth cannot continue indefinitely.

Several nations have instituted policies aimed at influencing population. Efforts to control population growth led to the one-child policy in China, which is now being phased out. India also implements national and regional populations to encourage family planning. On the other hand, Japan, Spain, Russia, Iran, and other countries have made efforts to increase population growth after birth rates dipped. Such policies are controversial, and the human population continues to grow. At some point the food supply may run out, but the outcomes are difficult to predict. The United Nations estimates that future world population growth may vary from 6 billion (a decrease) to 16 billion people by the year 2100.

Another result of population growth is the endangerment of the natural environment. Many countries have attempted to reduce the human impact on climate change by reducing their emission of the greenhouse gas carbon dioxide. However, these treaties have not been ratified by every country. The role of human activity in causing climate change has become a hotly debated socio-political issue in some countries, including the United States. Thus, we enter the future with considerable uncertainty about our ability to curb human population growth and protect our environment.

### Link to Learning

Visit this website and select “Launch movie” for an animation discussing the global impacts of human population growth.

## Biological exponential growth

**Biological exponential growth** is the exponential growth of biological organisms. When the resources availability is unlimited in the habitat, the population of an organism living in the habitat grows in an exponential or geometric fashion. Population growth in which the number of individuals increase by a constant multiple in each generation. The potential for population growth can be demonstrated in the laboratory under conditions that provide abundant resources and space. For example, a few fruit flies in a large culture jar containing an abundant food source may reproduce rapidly. One female fruit fly may lay more than 50 eggs. Reproductive adults develop in about 14 days, with approximately equal numbers of male and female offspring. For each female that began the population, 50 flies are expected 2 weeks later. Each female in the second generation produces 50 more flies after 2 more weeks, and so on. In other words, the population is experiencing exponential growth. [1] Slow exponential growth is when a population grows slowly yet exponential because the population has long live spans. While a rapid exponential growth refers to a population that grows ( and dies ) rapidly because the population has short life spans.

Resource availability is obviously essential for the unimpeded growth of a population. Ideally, when resources in the habitat are unlimited, each species has the ability to realise fully its innate potential to grow in number, as Charles Darwin observed while developing his theory of natural selection.

If, in a hypothetical population of size *N*, the birth rates (per capita) are represented as *b* and death rates (per capita) as *d*, then the increase or decrease in *N* during a time period *t* will be:

(b-d) is called the 'intrinsic rate of natural increase' and is a very important parameter chosen for assessing the impacts of any biotic or abiotic factor on population growth.

Any species growing exponentially under unlimited resource conditions can reach enormous population densities in a short time. Darwin showed how even a slow growing animal like the elephant could reach an enormous population if there were unlimited resources for its growth in its habitat.

## Population size is regulated by factors that are dependent or independent of population density

Biological and non-biological factors can influence population size. Biological factors include interspecific interactions like predation, competition, parasitism, and mutualism, as well as disease. Non-biological factors are environmental variables like temperature, precipitation, disturbance, pollution, salinity, and pH. All of these factors can *change* population size, but only the biological factors (except mutualism) can “*regulate*” a population, meaning they push the population to an equilibrium density, or carrying capacity. Of the biological factors, mutualism does not regulate population size because mutualisms promote population increase through beneficial interactions with another species.

The biological factors of competition, predation, etc. that regulate population growth affect dense versus sparse populations differently. For instance, communicable disease does not spread quickly in a sparsely packed population, but in a dense population, like humans living in a college residence hall, disease can spread quickly through contact between individuals. Density plays a key role in population regulation in the following ways:

- Territoriality: Maintaining a territory will enable an individual to capture enough food to reproduce, where space is a limiting resource.
- Disease: Transmission rate often depends on population density
- Predation: Predators may concentrate on the most abundant prey
- Toxic Wastes: Metabolic by-products accumulate as populations grow

Identifying evidence of density regulation requires a field or lab experiment that manipulates density and quantifies the response in population growth. Often an (easier to measure) proxy of population growth, like survival or reproductive output, stands in as a quick metric of the births and deaths that will impact population growth. The characteristic negative correlation in the image below is evidence of density-dependent population regulation: higher densities yield lower survival.

Van Burkirk and Smith (1991) manipulated salamander (*Ambystoma laterale)* larval density in the field and found lower salamander survival in the high density treatments on North Government Island, Isle Royale, Michigan. (Image after van Burkirk and Smith. 1991. Ecology 72(5): 1747-1756.)

Here’s Hank Green’s take on Population Growth to help you review these ideas:

## Exponential Model of Population Growth

The revenue of Red Rocks, Inc., in millions of dollars, is given by the function $R(t)=frac<4000><1+1999 e^<-0.5 t>>$ where $t$ is measured in years.

a) What is $R(0),$ and what does it represent?

b) Find $lim _

it means.

c) Find the value of $t$ (to the nearest integer) for which $R(t)=0.99 R_

Describe the differences in the graphs of an exponential function and a logistic function.

Complete the table below, which relates growth rate $k$ and doubling time $T$

(TABLE CANNOT COPY)

Graph $T=(ln 2) / k .$ Is this a linear relationship? Explain.

Suppose that $$ 100$ is invested at $7 \%,$ compounded continuously, for 1 yr. We know from Example 4 that the ending balance will be $$ 107.25 .$ This would also be the ending balance if $$ 100$ were invested at 7.25 $\%,$ compounded once a year (simple interest). The rate of

$7.25 \%$ is called the effective annual yield. In general, if $P_<0>$ is invested at interest rate $k,$ compounded continuously, then the effective annual yield is that number i satisfying $P_<0>(1+i)=P_ <0>e^

The effective annual yield on an investment compounded continuously is $6.61 \% .$ At what rate was it invested?

Suppose that $$ 100$ is invested at $7 \%,$ compounded continuously, for 1 yr. We know from Example 4 that the ending balance will be $$ 107.25 .$ This would also be the ending balance if $$ 100$ were invested at 7.25 $\%,$ compounded once a year (simple interest). The rate of

$7.25 \%$ is called the effective annual yield. In general, if $P_<0>$ is invested at interest rate $k,$ compounded continuously, then the effective annual yield is that number i satisfying $P_<0>(1+i)=P_ <0>e^

The effective annual yield on an investment compounded continuously is $9.42 \% .$ At what rate was it invested?

Suppose that $$ 100$ is invested at $7 \%,$ compounded continuously, for 1 yr. We know from Example 4 that the ending balance will be $$ 107.25 .$ This would also be the ending balance if $$ 100$ were invested at 7.25 $\%,$ compounded once a year (simple interest). The rate of

$7.25 \%$ is called the effective annual yield. In general, if $P_<0>$ is invested at interest rate $k,$ compounded continuously, then the effective annual yield is that number i satisfying $P_<0>(1+i)=P_ <0>e^

An amount is invested at $8 \%$ per year compounded continuously. What is the effective annual yield?

Suppose that $$ 100$ is invested at $7 \%,$ compounded continuously, for 1 yr. We know from Example 4 that the ending balance will be $$ 107.25 .$ This would also be the ending balance if $$ 100$ were invested at 7.25 $\%,$ compounded once a year (simple interest). The rate of

$7.25 \%$ is called the effective annual yield. In general, if $P_<0>$ is invested at interest rate $k,$ compounded continuously, then the effective annual yield is that number i satisfying $P_<0>(1+i)=P_ <0>e^

An amount is invested at $7.3 \%$ per year compounded continuously. What is the effective annual yield?

To what exponential growth rate per hour does a growth rate of $100 \%$ per day correspond?

A quantity $Q_<1>$ grows exponentially with a doubling time of 1 yr. A quantity $Q_<2>$ grows exponentially with a doubling time of 2 yr. If the initial amounts of $Q_<1>$ and $Q_<2>$ are the same, how long will it take for $Q_<1>$ to be twice the size of $Q_ <2>?$

Find an expression relating the exponential growth rate $k$ and the tripling time $T_<3>$.

Find an expression relating the exponential growth rate $k$ and the quadrupling time $T_<4>$

We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. Identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function.

The life expectancy of the average American