Population Dynamics
Population Dynamics
A population describes a group of individuals of the same species occupying a specific area at a specific time. Some characteristics of populations that are of interest to biologists include the population density , the birthrate , and the death rate . If there is immigration into the population, or emigration out of it, then the immigration rate and emigration rate are also of interest. Together, these population parameters, or characteristics, describe how the population density changes over time. The ways in which population densities fluctuate—increasing, decreasing, or both over time—is the subject of population dynamics.
Population density measures the number of individuals per unit area, for example, the number of deer per square kilometer. Although this is straightforward in theory, determining population densities for many species can be challenging in practice.
Measuring Population Density
One way to measure population density is simply to count all the individuals. This, however, can be laborious. Alternatively, good estimates of population density can often be obtained via the quadrat method. In the quadrat method, all the individuals of a given species are counted in some subplot of the total area. Then that data is used to figure out what the total number of individuals across the entire habitat should be.
The quadrat method is particularly suited to measuring the population densities of species that are fairly uniformly distributed over the habitat. For example, it has been used to determine the population density of soil species such as nematode worms. It is also commonly used to measure the population density of plants.
For more mobile organisms, the capture-recapture method may be used. With this technique, a number of individuals are captured, marked, and released. After some time has passed, enough time to allow for the mixing of the population, a second set of individuals is captured. The total population size may be estimated by looking at the proportion of individuals in the second capture set that are marked. Obviously, this method works only if one can expect individuals in the population to move around a lot and to mix. It would not work, for example, in territorial species, where individuals tend to remain near their territories.
The birthrate of a population describes the number of new individuals produced in that population per unit time. The death rate, also called mortality rate, describes the number of individuals who die in a population per unit time. The immigration rate is the number of individuals who move into a population from a different area per unit time. The emigration rates describe the numbers of individuals who migrate out of the population per unit time.
The values of these four population parameters allow us to determine whether a population will increase or decrease in size. The "intrinsic rate of increase r " of a population is defined as r = (birth rate immigration rate +)-(death rate + emigration rate ).
If r is positive, then more individuals will be added to the population than lost from it. Consequently, the population will increase in size. If r is negative, more individuals will be lost from the population than are being added to it, so the population will decrease in size. If r is exactly zero, then the population size is stable and does not change. A population whose density is not changing is said to be at equilibrium .
Population Models
We will now examine a series of population models, each of which is applicable to different environmental circumstances. We will also consider how closely population data from laboratory experiments and from studies of natural populations in the wild fit these models.
Exponential growth.
The first and most basic model of population dynamics assumes that an environment has unlimited resources and can support an unlimited number of individuals. Although this assumption is clearly unrealistic in many circumstances, there are situations in which resources are in fact plentiful enough so that this model is applicable. Under these circumstances, the rate of growth of the population is constant and equal to the intrinsic rate of increase r. This is also known as exponential growth .
What happens to population size over time under exponential growth? If r is negative, the population declines quickly to extinction. However, if r is positive, the population increases in size, slowly at first and then ever more quickly. Exponential growth is also known as "J-shaped growth" because the shape of the curve of population size over time resembles the letter "J." Also, because the rate of growth of the population is constant, and does not depend on population density, exponential growth is also called "density-independent growth." Exponential growth is often seen in small populations, which are likely to experience abundant resources. J-shaped growth is not sustainable however, and a population crash is ultimately inevitable.
There are numerous species that do in fact go through cycles of exponential growth followed by population crashes. A classic example of exponential growth resulted from the introduction of reindeer on the small island of Saint Paul, off the coast of Alaska. This reindeer population increased from an initial twenty-five individuals to a staggering two thousand individuals in twenty-seven years. However, after exhausting their food supply of lichens, the population crashed to only eight. A similar pattern was seen following the introduction of reindeer on Saint Matthew Island, also off the Alaskan coast, some years later. Over the course of history, human population growth has also been J-shaped.
Logistic growth.
A different model of population increase is called logistic growth . Logistic growth is also called "S-shaped growth" because the curve describing population density over time is S-shaped. In S-shaped growth, the rate of growth of a population depends on the population's density. When the population size is small, the rate of growth is high. As population density increases, however, the rate of growth slows. Finally, when the population density reaches a certain point, the population stops growing and starts to decrease in size. Because the rate of growth of the population depends on the density of the population, logistic growth is also described as "density-dependent growth".
Under logistic growth, an examination of population size over time shows that, like J-shaped growth, population size increases slowly at first, then more quickly. Unlike exponential growth, however, this increase does not continue. Instead, growth slows and the population comes to a stable equilibrium at a fixed, maximum population density. This fixed maximum is called the carrying capacity , and represents the maximum number of individuals that can be supported by the resources available in the given habitat. Carrying capacity is denoted by the variable K.
The fact that the carrying capacity represents a stable equilibrium for a population means that if individuals are added to a population above and beyond the carrying capacity, population size will decrease until it returns to K. On the other hand, if a population is smaller than the carrying capacity, it will increase in size until it reaches that carrying capacity. Note, however, that the carrying capacity may change over time. K depends on a wealth of factors, including both abiotic conditions and the impact of other biological organisms.
Logistic growth provides an accurate picture of the population dynamics of many species. It has been produced in laboratory situations in single-celled organisms and in fruit flies, often when populations are maintained in a limited space under constant environmental conditions.
Perhaps surprisingly, however, there are fewer examples of logistic growth in natural populations. This may be because the model assumes that the reaction of population growth to population density (that is, that population growth slows with greater and greater population densities, and that populations actually decrease in size when density is above the carrying capacity) is instantaneous. In actuality, there is almost always a time lag before the effects of high population density are felt. The time lag may also explain why it is easier to obtain logistic growth patterns in the laboratory, since most of the species used in laboratory experiments have fairly simple life cycles in which reproduction is comparatively rapid.
Biological species are sometimes placed on a continuum between r -selected and k -selected, depending on whether their population dynamics tend to correspond more to exponential or logistic growth. In r -selected species, there tend to be dramatic fluctuations, including periods of exponential growth followed by population crashes. These species are particularly suited to taking advantage of brief periods of great resource abundance, and are specialized for rapid growth and reproduction along with good capabilities for dispersing.
In k -selected species, population density is more stable, often because these species occupy fairly stable habitats. Because k -selected species exist at densities close to the carrying capacity of the environment, there is tremendous competition between individuals of the same species for limited resources. Consequently, k -selected individuals often have traits that maximize their competitive ability. Numerous biological traits are correlated to these two life history strategies .
Lotka-Volterra models.
Up to now we have been focusing on the population dynamics of a single species in isolation. The roles of competing species, potential prey items, and potential predators are included in the logistic model of growth only in that they affect the carrying capacity of the environment. However, it is also possible directly to consider between-species interactions in population dynamics models. Two that have been studied extensively are the Lotka-Volterra models, one for competition between two species and the other for interactions between predators and prey.
Competition describes a situation in which populations of two species utilize a resource that is in short supply. The Lotka-Volterra models of the population dynamics of competition show that there are two possible results: either the two competing species are able to coexist , or one species drives the other to extinction. These models have been tested thoroughly in the laboratory, often with competing yeasts or grain beetles.
Many examples of competitive elimination were observed in lab experiments. A species that survived fine in isolation would decline and then go extinct when another species was introduced into the same environment. Coexistence between two species was also produced in the laboratory. Interestingly, these experiments showed that the outcome of competition experiments depended greatly on the precise environmental circumstances provided. Slight changes in the environment—for example, in temperature—often affected the outcome in competitions between yeasts.
Studies in natural populations have shown that competition is fairly common. For example, the removal of one species often causes the abundance of species that share the same resources to increase. Another important result that has been derived from the Lotka-Volterra competition equations is that two species can never share the same niche . If they use resources in exactly the same way, one will inevitably drive the other to extinction. This is called the competitive exclusion principle . The Lotka-Volterra models for the dynamics of interacting predator and prey populations yields four possible results. First, predator and prey populations may both reach stable equilibrium points. Second, predators and prey may each have never-ending, oscillating (alternating) cycles of increase and decrease. Third, the predator species can go extinct, leaving the prey species to achieve a stable population density equal to its carrying capacity. Fourth, the predator can drive the prey to extinction and then go extinct itself because of starvation.
As with competition dynamics, biologists have tried to produce each of these effects in laboratory settings. One interesting result revealed in these experiments was that with fairly simple, limited environments, the predator would always eliminate the prey, and then starve to death. The persistence of both predator and prey species seemed to be dependent on living in a fairly complex environment, including hiding places for the prey.
In natural populations, studies of predator-prey interactions have involved predator removal experiments. Perhaps surprisingly, it has often proven difficult to demonstrate conclusively that predators limit prey density. This may be because in many predator-prey systems, the predators focus on old, sick, or weak individuals. However, one convincing example of a predator limiting prey density involved the removal of dingoes in parts of Australia. In these areas, the density of kangaroos skyrocketed after the removal of the predators.
Continuing oscillations between predators and prey do not appear to be common in natural populations. However, there is one example of oscillations in the populations of the Canada lynx and its prey species, the snowshoe hare. There are peaks of abundance of both species approximately every ten years.
see also Populations.
Jennifer Yeh
Bibliography
Curtis, Helena. Biology. New York: Worth Publishers, 1989.
Gould, James L., and William T. Keeton, and Carol Grant Gould. Biological Science, 6th ed. New York: W. W. Norton & Co., 1996.
Krebs, Charles J. Ecology: The Experimental Analysis of Distribution and Abundance. New York: Harper Collins College Publishers, 1994.
Murray, Bertram G., Jr. Population Dynamics: Alternative Models. New York: Academic Press, 1979.
Pianka, Eric R. Evolutionary Ecology. New York: Addison Wesley Longman, 2000.
Ricklefs, Robert E., and Gary L. Miller. Ecology, 4th ed. New York: W. H. Freeman, 2000.
Soloman, Maurice E. Population Dynamics. London: Edward Arnold, 1969.
Population Dynamics
POPULATION DYNAMICS
Population dynamics refer to the way in which the size and age structure of populations change over time and the characterization of that change in mathematical terms. This article is a basic introduction to the topic, largely avoiding the mathematics.
Population Growth/Decline
Population growth (or decline) is the net balance of births, deaths, in-migration and out-migration. If growth is assumed to occur on a discrete basis, usually annually, and if the rate of growth is constant, population size at a future time t, P(t), can be related to population size at time 0, P(0), by the geometricequation P(t) = (0)(1 + r)t where is the rate of growth. In practice, demographic events are spread throughout the year so that growth is usually smooth enough to be described by a continuous variable over time. If the rate of growth is constant, P(t) can be related to (0) by the simple exponentialequation P(t) = (0)ert.
A convenient measure of population growth is doubling time, or the time that would be needed for population size to double at the current rate of growth. The doubling time is calculated by solving the above equation for in the case where P() = 2P(0). Hence the doubling time is the natural logarithm of 2 (about 0.693) divided by the rate of growth (equivalent to dividing 69.3 by the percentage rate of growth). For example, if the annual rate of growth is 2 percent, the doubling time is 69.3 ÷ 2 or just under 35 years. The higher the rate of growth, the shorter the doubling time.
Balance Equation
Changes in population size are a function of the three constituent components of population change: birth, death, and migration. Change is expressed in discrete terms by the balance equation:
P(t+1)–) = B, D+1) + I(t, +1)–O+1)
where B(t, +1) is the number of births occurring during the interval t to +1 (usually a year), and D, I, and O are corresponding numbers of deaths, inmigrants, and out-migrants. When age is taken into account, this equation expresses the changes occurring to cohorts (persons of the same age in years at time) alive at time. For example, the number of
children in the cohort aged 2 at time t+1 will be equal to the number aged 1 at t, minus deaths occurring to the cohort during the interval from t to +1, plus net migrants of the same age cohort arriving or departing during this interval (note that B is zero). These demographic accounting equations are used in assessing data quality, in estimating net migration, and, in populations closed to migration, as the basis for demographic estimation using indirect methods.
Age Structure
The age structure of a population is determined by the relative size of past successive cohorts at birth and subsequent changes in size because of death and migration. The numbers of births, deaths, and migrants in the present year (or other time interval) are determined by prevailing fertility, mortality, and migration rates and the existing age structure of the population, which in turn is a result of previous demographic rates and structures. Thus the present age structure is a function of current and past demographic rates–that is, of the population's demographic history. Past changes in demographic rates lead to changes in age structure over time.
Usually, the most significant factor in determining population structure is fertility and, in particular, changes in fertility over time. High-fertility populations are characterized by a population pyramid with a wide base relative to older ages, whereas lowfertility populations have a much narrower base. A rapid change in fertility, whether an increase or a decrease, can lead to dramatic changes in both population structure and growth. An example of such a change is the post–World War II increase and subsequent decrease in fertility in many Western countries, resulting in the period of large birth cohorts known as the "baby boom." A large birth cohort appears as a bulge that "ripples" through the population age structure toward older ages over time. As this cohort passes through the childbearing ages, it produces a secondary large birth cohort (large relative to adjacent cohorts and for given fertility rates), which will itself produce a further large cohort a generation later. The effect–sometimes described as an echo–is progressively dampened as a consequence of the wide age range of childbearing and gradually disappears. Thus, through the initial effect on population structure, an increase in fertility, even if temporary, affects population size and structure for many years into the future.
More permanent changes in fertility and changes in mortality and migration have similar effects. Age-and sex-specific effects caused by war, major epidemics, and large-scale migration can have a marked effect on population structure, with consequent effects on future births and hence on population growth and structure.
Population Momentum and Aging
The effect of age structure on population growth is referred to as population momentum. Even if fertility were to change immediately to the level that would just ensure the replacement of each generation (that is, a net reproduction rate of 1), the population would continue to grow (or decline) for as long as structural effects remain in force. Thus, for populations that experienced fertility declines from high levels to near replacement levels in the latter decades of the twentieth century, the age structures are such that substantial further growth will occur for many decades into the twenty-first century. (Momentum effects can also perpetuate population decline, despite a fertility increase, after a long period of below-replacement fertility. This has not yet been observed.)
A consequence of changing age structure is uneven growth across age groups. This is easily seen in the ripple effect of the baby boom. In many populations, past declines in fertility accompanied by increases in longevity have resulted in the aging of the population. An aging population experiences increases in the size of the older population (for example, persons aged 65 and over) relative to the total. In contrast, population rejuvenation occurs when the younger population (for example, persons aged 0 to 14) increases in relative size, normally because of falling infant and child mortality or increased fertility. Growth in the size of different age groups can be compared across ages and populations. For example, during the period from 1950 to 2000, the population of India aged 0 to 14 years increased by a factor of 2.5 compared with a factor of 3.8 for the population aged 65 and over, while the total population increased by a factor of 2.8. This compares with the more rapidly aging population of Singapore, which has corresponding factors of 2.1 at age 0 to 14,12.0 at age 65 and over, and 3.9 for the total population. By 2050, the populations aged 65 and over in India and Singapore are expected to have increased (since 1950) by factors of 18 and 55 respectively, compared with factors of only 4 or 5 for the total populations, illustrating both the magnitude of growth at this age and the wide variation between populations. The disparity is caused by the different speeds at which fertility declined and mortality improved in the two populations: rapidly in Singapore and only moderately in India.
See also: Aging of Population; Age Structure and Dependency; Baby Boom; Estimation Methods, Demographic; Fertility Measurement; Momentum of Population Growth; Mortality Measurement; Renewal Theory and the Stable Population Model.
bibliography
Chesnais, Jean-Claude. 1990. "Demographic Transition Patterns and their Impact on the Age Structure." Population and Development Review 16:327–336.
Preston, Samuel H., Patrick Heuveline, and Michel Guillot. 2001. Demography: Measuring and Modeling Population Processes. Oxford: Blackwell.
Shryock, Henry S., and Jacob S. Siegal. 1973. The Methods and Materials of Demography. Washington, D.C.: U.S. Government Printing Office.
Heather Booth
Population Dynamics
Population Dynamics
A population is a collection of individual organisms of the same species that occupy some specific area. The term "population dynamics" refers to how the number of individuals in a population changes over time. Biologists study the factors that affect population dynamics because they are interested in topics such as conservation of endangered species (for example, the Florida panther) and management of fish and wildlife. In addition, basic knowledge about the processes that affect population dynamics can be used to predict future patterns of human population growth.
How Do Biologists Characterize Populations?
Biologists distinguish between two main types of populations: unstructured and structured. In an unstructured population, all individuals are subject to the same general ecological pressures. That is, the rates of growth, reproduction, and mortality are roughly the same for all individuals in the population. A bacterial colony is a good example of an unstructured population. Conversely, in structured populations, individuals can differ from one another in ways that make some individuals more susceptible to mortality or more likely to reproduce than others. Examples of structured populations include many insects, sea turtles, trees, and fish. In these cases, mortality is often much higher for younger (and/or smaller) individuals. In addition, reproduction is often delayed until individuals are older (and/or larger).
How Does Resource Abundance Affect Population Dynamics?
The abundance of environmental resources such as food, water, and space determines how population abundance changes over time. In the presence of unlimited resources, populations grow exponentially. If one plots the number of individuals in an exponentially growing population over time, one finds a J-shaped curve where the slope gets ever steeper. This curve is described by the following equation:
N t = N 0ert
Where N 0 is the initial number of individuals, N t is the number of individuals at a future time, r is the rate of increase, t is time, and e is the base of the natural logarithm (roughly 2.718). The rate of increase (r ) is determined by the difference between birth and death rates of the population. In 1999 the U.S. Bureau of the Census estimated the rate of population increase (r ) for the world human population to be 0.0129 (or 1.29 percent) per year. Few natural populations grow at exponential rates for extended periods of time because resources typically become limiting when population abundance is very high.
In an environment where resources become limited, populations exhibit a pattern of growth called logistic growth. In this case, if one plots the number of individuals in the population over time, one finds a sigmoidal, or S-shaped curve. When population abundance is low, the population grows exponentially. However, as population size increases, resources become limited, the population growth rate slows, and the population abundance curve flattens. The number of individuals present in the population when the growth rate slows to zero is referred to as K, the carrying capacity. The carrying capacity is the theoretical maximum number of individuals that the environment can support. Although estimates of K for humans are controversial, most are around 12 billion.
Using concepts from basic population biology, biologists have distinguished two strategies for population growth. Some species have characteristics that allow them to grow rapidly when an environment with abundant resources is newly created (for example, a new clearing in a forest). These species are referred to as r -selected species and typically reproduce at a young age and produce many offspring. Other species, called K -selected species, have characteristics that make them well suited for life in environments where there is intense competition for limited resources. These species are often strong competitors, reproduce later in life, and produce fewer offspring than r -selected species.
How Does Variability in Environmental Conditions Affect Population Dynamics?
A key assumption of the logistic population growth model for environments where resources are limiting is that environmental conditions are constant. In nature, environmental conditions may vary substantially over time. In such variable environments, the abundance of individuals in a population may also fluctuate over time. Some populations cycle in a predictable manner. Populations that fluctuate widely or have low abundance are especially vulnerable to extinction, an event in which population abundance declines to zero. Extinctions may be local (a population in a particular area is lost) or global (all populations of a species decline to zero and there are no living individuals of the species left on the planet). For example, the passenger pigeon, which was once one of the most numerous birds on Earth, went globally extinct in 1914 due to overhunting and habitat loss.
How Do Physical and Biological Factors Regulate Population Dynamics?
Patterns of population abundance are affected by a variety of biological and physical factors. For example, the abundance of a given species (for example, snails) might be controlled by the abundance of organisms that have a negative effect on the species of interest, such as competitors, predators, and diseases. Similarly, population abundance could be limited by the abundance of organisms that benefit the species of interest (for example, algae consumed by the snails).
In fact, some organisms require the presence of other species called symbionts with whom they live in direct contact. For example, corals use food molecules synthesized by symbiotic zooxanthellae (a type of algae), and zooxanthellae receive nutrients and protection from corals. However, not all populations are regulated by biological factors involving interactions with other species. Physical factors like water availability and temperature can control population abundance of some species.
Which type of factor (biological or physical) has a stronger effect on population dynamics? As one might suspect, the answer depends largely on the population that is studied. Some populations are regulated mostly by biological factors, others are controlled by physical factors, and most populations are affected by both biological and physical factors.
see also Algae; Coral Reef; Extinction; Human Population; Theoretical Ecology
Janet M. Fischer
Bibliography
Cohen, Joel E. "Population Growth and Earth's Human Carrying Capacity." Science 269, no. 5222 (1995): 341–346.
Ehrlich, Paul R. The Population Bomb. San Francisco, CA: Sierra Club, 1969.
Molles, Manuel C. Ecology: Concepts and Applications. Boston: McGraw-Hill, 1999.
U.S. Bureau of the Census, International Data Base. <http://www.census.gov/pub/ipc/www/worldpop.html>.