Population Sampling
CHAPTER 9
Smith, T. M., & Smith, R. L. (2015). Elements of Ecology (9th ed.). Boston, MA: Pearson.
9.1 Population Growth Reflects the Difference between Rates of Birth and Death
Suppose we were to undertake an experiment in which we monitor a population of an organism that has a very simple life cycle, such as a population of freshwater hydra (Figure 9.2) growing in an aquarium in the laboratory. Most reproduction is asexual, involving a process termed budding, in which a new hydra develops as a bud from the parent (see Figure 9.2). If we assume only asexual reproduction, then all individuals are capable of reproduction and produce a single offspring at a time.
We define the population size at a given time (t) as N(t), where N represents the number of individuals. Let us assume that the initial population is small, N(0) = 100 (where 0 refers to time zero at the start of the experiment), so that the food supply within the aquarium is much more than is needed to support the current population. How will the population change over time?
Because no emigration or immigration is allowed by the lab setting, the population is closed. The number of hydra will increase as a result of new “births.” Additionally, the population will decrease as a result of some hydra dying. Because the processes of birth and death in this population are continuous—no defined period of synchronized birth or death—we can observe the number of births that occur over some appropriate time interval. Given the rates of budding (reproduction) for freshwater hydra, an appropriate time unit (t) would be one day. We can define the number of new hydra produced by budding over the period of one day as B and the number of hydra dying over the same day as D. In our hypothetical experiment, let us assume that the initial population produced 40 new individuals (births) over the first day (B = 40) and that 10 of the original 100 hydra died (D = 10). The population size at the end of day 1, N(1), can then be calculated from the initial population size, N(0), and the observed numbers of births (B) and deaths (D):
N(0) + B − D = N(1) or 100 + 40 − 10 = 130N(0) + B − D = N(1) or 100 + 40 − 10 = 130
But what if we now want to predict what the population will be the following day, N(2)? How could we use the measures of B and D to determine the number of births and deaths that will occur in our population that is now composed of 130 hydra? Although B and D represent the measure of birth and death in the population, the actual values are dependent on the initial population size, N(0) = 100. For example, if the initial population size was 200 rather than 100, we could assume that the values of B and D would be twice as large. If we wish to calculate an estimate of birthrate that is independent of the initial population size we need to divide the number of hydra born during the day by the initial population size: B/N(0) or 40/100. We can now define the resulting value 0.4 as b, which is the per capita birthrate (per capita meaning per individual). The per capita birthrate is the average number of births per individual during the time period t (one day). Likewise, we can calculate the per capita death rate as D/N(0) of 10/100 = 0.1. The advantage of expressing the observed values of birth (B) and death (D) for the population as per capita rates (b and d) is that if we assume they are constant (do not change over time), we can use b and d to predict the growth of the population over time regardless of the population size N(t).
If we start with N(t) hydra at time t, then to calculate the total number of hydra reproducing over the following day (t + 1), we must simply multiply the per capita birthrate (b) by the total number of hydra at time t [N(t)], which is bN(t). The number of hydra dying over the time interval is calculated in a similar manner: dN(t).
The population size at the next time period (t + 1) would then be
N(t + 1) = N(t) + bN(t) − dN(t)N(t + 1) = N(t) + bN(t) − dN(t)
Applied to the hydra population:
N(0) = 100N(1) = 100 + 0.4(100) − 0.1(100) = 130N(2) = 130 + 0.4(130) − 0.1(130) = 169N(0) = 100N(1) = 100 + 0.4(100) − 0.1(100) = 130N(2) = 130 + 0.4(130) − 0.1(130) = 169
The resulting pattern of population size as a function of time is shown in Figure 9.3 and is referred to as geometric population growth.
We can calculate the rate of change in the population (the population growth rate) by subtracting N(t) from both sides of the preceding equation:
N(t + 1) − N(t) = bN(t) − dN(t)N(t + 1) − N(t) = bN(t) − dN(t)
or
N(t + 1) − N(t) = (b − d)N(t)N(t + 1) − N(t) = (b − d)N(t)
Applied to the hydra population over the first two days:
N(1) − N(0) = 0.4(100) − 0.1(100) = 30N(2) − N(1) = 0.4(130) − 0.1(130) = 39N(1) − N(0) = 0.4(100) − 0.1(100) = 30N(2) − N(1) = 0.4(130) − 0.1(130) = 39
The term on the left side of the equation [N(t + 1) − N(t)] is the change in the population size N over the time interval [(t + 1) − t]. If we represent the change in population size as ΔN and the change in time (the time interval) as Δt—the mathematical symbol Δ refers to a “change” in the associated variable—we can rewrite the equation for the rate of population change in a simplified form:
ΔNΔt= (b − d)N(t)ΔNΔt= (b − d)N(t)
Because per capita birthrates and death rates, b and d, are constants (fixed values), we can simplify the equation even further by defining a new parameter r = (b – d). The value r is the per capita growth rate.
ΔNΔt= rN(t)ΔNΔt= rN(t)
Thus, the population growth rate (ΔN/Δt) defines the unit change in population size per unit change in time, or the slope of the relationship between N(t) and t (the “rise” over the “run”) presented in Figure 9.3. Note that because the pattern of population growth is an upward sloping curve, the rate of population change depends on the time interval being viewed (see Figure 9.3 and preceding calculations). With a population that is growing geometrically, the rate of population growth is continuously increasing as the population size increases.
It is important to remember that the model of geometric growth that we have developed for the hydra population can only predict changes in population size on discrete time intervals of one day. This is because the estimates of b and d (and therefore, r) were estimated using observations of birth and death over a one-day period. For populations, such as the hydra, where birth and death are occurring continuously (not daily intervals), population ecologists often represent the processes of birth and death as instantaneous rates and population growth as a continuous process rather than on defined time steps (such as one day). The model is then presented as a differential equation:
dNdt= rNdNdt= rN
The term ΔN/Δt is replaced by dN/dt to express that Δt (the time interval) approaches a value of zero, and the rate of change becomes instantaneous. The value r is now the instantaneous per capita rate of growth (sometimes called the intrinsic rate of population growth), and the resulting equation is referred to as the model of exponential population growth (in contrast to geometric population growth based on discrete time steps).
The model of exponential growth (dN/dt = rN) predicts the rate of population change over time. If we wish to define the equation to predict population size, N(t), under conditions of exponential growth [N(t) at any given value of t], it is necessary to integrate the differential equation presented previously. The result is:
N(t) = N(0)ertN(t) = N(0)ert
where N(0) is the initial population size at t = 0, and e is the base of the natural logarithms; its value is approximately 2.72.
Examples of exponential growth for differing values of r are shown in Figure 9.4. Note that when r = 0 (when b = d), there is no change in population size. For values of r > 0 (when b > d) the population increases exponentially, whereas values of r < 0 (when b < d) result in an exponential decline in the population. As with the pattern of geometric population growth, exponential growth results in a continuously accelerating (or decelerating) rate of population growth as a function of population size.
Exponential (or geometric) growth is characteristic of populations inhabiting favorable environments at low population densities, such as during the process of colonization and establishment in new environments. An example of a population undergoing exponential growth is the rise of the reindeer herd introduced on St. Paul, one of the Pribilof Islands, Alaska (Figure 9.5). In the fall of 1911, the United States government introduced 25 reindeer on the island of St. Paul to provide the native residents with a sustained source of fresh meat. Over the next 30 years, the original herd of 4 males and 21 females grew to a herd of more than 2000 individuals.
The whooping crane (Grus americana) provides another example of a population exhibiting exponential growth (Figure 9.6). At the time of European settlement of North America, the population of the whooping crane was estimated at more than 10,000. That number dropped to between 1300 and 1400 individuals by 1870, and by 1938, only 15 birds existed. The species was declared endangered in 1967, and thanks to conservation efforts, the 2011 population was estimated at more than 300 individuals. The whooping crane breeds in the Northwestern Territories of Canada and migrates to overwinter on the Texas coast at the Aransas National Wildlife Refuge. Counts of the entire population from the period of 1938 to 2013 have provided the data presented in Figure 9.6. (For an example of exponential population growth for humans, see Chapter 10, Ecological Issues & Applications).
9.2 Life Tables Provide a Schedule of Age-Specific Mortality and Survival
As we established in the previous section, change in population abundance over time is a function of the rates of birth and death, as represented by the per capita growth rate r. But how do ecologists estimate the per capita growth rate of a population? For the hydra population, where all individuals can be treated as identical, the rates of birth and death for the population were estimated by counting the number of individuals in the population either giving birth or dying per unit of time. This simple approach was possible because each individual within the population could be treated as identical with respect to its probability of giving birth or dying. But when birthrates and death rates vary with age, a different approach must be used.
To obtain a clear and systematic picture of mortality and survival within a population, ecologists use an approach involving the construction of life tables. The life table is simply an age-specific account of mortality. Life insurance companies use this technique, first developed by students of human populations, as the basis for evaluating age-specific mortality rates. Now, however, population ecologists are using life tables to examine systematic patterns of mortality and survivorship within animal and plant populations.
The construction of a life table begins with a cohort, which is a group of individuals born in the same period of time. For example, data presented in the following table represent a cohort of 530 gray squirrels (Sciurus carolinensis) from a population in northern West Virginia that was the focus of a decade-long study. The fate of these 530 individuals was tracked until all had died some six years later. The first column of numbers, labeled x, represents the age classes; in this example, the age classes are in units of years. The second column, nx, represents the number of individuals from the original cohorts that are alive at the specified age (x).
Of the original 530 individuals born (age 0), only 159 survived to an age of 1 year, whereas of those 159 individuals, only 80 survived to age 2. Only 5 individuals survived to age 5, and none of those individuals survived to age 6 (that is why there is no age class 6).
When constructing life tables, it is common practice to express the number of individuals surviving to any given age as a proportion of the original cohort size (nx/n0). This value, lx, referred to as survivorship, represents the probability at birth of surviving to any given age (x).
The difference between the number of individuals alive for any age class (nx) and the next older age class (nx+1) is the number of individuals that have died during that time interval. We define this value as dx, which gives us a measure of age-specific mortality.
The number of individuals that died during any given time interval (dx) divided by the number alive at the beginning of that interval (nx) provides an age-specific mortality rate, qx.
A complete life table for the cohort of gray squirrels, including all of the preceding calculations, is presented in Table 9.1. In addition, the calculation of age-specific life expectancy, ex, which is the average number of years into the future that an individual of a given age is expected to live, is presented in Quantifying Ecology 9.1.
Quantifying Ecology 9.1 Life Expectancy
Most of us are not familiar with the concept of life tables; however, almost everyone has heard or read statements like, “The average life expectancy for a male in the United States is 72 years.” What does this mean? What is life expectancy? Life expectancy (e) typically refers to the average number of years an individual is expected to live from the time of its birth. Life tables, however, are used to calculate age-specific life expectancies (ex), or the average number of years that an individual of a given age is expected to live into the future. We can use the life table for the cohort of female gray squirrels presented in Table 9.1 to examine the process of calculating age-specific life expectancies for a population.
The first step in estimating ex is to calculate Lx using the nx column of the life table. Lx is the average number of individuals alive during the age interval x to x + 1. It is calculated as the average of nx and nx + 1. This estimate assumes that mortality within any age class is distributed evenly over the year.
In the example of the gray squirrel, the value of T0 is 578. This means that the 530 individuals in the cohort lived a total of 578 years (some only 1 year, whereas others lived to age 5).
The life expectancy for each age class (ex) is then calculated by dividing the value of Tx by the corresponding value of nx. In other words, it is calculated by dividing the total number of years lived into the future by individuals of age x by the total number of individuals in that age group.
Note that life expectancy changes with age. On average, at birth gray squirrel individuals can expect to live for only 1.09 years. However, for those individuals that survive past their first birthday, life expectancy increases to 1.47. Life expectancy remains high for age class 2 and then declines for the remainder of the age classes.
Why does life expectancy increase for those individuals that survive to age 1 (1.47 as compared to 1.09 for newborn individuals)?
Which would have a greater influence on the life expectancy of a newborn (age 0): a 20 percent decrease in mortality rate for individuals of age class 0 (x = 0) or a 20 percent decrease in the mortality rate of age class 4 (x = 4) individuals? Why?
There are two basic kinds of life tables. The first type is the cohort or dynamic life table. This is the approach used in constructing the gray squirrel life table presented in Table 9.1. The fate of a group of individuals, born at a given time, is followed from birth to death—for example, a group of individuals born in the year 1955. A modification of the dynamic life table is the dynamic composite life table. This approach constructs a cohort from individuals born over several time periods instead of just one. For example, you might follow the fate of individuals born in 1955, 1956, and 1957.
The second type of life table is the time-specific life table. This approach does not involve following a single or group of cohorts, but rather it is constructed by sampling the population in some manner to obtain a distribution of age classes during a single time period. Although it is much easier to construct, this type of life table requires some crucial assumptions. First, it must be assumed that each age class was sampled in proportion to its numbers in the population. Second, it must be assumed that the age-specific mortality rates (and birthrates) have remained constant over time.
Most life tables have been constructed for long-lived vertebrate species having overlapping generations (such as humans). Many animal species, especially insects, live through only one breeding season. Because their generations do not overlap, all individuals belong to the same age class. For these species, we obtain the values of nx by observing a natural population several times over its annual season, estimating the size of the population at each time. For many insects, the nx values can be obtained by estimating the number surviving from egg to adult. If records are kept of weather, abundance of predators and parasites, and the occurrence of disease, death from various causes can also be estimated.
Table 9.2 represents the fate of a cohort from a single gypsy moth egg mass. The age interval, or x column, indicates the different life stages, which are of unequal duration. The nx column indicates the number of survivors at each stage. The dx column gives an accounting of deaths in each stage.
In plant demography, the life table is most useful in studying three areas: (1) seedling mortality and survival, (2) population dynamics of perennial plants marked as seedlings, and (3) life cycles of annual plants. An example of the third type is Table 9.3, showing a life table for the annual elf orpine (Sedum smallii). The time of seed formation is the initial point in the life cycle. The lx column indicates the proportion of plants alive at the beginning of each stage; the dx column indicates the proportion dying, rather than the actual number of individuals (as in the other examples).
9.4 Life Tables Provide Data for Mortality and Survivorship Curves
Although we can graphically display data from any of the columns in a life table, the two most common approaches are the construction of (1) a mortality curve based on the qx column and (2) a survivorship curve based on the lx column. A mortality curve plots mortality rates in terms of qx as a function of age. Mortality curves for the life tables presented in Table 9.1 (gray squirrel) and Table 9.3 (S. smallii) are shown in Figure 9.7. For the gray squirrel cohort (Figure 9.7a), the curve consists of two parts: a juvenile phase, in which the rate of mortality is high, and a post-juvenile phase, in which the rate decreases with age until mortality reaches some low point, after which it increases again. For plants, the mortality curve may assume various patterns, depending on whether the plant is annual or perennial and how we express the age structure. Mortality rates for the Sedum population (Figure 9.7b) are initially high, declining once seedlings are established.
Survivorship curves plot the lx from the life table against time or age class (x). The time interval is on the horizontal axis, and survivorship is on the vertical axis. Survivorship (lx) is typically plotted on a log10 scale. Survivorship curves for the life tables presented in Table 9.1 (gray squirrel) and Table 9.3 (S. smallii) are shown in Figure 9.8.
Life tables and survivorship curves are based on data obtained from one population of the species at a particular time and under certain environmental conditions. They are like snapshots. For this reason, survivorship curves are useful for comparing one time, area, or sex with another (Figure 9.9).
Survivorship curves fall into three general idealized types ( Figure 9.10). When individuals tend to live out their physiological life span, survival rate is high throughout the life span, followed by heavy mortality at the end. With this type of survivorship pattern, the curve is strongly convex, or Type I. Such a curve is typical of humans and other mammals and has also been observed for some plant species. If survival rates do not vary with age, the survivorship curve will be straight, or Type II. Such a curve is characteristic of adult birds, rodents, and reptiles, as well as many perennial plants. If mortality rates are extremely high in early life—as in oysters, fish, many invertebrates, and many plant species, including most trees—the curve is concave, or Type III. These generalized survivorship curves are idealized models to which survivorship of a species can be compared. Many survivorship curves show components of these three generalized types at different times in the life cycle of a species ( Figure 9.11).
9.5 Birthrate Is Age-Specific
A standard convention in demography (the study of populations) is to express birthrates as births per 1000 individuals of a population per unit of time. This figure is obtained by dividing the number of births that occurred during some period of time (typically a year) by the estimated population size at the beginning of the time period and multiplying the resulting number by 1000. This figure is the crude birthrate.
This estimate of birthrate can be improved by taking two important factors into account. First, in a sexually dimorphic population (separate male and female individuals), only females within the population give birth. Second, the birthrate of females generally varies with age. Therefore, a better way of expressing birthrate is the number of births per female of age x. Because in sexually dimorphic species population increase is a function of the number of females in the population, the age-specific birthrate can be further modified by determining only the mean number of females born to a female in each age group, bx. Following is the table of age-specific birthrates for the gray squirrel population used to construct the life table (Table 9.1):
At age 0, females produce no young; thus, the value of bx is 0. The average number of female offspring produced by a female of age 1 is 2. For females of ages 2 and 3, the bx value increases to 3 and then declines to 2 at age 4. By age 5 the females no longer reproduce; thus, the value of bx is 0.
The sum—represented by the Greek letter sigma, Σ—of the bx values across all age classes provides an estimate of the average number of female offspring born to a female over her lifetime; this is the gross reproductive rate. In the example of the squirrel population presented previously, the gross reproductive rate is 10. However, this value assumes that a female survives to the maximum age of 5 years. What we really need is a measure of net reproductive rate that incorporates the age-specific birthrate as well as the probability of a female’s surviving to any specific age.
9.6 Birthrate and Survivorship Determine Net Reproductive Rate
We can use the gray squirrel population as the basis for constructing a fecundity, or fertility, table (Table 9.4). The fecundity table uses the survivorship column, lx, from the life table together with the age-specific birthrates (bx) described previously. Although bx may initially increase with age, survivorship (lx) in each age class declines. To adjust for mortality, we multiply the bx values by the corresponding lx (the survivorship values). The resulting value, lxbx, is the mean number of females born in each age group, adjusted for survivorship.
Thus, for 1-year-old females, the bx value is 2; but when adjusted for survival (lx), the value drops to 0.6. For age 2, the bx is 3; but lxbx drops to 0.45, reflecting poor survival of adult females. The values of lxbx are summed over all ages at which reproduction occurs. The result represents the net reproductive rate, R0, defined as the average number of females that are produced during a lifetime by a newborn female. If the R0 value is 1, on average females will replace themselves in the population (produce one daughter). If the R0 value is less than 1, the females do not replace themselves. If the value is greater than 1, females are more than replacing themselves. For the gray squirrel, an R0 value of 1.4 suggests a growing population of females. Note the significant difference between the gross and net reproductive rates (10 and 1.4, respectively). The difference reflects the fact that only a small proportion of the females born will survive to the maximum age and produce 10 female offspring.
Because the value of R0 is a function of the age-specific patterns of birth and survivorship, it is a product of the life history characteristics: the allocation of resources to reproduction, the timing of reproduction, the trade-off between the size and number of offspring produced, and the degree of parental care. The net reproductive rate (R0), therefore, provides a means of evaluating both the individual (fitness) and the population consequences of specific life history characteristics. We will discuss this topic in detail in the following chapter (Chapter 10).
9.7 Age-Specific Mortality and Birthrates Can Be Used to Project Population Growth
Age-specific mortality rates (qx) from the life table together with the age-specific birthrates (bx) from the fecundity table can be combined to project changes in the population into the future. To simplify the process, the values for age-specific mortality are converted to age-specific survival. If qx is the proportion of individuals alive at the beginning of an age class that die before reaching the next age class, then 1 – qx is the proportion that survive to the next age class ( Table 9.5) and is designated as sx. With age-specific values of sx and bx, we can project the growth of a population by constructing a population projection table.
In year 1, we now have six 1-year-olds and five 2-year-olds, and we are now ready to calculate reproduction (recruitment into age class 0).
The bx value of the six 1-year-olds is 2, so they produce 12 offspring. The five 2-year-olds have a bx value of 3, so they produce 15 offspring. Together, the two age classes produce 27 young for year 1, and they now make up age class 0. The total population for year 1 [N(1)] is 38. Survivorship and fecundity are determined in a similar manner for each succeeding year (Table 9.6). Survival is tabulated year by year diagonally down the table to the right through the years, while new individuals are being added each year to age class 0.
As we can see, the process of calculating a population projection table from a life table in which life stages are defined by age (or age classes) is relatively straightforward. For many populations, such as perennial plants or fish, however, survival and birth are better described in terms of size rather than age, and rates are expressed as the probability of survival or number of offspring produced per individual of a given size. For these species, the process of developing a population projection table is similar to that presented previously if the size of an individual increases continuously through time. If, however, the size of an individual can either increase or decrease from one time to the next, as is the case with most perennial herbaceous plants, a more complex approach must be taken (see Quantifying Ecology 9.2).
Given the population projection table presented in Table 9.6, we can calculate the age distribution for each successive year—the proportion of individuals in the various age classes for any one year—by dividing the number in each age class (x) by the total population size for that year [N(t)] (see Section 8.5). In comparing the age distribution of the squirrel population over time (Table 9.7), we observe that the population attains an unchanging or stable age distribution by year 7. From that year on, the proportions of each age group in the population remain the same year after year, even though the population [N(t)] is increasing. Another piece of information that can be derived from the population projection shown in Table 9.6 is an estimate of population growth. By dividing the total number of individuals in year t + 1, N(t + 1), by the total number of individuals in the previous year, N(t), we can arrive at the finite multiplication rate—? (Greek letter lambda)—for each time period.
N(t + 1)/N(t) = λN(t + 1)/N(t) = λ
The rate λ has been calculated for each time interval and is shown at the bottom of each column (year) in Table 9.6. Note that initially λ varies between years, but once the population has achieved a stable age distribution, the value of λ remains constant. Values of λ greater than 1.0 indicate a population that is growing, whereas values less than 1.0 indicate a population in decline. A value of λ = 1.0 indicates a stable population size—neither increasing nor decreasing through time.
Quantifying Ecology 9.2 Life History Diagrams and Population Projection Matrices
The construction of life tables and their use in the development of population projection tables are important approaches in studying the dynamics of age-structured populations. We can represent the steps involved in the construction of the population projection table presented in Table 9.6 graphically using a life history diagram.
Each of the circles represents an age class. The red arrows pointing from each age class to the next represent the values of sx, the age-specific values of survival in Table 9.5. The blue arrows from age classes 1 to 4 that point to age class 0 are the age specific birth rates, bx.
Although the life history diagram provides a convenient way of graphically representing the age-specific rates of survival and birth from a life table, the real value of this approach is that it summarizes the changes that occur in populations (species) that exhibit a more complex pattern of transition between different developmental stages or size classes. For example, in perennial herbaceous plant species, birthrates (seed or seedling production) are a function of size rather than age. In addition, the die-back and regrowth each year of aboveground tissues (stem, leaves, and flowers) can result in an individual remaining in the same size class (or developmental stage) for multiple years or even reverting to a smaller size class. The population structure of the perennial herb Prinmila vulgaris presented in Figure 8.20 provides an example.
In their analysis of the population growth of the forest herb Prinmila vulgaris, Teresa Valverde and Jonathan Silverston represent the population structure using five developmental stage classes based on reproductive stage and size (see legend to figure). The resulting life history diagram is presented here.
In the preceding diagram, the arrows labeled with the letter F represent fecundity or birthrates (the number of seedlings produced by the various adult stage classes: Adults 1, 2, and 3). These values are equivalent to the bx values used in the analysis of the grey squirrel population (see Table 9.5). The arrows labeled with the letter G represent the probability of an individual in that stage class growing into the next larger stage class the following year. These values are equivalent to the sx values in Table 9.5. The arrows labeled S represent the probability that an individual of a given stage class will either stay in the same age class or revert into a previous stage class the following year (smaller in size than the previous year). These values do not exist for a population that is described in terms of age classes because an individual cannot revert to a previous age class.
Because of the greater complexity of possible transitions, a system of subscripts is required to identify the transitions. Each transition has a two-number subscript, i and j, that represents the probability that an individual of stage class j at year t will move into stage class i the following year (calculated as t + 1). For example, there are three possible transitions for individuals currently in the Stage Class 3. The transition G43 is the probability that an individual in Stage Class 3 will move into Stage Class 4 the following year, whereas the transition S23 is the probability that the individual will revert to Stage Class 2 the following year. The transition labeled S33 is the probability that the individual will remain in Stage Class 3 the following year.
The values of fecundity (F) and the transition probabilities (G and S) are organized in the form of a population projection matrix.
The top row of the matrix contains the values of fecundity for each of the stage classes (Fij ). The other elements of the matrix contain the transition probabilities between stage classes (Gij and Sij ). The elements of the matrix that have a value of zero represent rates or transitions that are either not possible or do not exist for the population under study.
Combined with estimates of the current population structure (number of individuals in each of the stage classes), the population projection matrix can be used to project patterns of population growth and structure in the future, just as was done in Table 9.6 for the grey squirrel population. The procedure involves matrix multiplication—multiplying the preceding matrix by a vector representing the current population structure. To further investigate this technique and apply it to predict patterns of population growth for the population of P. vulgaris studied by Teresa Valverde and Jonathan Silverston and discussed previously, go to Analyzing Ecological Data at www.masteringbiology.com.
The population projection table demonstrates two important concepts of population growth: (1) the rate of population growth, as estimated by λ, is a function of the age-specific rates of survival (sx) and birth (bx), and (2) the constant rate of increase of the population from year to year and the stable age distribution are results of survival and birthrates for each age class that are constant through time.
Given a stable age distribution in which λ does not vary, λ can be used as a multiplier to project population size into the future (t + 1). This can be shown simply by multiplying both sides of the equation for λ shown previously by the current population size, N(t), giving:
N(t + 1) = N(t)λN(t + 1) = N(t)λ
We can predict the population size at year 1 by multiplying the initial population size N(0) by λ, and for year 2 by multiplying N(1) by λ:
N(1) = N(0)λN(2) = N(1)λN(1) = N(0)λN(2) = N(1)λ
Note that by substituting N(0)λ for N(1), we can rewrite the equation predicting N(2) as:
N(2) = [N(0)λ]λ = N(0)λ2N(2) = [N(0)λ]λ = N(0)λ2
In fact, we can use λ to project the population at any year into the future using the following general form of the relationship developed previously:
N(t) = N(0)λtN(t) = N(0)λt
For our squirrel population, we can multiply the population size at year 0 — N(0) = 30 — by λ = 1.2, which is the value derived from the population projection table, to obtain a population size of 36 for year 1. If we again multiply 36 by 1.20, or the initial population size 30 by λ 2 (1.202), we get a population size of 43 for year 2; and if we multiply the initial population size of 30 by λ10, we arrive at a projected population size of 186 for year 10 (Figure 9.12). These population sizes do not correspond exactly to the population sizes calculated in the population projection table because λ fluctuates above and below the eventual value attained at stable age distribution. Only after the population achieves a stable age distribution does the λ value of 1.20 project future population size.
The equation N(t) = N(0)λt describes a pattern of population growth (see Figure 9.12) similar to that presented for the exponential growth model developed in Section 9.1. Recall that when described over discrete time intervals, however, the pattern of growth is termed geometric population growth (see discussion in Section 9.1). In this example, the time interval (Δt) is 1 year, the interval (x) used in constructing the life and fecundity tables from which λ is derived.
Note that the equation predicting population size through time using the finite growth multiplier λ is similar to the corresponding equation describing conditions of exponential growth developed in Section 9.1:
N(t) = N(0)λt (geometric population growth)N(t) = N(0)ert (exponential population growth)N(t) = N(0)λt (geometric population growth)N(t) = N(0)ert (exponential population growth)
In fact, the two equations (finite and continuous) illustrate the relationship between λ and r:
λ = er or r = ln λλ = er or r = ln λ
For the gray squirrel population, we can calculate the value of r = ln(1.20), or 0.18.
Unlike the original calculation of r for the hydra population in Section 9.1, this estimate of the per capita population growth rate does not assume that all individuals in the population are identical. It is derived from λ, which as we have seen is an estimate of population growth based on the age-specific patterns of birth and death for the population. This estimate does, however, assume that the age-specific rates of birth and death for the population are constant; that is, they do not change through time. It is this assumption that results in the population converging on a stable age distribution and constant value of λ.
The geometric and exponential models developed thus far provide an important theoretical framework for understanding the demographic processes governing the dynamics of populations. But nature is not constant; systematic and stochastic (random) processes, both internal (demographic) and external (environmental), can influence population dynamics.
9.8 Stochastic Processes Can Influence Population Dynamics
Thus far we have considered population growth as a deterministic process. Because the rates of birth and death are assumed to be constant for a given set of initial conditions — values of r or λ and N(0) — both the exponential and geometric models of population growth will predict only one exact outcome. Recall, however, that the age-specific values of survival and birth in the life and fecundity tables (Tables 9.1 and 9.4) represent probabilities and averages derived from the cohort or population under study. For example, the values of bx are the average number of females produced by a female of that age group. For the 1-year-old females, the average value is 2.0; however, some female squirrels in this age class may have given birth to four female offspring, whereas others may not have given birth at all. The same holds true for the age-specific survival rates (sx), which represent the probability of a female of that age surviving to the next age class. For example, in Table 9.5 the probability of survival for a 1-year-old female gray squirrel is 0.5—the same probability of getting a heads or tails in a coin toss. Although survival (and mortality) is expressed as a probability, it is a discrete event for any individual—it either survives to the next year or not, just as the outcome of a single coin toss will be either heads or tails. If we toss a coin 10 times, however, we expect to get on average an outcome of 5 heads and 5 tails. This is in fact what we assume when we multiply the probability of survival (0.5) by the number of females in an age class (10) to project the number surviving to the next year (5) in Table 9.6. But each individual outcome in the 10 coin tosses is independent, and there is a possibility of getting 4 heads and 6 tails (probability p = 0.2051), or even 0 heads and 10 tails (p = 9.765 × 10–4). The same is true for the probability of survival when applied to individuals in a specific age class. The realization that population dynamics represent the combined outcome of many individual probabilities has led to the development of probabilistic, or stochastic, models of population growth. These models allow the rates of birth and death to vary about the mean estimate represented by the values of bx and sx.
The stochastic (or random) variations in birthrates and death rates occurring in populations from year to year are called demographic stochasticity, and they cause populations to deviate from the predictions of population growth based on the deterministic models discussed in this chapter.
Besides demographic stochasticity, random variations in the environment, such as annual variations in climate (temperature and precipitation) or the occurrence of natural disasters, such as fire, flood, and drought, can directly influence birthrates and death rates within the population. Such variation is referred to as environmental stochasticity. We will discuss the role of environmental stochasticity in controlling the growth of populations later in Chapter 11 (Section 11.13).
9.9 A Variety of Factors Can Lead to Population Extinction
When deaths exceed births, populations decline. R0 becomes less than 1.0, and r becomes negative. Unless the population reverses the trend, it may become so low that it declines toward extinction (see Figure 9.4).
Small populations—because of their greater vulnerability to demographic and environmental stochasticity (see Section 9.8) and loss of genetic variability (see Section 5.7)—are more susceptible to extinction than larger populations (we shall explore this issue at greater length in Chapter 11). However, a wide variety of factors can lead to population extinction regardless of population size.
Extreme environmental events, such as droughts, floods, or extreme temperatures (heat waves or frosts), can increase mortality rates and reduce population size. Should the environmental conditions exceed the bounds of tolerance for the species, the event could well lead to extinction (for examples of environmental tolerances see Figures 5.7, 6.6, and 7.14a). Changes in regional and global climate over the past century (see Chapter 2, Ecological Issues & Applications) have led to a decline in many plant and animal species, and projected future climate change could result in the extinction of many species (see Chapter 27).
A severe shortage of resources, caused by either environmental extremes (as discussed previously) or overexploitation, could result in a sharp population decline and possible extinction should the resource base not recover in time to allow for adequate reproduction by survivors. In the example of exponential growth in the population of reindeer introduced on St. Paul in 1911 (see Figure 9.5), the reindeer overgrazed their range so severely that the herd plummeted from its high of more than 2000 in 1938 to 8 animals in 1950 (Figure 9.13). The decline produced a curve typical of a population that exceeds the resources of its environment. Growth stops abruptly and declines sharply in the face of environmental deterioration. From a low point, the population may recover to undergo another phase of exponential growth or it may decline to extinction.
(Adapted from Scheffer 1951.)
As we discussed in Chapter 8 (Ecological Issues & Applications), when a nonnative species (invasive species) is introduced to an ecosystem through human activity, the resulting interactions with species in the community can often be detrimental. The introduction of a novel predator, competitor, or parasite (disease) can increase mortality rates, having a devastating effect on the target population and causing population decline or even extinction.
Ecological Issues & Application The Leading Cause of Current Population Declines and Extinctions Is Habitat Loss
A multitude of ecological studies over the past several decades have documented a pattern of population decline and extinction for an ever-growing number of plant and animal species across the planet ( Figure 9.14 ). The primary cause of current population extinctions is the loss of habitat as a result of human activities. The cutting of forests, draining of wetlands, clearing of lands for agriculture, and damming of rivers have resulted in a significant decline in the available habitat for many species and are currently the leading causes of species extinctions on a global scale.
Freshwater ecosystems have been particularly vulnerable to habitat destruction over the past century. In the United States alone, there are approximately 75,000 dams impounding some 970,000 km of river, or about 17 percent of the rivers in the nation ( Figure 9.15 ). Dams remove sections of turbulent river and create standing bodies of water (lakes and reservoirs), affecting flow rates, temperature and oxygen levels, and sediment transport. Dams have a particularly negative impact on migratory species, restricting movement upriver to breeding areas. As a result of the degradation of freshwater habitats, between the years 1900 and 2010, 57 species and subspecies of North American freshwater fish have become extinct ( Figure 9.16 ).
Birds are perhaps the most extensively monitored group of terrestrial species in North America over the past 50 years and therefore provide some of the best examples of population declines resulting from human activity and land-use change. The North American Breeding Bird Survey (a joint effort between the United States Geological Service and the Canadian Wildlife Service) has conducted annual surveys in the United States and Canada since its initial launch in 1966. These data provide a basis for evaluating population trends that can be related to changes in land use and habitat decline over the same period.
Data from the Breeding Bird Survey show that one of the most negatively impacted groups of birds over the past 50 years has been species that inhabit the grassland habitats of the Great Plains of central North America. Beginning in the latter half of the 19th century, the expansion of agriculture west of the Mississippi River has led to the decline of native grasslands (prairies) as land has been converted to cropland ( Figure 9.17 ). More than 80 percent of North American grasslands have been converted to agriculture or other land uses. The loss of habitat has led to a steady decline in grassland bird species in both the United States and Canada ( Figure 9.18 ). The observed decline in populations is not only a result of a reduction in the area of native grasslands to support local populations but also because of declines in local population growth rates as a result of habitat degradation on remaining grassland areas (quality of habitat). Kimberly With of the University of Kansas and colleagues examined the growth rates of the three dominant grassland bird species in the Flint Hills region of the central Great Plains: Dickcissel (Spiza Americana), Grasshopper Sparrow (Ammodramus savannarum), and Eastern Meadowlark (Sturnella magna). Using estimates of annual population growth rate (λ; see Section 9.7) for numerous local populations across the region, the researchers determined that the mean annual growth rates for all three species on remaining habitats are negative (λ < 1). These results indicate that the observed decline in regional populations of these three species is a result of both decreasing area of habitat and negative growth rates for populations on the remaining areas of grassland.
Not all species are equally susceptible to extinction from habitat decline. One group of species that are particularly vulnerable is migratory species. Species that migrate seasonally depend on two or more distinct habitat types in different geographic regions (see Section 8.7). If either of these habitats is altered or destroyed, the species will not persist. The more than 120 species of neotropical birds that migrate each year between the temperate zone of eastern North America and the tropics of Central and South America (and the islands of the Caribbean) depend on suitable habitat in both locations ( Figure 9.19 ) as well as stopover habitat in between. An analysis of the North American Breeding Bird Survey data for Canada during the period of 1970 to 2010 shows a decline of more than 50 percent in the populations of bird species that breed in Canada (spring and summer months) and migrate to South America for the winter. The primary reason for this decline in migratory bird populations is the destruction of rain forest habitats in South America.
The species most vulnerable to extinction are endemics, which are species found only in a particular locality or localized habitat. Endemic species are particularly susceptible to extinction because of their limited geographic distribution (see Chapter 8, Section 8.2 and Figure 8.6). Environmental changes, disturbances, or human activities within their limited range could result in a complete loss of habitat for the species. For example, the island of Madagascar off the east coast of Africa is home to a diverse flora and fauna, of which approximately 90 percent are endemics and found only on Madagascar. The majority of these species inhabit the island’s tropical rain forest habitats, which have declined in extent steadily over the past 50 years ( Figure 9.20 ). More than 90 percent of the original rain forest has been cleared, and as a result, Madagascar has the largest percentage of plant and animal species listed as threatened or endangered compared to any other geographic region in the world. For example, lemurs are a group of primates endemic to the island of Madagascar that depend on the rain forest habitat. As a result of forest clearing and habitat loss, 91 percent of the known lemur species are threatened. Twenty-three of the species are now considered “critically endangered,” 52 are “endangered,” and 19 are listed as “vulnerable” on the International Union for Conservation of Nature’s (IUCN) Red List of Threatened Species. At least 17 species and 8 genera are believed to have become extinct in the 2000 years since humans first arrived in Madagascar.
Summary
Population Growth 9.1
In a population with no immigration or emigration, the rate of change in population size over time over a defined time interval is a function of the difference between the rates of birth and death. When the birthrate exceeds the death rate, the rate of population change increases with population size. As the time interval over which population change is evaluated decreases, approaching zero, the change in population size is expressed as a continuous function, and the resulting pattern is termed exponential population growth. The difference in the instantaneous per capita rates of birth and death is defined as r, which is the instantaneous per capita growth rate.
Life Table 9.2
Mortality and its complement, survivorship, are best analyzed by means of a life table—an age-specific summary of mortality. By following the fate of a cohort of individuals until all have died, we can calculate age-specific estimates of mortality and survival.
Types of Life Tables 9.3
We can construct a cohort or dynamic life table by following one or more cohorts of individuals over time. A time-specific life table is constructed by sampling the population in some manner to obtain a distribution of age classes during a single time period.
Mortality and Survivorship Curves 9.4
From the life table, we derive both mortality curves and survivorship curves. They are useful for comparing demographic trends within a population and among populations under different environmental conditions and for comparing survivorship among various species. Survivorship curves fall into three major types: Type I, in which individuals tend to live out their physiological life span; Type II, in which mortality and thus survivorship are constant through all ages; and Type III, in which the survival rate of the young is low.
Birthrate 9.5
Birth has the greatest influence on population increase. Like mortality rate, birthrate is age-specific. Certain age classes contribute more to the population than others do.
Net Reproductive Rate 9.6
The fecundity table provides data on the gross reproduction, bx, and survivorship, lx, of each age class. The sum of these products gives the net reproductive rate, R 0, which is defined as the average number of females that will be produced during a lifetime by a newborn female.
Population Projection Table 9.7
We can use age-specific estimates of survival and birthrates from the fecundity table to project changes in population density. The procedure involves using the age-specific survival rates to move individuals into the next age class and age-specific birthrates to project recruitment into the population. The resulting population projection table provides future estimates of both population density and age structure. Estimates of changes in population density can be used to calculate λ (lambda), which is a discrete estimate of population growth rate. This estimate can be used to predict changes in population size through time (geometric growth model). In addition, λ can be used to estimate r, which is the instantaneous per capita growth rate. The estimate of r, based on λ, accounts for differences in the age-specific rates of birth and death.
Stochastic Processes 9.8
Because the age-specific values of survival and birth derived from the life and fecundity tables represent average values (probabilities), actual values for individuals within the population can vary. The random variations in birthrates and death rates that occur in populations from year to year are called demographic stochasticity. Random variations in the environment that directly influence rates of birth and death are termed environmental stochasticity.
Extinction 9.9
A variety of factors can result in a population declining to extinction, including environmental stochasticity, the introduction of new species, and habitat destruction.
Habitat Loss and Extinction Ecological Issues & Applications
The primary cause of species extinctions is habitat destruction resulting from the expansion of human populations and activities. Declining populations are a result of both a reduction in the area of habitat available to support populations and a decline in the growth rate of populations that inhabit remaining areas of habitat. The latter is the result of the degradation of remaining habitats because of human activities.
