1 BIOL 217 DEMOGRAPHY Introduction Demography is the study of populations, especially their size, density, age and sex. The intent of this lab is to give you some practices working on demographics, and to help you review material that you have already been exposed to in lecture. A fair amount of background information is included along the way to help you get a better handle on the topic. Four factors affect population size: birth (B), death (D), immigration (I), and emigration (E). Birth here refers to the number of new individuals entering the population due to reproduction within the population. The new individuals may be born, hatched, budded, germinated, or produced by any method, but we still usually use B to symbolize the number of newly reproduced individuals. Death is the number of individuals in the population that die. Immigration is one-way movement into the population, i.e., to an area outside the boundaries of the study area, so I is the number of individuals that immigrate. Emigration is one-way movement out of the population from another area, so E is the number of individuals that emigrate. Migration sometimes refers to either immigration or emigration, but is often (and perhaps more usefully) used to describe round-trip movements and is sometimes restricted to those round trips. The size of or number of individuals in a population at time t is N t. N t = N t-1 + B - D + I - E In many demographic studies I and E are not measured because it is much easier to collect data on individuals that are present than ones that move into or out of an area. Populations must be studied in larger areas and more detailed data must be collected for individuals to determine immigration and emigration rates than population-wide birth and death rates. When I and E are not measured, they must be assumed to be negligible or to be approximately equal. If these assumptions are false, measurements of birth and death rates will be inaccurate. Fortunately, there are some circumstances in which the assumptions are very likely to be approximately correct. Furthermore, when they might be incorrect, we have techniques that allow us to determine the rate of increase due to the combined effects of birth and immigration and the rate of decrease due to combined effects of death and emigration. Life Tables and Fecundity Schedules Many important aspects of a population s demography can be summarized in a life table. A life table is an age-specific summary of mortality rates operating in a population. Several methods are used to estimate the variables in life table and fecundity schedules. Fecundity is a general term for reproductive output, regardless of
2 the method of reproduction. It applies to production of new individuals by live-bearing, egg-laying, fission, or whatever method. A fecundity schedule is an age-specific summary of the reproductive rates operating in a population. By combining the information in life tables and fecundity schedules, we can calculate population growth rates and generation times. Types of Life Tables and Fecundity Schedules By far the best of constructing a life table is the cohort method. A cohort is a group of individuals born within a short time. For example, it might consist of all people born in one particular year. For a population of animals that produces three clutches of eggs each year, with all females laying each clutch synchronously, perhaps within two weeks of each other, offspring produced by clutches 1, 2, and 3 would form three successive cohorts. Those of you who took BIOL 117 did a laboratory exercise in which you constructed a life table for human beings based on data taken from gravestones. From the gravestones you could determine the dates of birth and death, and then calculate the age of each person at death. By compiling data on the frequency of death at each age, you were able to estimate age-specific death rates. You constructed what is sometimes called a doomsday life table, which is an example of a static life table. A static life table is constructed from data collected at a single time and including data for individuals from several cohorts. A cohort life table is a life table generated using data for a single cohort. This is the best type of life table. Its advantage over static life tables is that the cohort life table is constructed by following the entire group born in the same interval from birth to the death of the last individual. The cohort life table should be used whenever possible. Although cohort life tables are preferred, the data may be prohibitively difficult to collect. If cohort life tables are better, why would we bother to construct static life tables? Static life tables are frequently used when we need to estimate demographic parameters, but lack the time or resources to follow an entire cohort from birth to death. Collecting the information for a whole cohort is especially difficult for long-lived, continuously breeding species. Imagine trying to collect data for a whole cohort of box turtles, which can live over 100 years. It would take several generations of biologists to complete the project. When we can t practically track the whole cohort until all individuals die, we can often determine the ages of all individuals in a population at a particular time and estimate the mortality and fecundity rates for each age group. If it is possible to determine the age of each individual, we can calculate the life table and fecundity schedule variables by making the assumption there is no variation in survival or fecundity over time. Stated differently, the assumption is that the static life table based on data collected at one time is the same as would have been obtained by following a cohort through time.
3 This assumption is the crucial one that allows a static life table to be constructed, but it can be a serious weakness. Mortality rates can vary over time in local populations for many reasons, such as epidemics, floods, famine, and others. Similarly, fecundity rates may be affected by food supply and other factors. To the extent that mortality and fecundity rates vary over time, static life tables can give distorted estimates. When the assumption of constant rates is false, estimated mortality rates can sometimes be negative, which is clearly impossible short of resurrection. Given the conceptual flaw of static life tables and fecundity schedules, why bother? As noted already, they may be the only practical way to estimate mortality and fecundity schedules. Furthermore, unless large changes have occurred over time, they often give at least a rough idea of age-specific survival and fecundity rates as measured for cohorts. For short-lived organisms under relatively constant conditions, static life tables may give reasonably good estimates. Cohort Life Table Age or # at start of Proportion of cohort alive at start of age group Proportion dying during age-specific mortality = n x /n 0 = l x - l x+1 = d x /l x x n x l x d x q x 0 1000 1.000 0.500 0.500 1 500 0.500 0.100 0.200 2 400 0.400 0.100 0.250 3 300 0.300 0.200 0.667 4 100 0.100 0.100 1.000 5 0 0.000 The first column gives the age, x, of each age in the cohort. Age might represent years for people, some other time interval more appropriate for other organisms, or life history s for insects. When the cohort is born, x = 0. The second column, n x, gives the number of individuals that are alive at the beginning of the interval x to x+1. For example, n 1 might be the number of individuals that survive at least 1 year. Before calculating mortality rates, we usually convert the number surviving to a proportion of the initial cohort surviving. This is l x, the proportion of the entire cohort that is alive at time x. From successive values of l x we can calculate mortality rates.
We show two different variables that measure mortality rates. 1) The first mortality rate in the life table is d x, which is the proportion of the entire cohort that dies during the interval x to x+1. It is calculated by subtracting the proportion surviving at the end of an interval from the proportion surviving at the start of the same interval. It could also be calculated by subtracting the number alive at the end of an interval from the number alive at the beginning of the interval and then dividing the difference by the initial number of individuals in the cohort. This is rarely done because l x is itself a useful survival rate and calculations of all of the mortality rates from l x are simple. 2) The other traditional mortality rate is q x, the proportion of the individuals surviving to age x that die between x and x+1. Thus, q x is an age-specific measure of the intensity of mortality. This is in contrast to d x, which measures mortality in relation to the initial population size. Cohort Life Table Problem 4 Age or # at start of Proportion of cohort alive at start of age group Proportion dying during age specific mortality = nx/n0 = lx lx+1 = dx/lx x n x l x d x q x 0 2000 1 400 2 200 3 100 4 0 Mortality and Survivorship Although combinations of species response and environmental conditions can lead to a wide variety of survivorship and mortality patterns, we can talk about the three idealized types. Type I: low death rate through most of life span High mortality in old age Example humans in industrial nations Type II: straight line on semilog plot implies constant mortality rate, independent of age log l x. Examples - many adult birds, seed banks, hydra.
5 Type III: very high initial death rate, but individuals that survive the initial period have higher and relatively constant survival rate Examples - many marine fish and invertebrates, animals age with external fertilization and vulnerable juvenile s Figure 1. The Three Idealized Types of Survivorship and Mortality Curves. From Figure 8.6 in Krebs. Note that the survivorship curve is a semilog plot. With a standard numerical plot, the type II survivorship curve shows a nonlinear decrease in l x. The graph shows an exponential decrease in l x with age. What if the survivorship curve looked like a set of descending steps what would the mortality curve look like? What might you surmise about the demography of the organism? Construct mortality curves using q x (not d x as in Fig. 1b) and a survivorship curve using the following data. x n l x d x q x 0 1000 1 800 2 700 3 500 4 450 5 400 6 300 7 150 8 50 9 0
6 Fecundity Schedules Fecundity is measured by the number of offspring per female of a given age, b x. Demographers typically view populations as if they consisted only of females giving rise to female offspring. To construct a fecundity schedule, we start with the same information given in a life table, i.e., the ages, and numbers of individuals surviving to each age, and calculate the l x values. To proceed we need only one new variable, the total number of offspring produced by the population at each age, f x. From f x we calculate the number of female offspring produced per female during the interval x to x+1 by females of age x. This is b x, the age-specific natality rate. Fecundity Schedule Example x n x l x f x b x l x b x 0 1000 1 0 0 0 1 500 0.5 1500 3 1.5 2 400 0.4 2400 6 2.4 3 300 0.3 1800 6 1.8 4 100 0.1 200 2 0.2 5 0 0 0 R 0 = Σ l x b x = 5.9 The fecundity schedule is completed by calculating the product of l x and m x for each age group. Each l x m x gives the number of offspring per female times fraction of cohort surviving to age x. Thus, each l x m x is the contribution of offspring by females of age x expressed as a multiple of the total cohort size in the current generation. Both l x m x and R 0 are dimensionless (i.e., they have no units). As a sum of l x m x values, i.e., of age-specific multiples of current population size, R 0 gives the total offspring produced by the current generation in multiples of the initial cohort size. Therefore, R 0 is a measure of population growth rate for the whole cohort called the net reproductive rate or sometimes the basic reproductive rate. R 0 is the multiplication rate per generation. An alternative way of calculating it is R 0 = ΣF x /n 0, the total number of offspring produced by the cohort divided by the number of females in the cohort. This can be a handy shortcut method of calculating the growth rate if no information about the age distribution of reproductive output is needed. R 0 = Σl x b x
7 What is the value of R 0 if: population size is stable? population is going extinct? population declines by half? population triples? What is the range of R 0? Fecundity Schedule Example x n x l x f x b x l x b x 0 1000 0 1 100 1500 2 10 2400 3 0 1800 R 0 = Σ l x b x = Cohort Generation Time and Innate Capacity for Increase The length of one generation is T, which is the mean time between births of parent and offspring for the whole cohort. Therefore, T is given by the total of cohort birth-to-birth times divided by the total number of offspring for the cohort at all ages: Σxl x b x Σxl x b x T = ----------- = ----------- Σl x b x R 0 The rate of change in population size per individual per unit time, i.e., the individualspecific rate of increase is r. Little r is called the biotic potential, intrinsic rate of increase, or innate capacity for increase. This differs from R O, which gives a per generation rate of increase for the population, but is related to R O as follows: r = ln R O / T
8 x n x l x f x b x l x b x xl x b x 0 1000 1 0 0 0 0 1 500 0.5 1500 3 1.5 1.5 2 400 0.4 2400 6 2.4 4.8 3 300 0.3 1800 6 1.8 5.4 4 100 0.1 200 2 0.2 0.8 5 0 0 0 ---- ---- R 0 = Σ l x b x = 5.9 T = 12.5/5.9 = 2.1 r = ln R 0 /T = 1.77/2.1 = 0.85 LIFE TABLE, FECUNDITY SCHEDULE, & T - EXAMPLE x n x l x d x q x f x b x l x b x xl x b x 0 1000 5000 1 100 2000 2 10 100 3 0 R 0 = T = r =