A population is a group of similar individuals of the same kind living in the same area at the same time. Each population has characteristics which are unique to that population, including age distribution, population density, population distribution in time and space, the birth rate, the death rate, and the population growth rate. Populations are interrelated with populations of other organisms. For example, a population of predators affects the mortality rate of a prey population.
Some organisms (populations) have annual life cycles with non-overlapping generations. Examples of these would include many herbaceous plants and insects. In populations of these organisms, nearly all members of the population are the same age at the same time. Organisms with overlapping generations and/or which are continuously breeding, tend towards a stable age distribution: the ratio of age groups remains the same as long as birth and death rates remain the same. Note that any influence which changes the death rate of a population will also affect the birth rate and age structure of the population. Organisms (populations) of species which live longer can be divided into three ecological periods: pre-reproductive, reproductive, and post-reproductive. In these organisms, the length of time in each stage depends on the overall life history of that species.
In nature, a number of factors determine the rate of increase (r) of a population of a given species. The maximal rate of increase under optimum conditions (the innate capacity for increase) is symbolized by rm. Birth rate (natality) and death rate (mortality) are influencing factors. Note that the birth rate can be less than, equal to, or greater than the death rate.
A life table gives the probability at birth of being alive at age x (designated as lx). At zero age, this is l0, which by definition, equals one (if lx is expressed as a fraction of the total — if lx is expressed in whole numbers, l0 will be equal to the total). For example, in a table where x = 4.5 weeks and lx = 0.87, this means that from a sample of 100 newly-laid eggs, 87 will survive for 4.5 weeks. Some other symbols used in these calculations are:
|x||=||a given age group within the population. This might be expressed in days, weeks, or years depending on the life span of the organism, or may be expressed as stages in the life cycle (such as in insects).|
|lx||=||the actual number or the proportion (as a decimal or percentage) of survivors at the beginning of age interval x. Note that since several samples are often averaged together, the lx values may not always be whole numbers.|
|Lx||=||the average () number of years lived by all individuals in each age category = (lx + lx + 1) ÷ 2.|
|Tx||=||total number of time units (years, weeks, months, etc.) left for all individuals to live from age x onward = Lx **FROM THE BOTTOM UP!**.|
|ex||=||life expectancy for each interval = Tx ÷ lx.|
|dx||=||the number of individuals that die during time interval x (expressed as an actual number or as a proportion of the total). Note that lx+1 = lx - dx.|
|dx||=||the overall number of individuals that died.|
|dxf||=||the cause of death (not a mathematical quantity).|
|qx||=||the mortality rate = dx/lx (often × 100 = percentage or 1000 = number per 1000). Optionally, qx may also be calculated based on the number surviving at the end of a given time period ÷ the number alive at the beginning of that time period.|
|Mx||=||total eggs or young produced per female at age x.|
|mx||=||the number of female births to each age group of mothers; the number of eggs or young which are female (in a species with a 1:1 sex ratio, this = Mx/2). Since, for most organisms, one male can fertilize a number of females, the size of the population is more dependent on the number of females present, and the calculations are usually done using only females.|
|lxmx||=||the number of females born to each age group, adjusted for survivorship, or (prob. of reaching age x) × (# of female eggs at age x) = # of births per female.|
|t||=||some other time interval.|
|lt/lx||=||the proportion of females living from age x to age t.|
|vx||=||the reproductive value of each age group = (lt/lx) × mx.|
|N0||=||the number of individuals at time zero; the number of females at the beginning of the experiment|
|Nt||=||the number of individuals after time t; the number of females at time t; the number of females after one or more generations|
|R0||=||lxmx (summed over all ages) = the net reproductive rate; the ratio of total female births in two successive generations, the ratio of offspring to parents, or the number of female offspring that will be left during her lifetime by one female. Also, R0 = N1/N0 = Nt + 1/Nt, and thus, N1 = N0 × R0 (the population is growing exponentially). The closer R0 is to 1, the slower the population growth, and if R0 is less than one, the population is declining.|
|T||=||the mean time from birth of parents to birth of offspring; the average length of time for one generation; average age of parents who had offspring|
If rm = maximal rate of increase and T = time for one generation, then rmT = # of individuals at time T.|
By definition, R0 = ermT or ln(R0) = rmT, thus rm = [ln(R0)]/T
From the information in life tables, various curves may be plotted. A mortality curve is a plot of qx vs. x. These are often J-shaped: sometimes there is a high, intitial mortality, but there is typically a period of low mortality followed by a period of highter mortality later in life (more die when they’re older). Survivorship curves may be a plot of lx vs. x, but are more often a plot of log(lx) vs. x. If the log(lx) is used, survivorship curves tend to fall into one of three general types, indicative of higher mortality later in life (I), constant mortality throughout life (II), or higher mortality early in life (III). Mortality and survivorship curves may be used to compare survival of the sexes or of populations existing in different places or at different times.
Table 2. Life Table for the Spruce Budworm in Experimental Plot A.
|¶||Since there are equal numbers of males and females, no females are left without mates. Note that in Table 3, the ratio is 54:46, and thus 8 females out of 100 are left without mates. Note that 8/100 × 1.29 = 0.1032, so its as though only 1.29 - 0.10 = 1.19 are remaining to potentially reproduce.|
|†||Expected number of eggs - actual, observed number of eggs = difference (in this case negative because the observed number was higher). This would be explained by migration of moths into or out of the study location, resulting in more or less eggs than expected.|
|*||Difference ÷ expected × 100 = % of expected number.|
|§||E = Expected # eggs ÷ starting # eggs × 100 gives the expected % increase in the population.|
|‡||O = Actual # eggs ÷ starting # eggs × 100 gives the actual, observed % increase in the population.|
Table 3. Life Table for the Spruce Budworm in Experimental Plot B.
Organisms with overlapping generations and/or which are continuously-breeding tend towards a stable age distribution. The ratio of age groups remains the same as long as birth and death rates remain the same. Any new influence that changes the death rate will also affect the birth rate and age structure.
Ages are easier to get for humans than wild plants and animals. Current data for wild organisms often are less predictive of the future population than in humans because growth is more dependent on local resources. Humans, however, can move resources to meet their needs. Typically rapidly-growing populations have generally low or decilining death rates at younger ages. A graph of such a population would be a broad-based pyramid indicative of many young in the population. In comparison, stable or declining populations have lower birth rates, fewer young, and more older members.
Table 5. Numbers of Confused Flour Beetles in 8 gm of Flour.
Table 6. Census Data from Mid-1980s
|Community||North Avondale||West Norwood||Active Members,|
Church in Norwood
From these data, similar graphs can be constructed, again with percent on the X-axis. Again, the percentage of males is plotted to the left of center and the percentage of females to the right of center. Age (or age groups) goes on the Y-axis. A graph of data from the community of Evanston would look like Figure 3.
Copyright © 1998 by J. Stein Carter. All rights reserved.
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