SUMMARY
CONCEPT 11.1 Populations can grow exponentially when conditions are favorable, but exponential growth cannot continue indefinitely.
11.1.1 Define geometric population growth.
Geometric growth occurs when a population with synchronous reproduction changes in size by a constant proportion from one discrete time period to the next.
Mathematically, geometric growth is Nt + 1 = λNt, where Nt is the population size at time t and λ is the geometric population growth rate, a constant whose value is determined by per capita birth and death rates.
11.1.2 Define exponential population growth.
Exponential growth occurs when a population with continuous reproduction changes in size by a constant proportion at each instant in time.
Mathematically, population growth is dN/dt = rN, where N is the population size and r is the exponential population growth rate, a constant whose value is determined by instantaneous per capita birth and death rates.
11.1.3 Describe the characteristics of geometric and exponential growth.
Populations undergoing geometric and exponential growth have the potential to increase rapidly in size because they grow by multiplication, not by addition.
Similar to the interest on a savings account, populations undergoing geometric and exponential growth can grow rapidly even when the growth rate is low.
All populations experience limits to growth, which ensure that geometric and exponential growth cannot continue indefinitely.
CONCEPT 11.2 Population size is determined by a combination of density-dependent and density-independent factors.
11.2.1 Define density-independent factors and describe how they affect population size and growth rate.
Density-independent factors affect population size and growth rate independent of population density and include abiotic factors such as weather and climate and biotic factors such as hunting.
In many species, density-independent factors play a major role in determining changes in population size.
11.2.2 Define density-dependent factors and describe how they affect population size and growth rate.
Density-dependent factors affect population size and growth rate as a consequence of population density and include factors such as food or habitat.
When population size becomes large enough, a lack of food, habitat, or other resources causes birth rates to decrease, death rates to increase, or dispersal to increase.
CONCEPT 11.3 The logistic equation incorporates limits to growth and shows how a population may stabilize at a maximum size, the carrying capacity.
11.3.1 Define logistic population growth and compare to exponential population growth.
Logistic growth occurs when population density slows exponential population growth, stabilizing that growth at a maximum level, known as the carrying capacity (K) or the density at which the population stops increasing in size.
Mathematically, the logistic growth equation (dN/dt = rN (1 - N/K)) is a modification of the exponential growth equation (dN/dt = rN), where the term (1 - N/K) represents the fraction of the carrying capacity that is available for population growth.
Logistic growth is similar to, but slightly slower than, exponential growth when densities are low, but as densities increase, population growth slows, and eventually becomes zero, as it reaches K.
11.3.2 Describe the growth patterns of the U.S. population.
Logistic population growth provides a close fit to the size of the U.S. population up to 1950; since that time, the growth rate of the U.S. population has been greater than expected in logistic growth and has not reached a carrying capacity.
CONCEPT 11.4 Life tables show how survival and reproduction vary with age or size structure, influencing population growth and size.
11.4.1 Justify the use of life tables to determine population growth and size.
Birth and death rates vary with the age, size, or life stage of individuals within a population.
Life tables provide a summary of how survival and reproductive rates vary with the age, size, or life stage of the individuals, allowing more accurate determinations of population growth rates and sizes.
11.4.2 Describe how age or size structure influences population growth and population size.
Age or size structure influences the survival and reproductive rates of populations, thus affecting population growth and size.
If one population has many older individuals, while the other has many younger individuals, the second population will likely grow faster because it contains more individuals of reproductive age and fewer individuals at the end of their life span.
11.4.3 Compare the three types of survivorship curves.
In a population with a type I survivorship curve, most individuals survive to old age; death rates do not begin to increase until old age.
In a population with a type II survivorship curve, individuals experience a constant chance of surviving from one age to the next throughout their lives.
In a population with a type III survivorship curve—the most common type in nature—death rates are very high for young individuals, but adults survive well later in life.
11.4.4 Analyze life table data and calculate a net reproductive rate (R0) and exponential growth rate (r).
Cohort life tables can be constructed from data on the fates of individuals born during the same time period (age class) and used to calculate age-specific survivorship (lx) and fecundity (Fx).
Multiplying survivorship (lx) by fecundity (Fx) and summing these values for all the age classes gives the net reproductive rate, R0, or the mean number of offspring produced per individual, adjusted for survival.
The per capita growth rate, r, can be estimated by dividing ln R0 by the generation time (G = (sum (xlxFx)∕R0)).
In highly mobile or long-lived organisms, a static life table may be constructed from data on the survival and fecundity of individuals of different ages during a single time period.
REVIEW QUESTIONS
1.
A population of insects triples every year. Initially, there were 40 insects.a. How many insects will there be after 4 years?
b. How many insects will there be after 27 years? (Write your answer to this question as an equation.)
c. The habitat of the insect is degraded such that the population growth rate (λ) changes from 3.0 to 0.75. If there were 100 insects in the population when its habitat became degraded, how many insects will there be after 3 years?
2. What is the distinction between factors that regulate population size and factors that determine population size?
3. For a field ecology project, you count the number of individuals of different ages found in a population during a single time period. There are 100 newborns, 40 1-year-olds, 15 2-year-olds, 5 3-year-olds, and 0 4-year-olds. These individuals produced the following number of offspring: newborns = 0 offspring, 1-year-olds = 100 offspring, 2-year- olds = 30 offspring, 3-year-olds = 25 offspring, 4-year-olds = 0 offspring.
a. Use these data to create a static life table and calculate R0 and r.
b. Explain the difference between a static life table and a cohort life table.
4. Calculate your ecological footprint at ® https://www.footprintcalculator.org.
HONE YOUR PROBLEM-SOLVING SKILLS
As discussed in this chapter, life table data can be used to estimate a population’s growth rate (r). Here we'll calculate r and future population size using life table data collected for the grass Poa annua. These data were collected by marking 843 naturally germinating seedlings and then following their fates over time.
| Age (χ) | Number of individuals (Nχ) | Number of offspring (Nχ offspring) | Survivorship (Iχ) | Fecundity (Fχ) | ∣χFχ | χ∣χFχ |
| 0 | 843 | 0 | ||||
| 1 | 722 | 216,600 | ||||
| 2 | 527 | 326,740 | ||||
| 3 | 316 | 135,880 | ||||
| 4 | 144 | 30,240 | ||||
| 5 | 54 | 3,240 | ||||
| 6 | 15 | 450 | ||||
| 7 | 3 | 30 | ||||
| 8 | 0 | 0 |
Source: Data in M.
Begon et al. 1996. Population Ecology: A Unified Study of Animals and Plants, 3rd ed. Blackwell Science: Oxford. Adapted from R. Law. 1975. Unpublished PhD thesis. University of Liverpool.1. Use the life table above and the method in Table 11.1 to fill in the columns and calculate R0 and r.
2. Using the exponential growth model, calculate the population size of the grass species 10 years from now assuming N0 = 100 individuals.
3. Now, using a logistic growth model, calculate the population size of the grass species 10 years from now assuming N0 = 100 individuals and K = 1,000,000.
4. Comparing the population sizes after 10 years, does the grass population reach its carrying capacity?
LIST OF KEY TERMS
age structure carrying capacity cohort life table, density-dependent density-independent doubling time (⅛) ecological footprint exponential growth exponential growth rate fecundity finite rate of increase generation time (G) geometric growth geometric population growth rate intrinsic rate of increase life table logistic growth net reproductive rate (R0)
population regulation static life table survival rates survivorship curve survivorship (Ix) type III survivorship curve type II survivorship curve type I survivorship curve