BIOL 4120 Principles of Ecology Phil Ganter 320 Harned Hall 963-5782 |
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The damaged
cactus arm above is home to a community of microbes and insects.
The populations of each species there are living in a temporary
habitat and face the certainty of extinction when the damaged
tissue has been consumed. |
Lecture 10 Population Growth
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Overview - Link to Course Objectives
For more on the model of growth, go to Modeling Exponential Growth page
Describing a Population
Demography is the science of life tables
Life Tables are accounting of births and death in a population
Usually the life span of an individual is broken into stages (egg, larvae, etc.) or by age intervals or classes (0-5 years, 5-10, etc.)
Cohort - all of the individuals born at the same time
- x = the interval
- nx = the number alive at the START OF THE INTERVAL
- lx = age specific survivorship -- fraction of cohort alive at the START OF THE INTERVAL
- dx = age specific mortality -- number dying in the interval
- qx = age specific mortality rate -- death rate for individuals of a specific age
- ex = age specific life expectancy - number of years left for an organism of a particular age.
- Life expectancy is not always maximal at birth!!
- bx = age specific natality -- number born in the interval
- mx = age specific birth rate - the per female rate of offspring production for females of a particular age.
- R0 = net replacement rate = the number of females alive in the next generation for each female alive in the present generation.
- Tc = generation time -- the average time between a birth of a member of the cohort and the birth of a member's offspring
- Age specific means that the parameter varies with the individual's age
By convention, the first age interval has the notation 0, so the number in the cohort at the start is n0.
- n0 is also the Cohort size - a Cohort is the group of newly born organisms that are followed throughout their lives in the life table
Another convention (especially for human life tables) is to multiply the number in the lx column by 1000 (if it is actually 0.562, then it reads 562).
However, if you wish to calculate the net replacement rate (equation given below), then you must either convert lx back to a proportion or you will be calculating the number of females in the next generation per 1000 females alive in the present generation!
If we plot log (nx) versus age, we can see where most of the mortality occurs by where the numbers drop
To calculate ex, we will need to tack on another pair of columns to make it easier to do. This column is Lx, the average number alive in interval x. This is simple to do:
From this column, we can get Tx, which is the sum of all of the Lx values from x until the age that the last member of the cohort dies (this is symbolized by L°).
Life expectancy at age x is simply the average number of years lived by members of the cohort that are age x.
Net replacement rate is a measure of relative growth. It is the number of females alive in generation x+1 for each female alive in generation x. If it is greater than 1, the population is growing as there are always more females alive in the next generation than in the present. If less than 1, the population is decreasing (less than 1 female in the next generation per each alive in the present generation), and a net replacement rate of 1 means each female is exactly replaced in the next generation, so the population is stable, neither growing nor declining.
If we eliminate the males and add a column containing births, then we can calculate the net replacement rate. First, we must calculate the per capita birth rate (mx) for females of age x (females only!)
The net replacement rate is the sum of age-specific birth rates times the age specific survivorship. This means that an age class's contribution to the replacement rate must take account of the probability of surviving to that age.
The net replacement rate is a measure of population growth rate.
If it is 1, then the population in neither growing or declining in size (a stable population).
If it is less than 1, the population is declining by that proportion each generation (if it is 0.5, then the population will be half as big each generation).
A value over 1 indicates a growing population (a value of 2 is a population that doubles each generation).
Assume that the net replacement rate is near 1. If so, we approximate the generation time as:
The approximation is due to the discrete nature of a life table, versus the continuous nature of time (an integral approach would get the exact figure)
Deterministic Models of Population Growth:
A model is a description of a natural phenomenon
Can be
Can be:
Geometric (discrete generations) and Exponential (overlapping generations) Population Growth
Populations with overlapping generations (us!) use continuous equations
The instantaneous rate of increase of a population (dN/dt) is the result
For more on this topic, go to Modeling Exponential Growth page
Stochastic Processes and Models:
Stochastic demographic process are random changes in birth and death rates from year to year
We can include stochastic change into our demographic models
Predictions of population size made using stochastic models are couched as probability distributions of possible population sizes
One would say that the size of a population is predicted to be 1232 organisms in six months if using the prediction from a deterministic model.
One would say that, in six months, there is a 95% chance that the population will be somewhere in the range of 985 to 1456 if using the prediction from a stochastic model.
In general, the longer in the future you try to predict population size, the larger the range of possible sizes
Populations go extinct for many reasons
Some populations are, by nature, destined for extinction because their habitat is temporary
- many microbes live in situations that are temporary (my yeast are an example)
- plants that colonize openings in the forest live in habitats that will return to forest and become unsuitable for them
- populations of protists in temporary ponds
Some populations go extinct because of some unusual climatic event (very cold spell, flooding, severe storm, volcanic eruption [not really climatic]
Some populations go extinct due to Habitat Alteration by human activity
Probability of extinction is related to population size
smaller populations have a higher probability
Allee effect - the negative effects felt by populations that are smaller than some critical value (which is specific to each species)
- failure to find a mate
- Genetic drift may fix alleles in the population that are harmful (increase death rate or decrease birth rate)
- Inbreeding will increase the frequency of homozygotes (and, therefore, reduce heterozygosity) at loci with more than one allele present in the population
- Homozygotes are sometimes less fit and may expose some recessive alleles, normally masked by the dominant gene in heterozygotes
- Individuals with one or more detrimental alleles made homozygous by inbreeding may not be able to survive and reproduce in the population's environment.
Demography, Life Tables, Cohort, nx, lx, dx, age specific mortality, qx, age specific mortality rate, ex, age specific life expectancy, bx, age specific natality, mx, age specific birth rate, R0, Tc, generation time, Age specific, n0, Type I Mortality curve, Type II Mortality curve, Type III Mortality curve, Life Expectancy, Net Replacement Rate, Generation Time, Model, Verbal Model, Pictorial Model, Mathematical Model, Deterministic Model, Stochastic Model, Geometric Population Growth, Non-overlapping generations, Discrete Equations, Exponential Population Growth, Overlapping generations, Continuous Equations, Intrinsic Rate of Natural Increase, Environmental Stochasticity, Demographic Stochasticity, Determinstic models, Stochastic models, Habitat Alteration, Allee effect, Genetic drift, Inbreeding
Last updated February 4, 2007