Population Growth I

 Organisms reproduce; for almost all organisms, reproduction is the raison d'etre.  Some organisms (single celled, for example) can reproduce asexually, but for many plants and animals, some form of sexual reproduction is required.  Species of plant or animal, by definition, breed among themselves, and a group of individuals belonging to a single species constitutes a population.

 Populations are dynamic, they change with time.  Populations will grow if conditions of reproduction and survivability are good (substantial habitat, food, lack of predators and disease, appropriate climate, etc.) and populations will decline under substantially adverse conditions.  Even so-called "stable" populations can show long term fluctations; some organisms, like lemmings and the organisms that prey on them (owls, fox, ermine, wolf), have boom-bust cycles with wild changes of population size.

 Difference equations are functions where the next value is dependent upon the previous.  My weight next month, for example, depends upon how much I weigh now and how much I gain or lose in the next month; that's a difference equation.  Difference equations can be readily applied to population growth as well, because the size of the future population depends upon the size of the previous.  Difference equations represent a series of values or numbers, each of which is dependent upon the previous value.

 For a simple population, the size of the population at some point in the future depends upon the present size (starting size), the rate of reproduction (birth rate), and the rate of death (death rate).  The birth rate minus the death rate is called the growth rate.  If the growth rate is zero, this simple population is static, and does not change in size with time.  If the birth rate exceeds the death rate, the population will grow.

 For present-day human populations, the crude birth rate (number of live births per 1000 individuals, regardless of sex or age) varies from continent to continent from 10-40.  Worldwide, the crude birth rate averages 23/1000 of 0.023.  The crude death rate ranges from 7-15 (9 on average), and consequently the human population is growing rather rapidly.  The present day, worldwide growth rate of humans is approximately  0.014 or 1.4 %, an extraordinarily high value.  The worldwide growth rate was even higher in 1963, running about 2.2%.

 For the more complex case of population growth, the future population size also depends upon the rate of immigration into the population and emigration out of the population.  For some species, immigration and emigration may not be significant; orca, for example, form rather closed pods of related individuals.  For other species, like humans, individuals may freely immigrate into the pod from great distances, and likewise leave the pod forever.  Immigration and emigration are dealt with explicitly in the second exercise on population dynamics.

 Population growth is often said to be "exponential."  If a population grows 1.4% every year, that is exponential or "compound" growth.   Strictly speaking, a simple exponential function has a constant rate of change, a constant percentage change.  The rate of change of worldwide population has decreased from 2.3% to 1.4%, therefore worldwide population growth has not been strictly exponential, a single exponential function or relation will not match the actual worldwide population values.  Many of the questions on this exercise relate to this non-constant rate of change.

 Logistic equations are a type of difference equation that have a non-constant rate of change.  Many natural populations show logistic growth, growing rapidly in the initial stage with few constraints on the population, and then growing more slowly and finally stabilizing as the carrying capacity of the habitat is reached.  This type of growth will produce an S-shaped curve on a plot of X = time and Y = population size.

Population Growth II

 Population changes are dependent upon birth and death rates.  Population size also depends upon the rate of immigration into the population and emigration out of the population.  For some species, immigration and emigration may not be significant; orca, for example, form rather closed pods of related individuals.  For other species, like humans, individuals may freely immigrate into the pod from great distances, and likewise emigrate from the pod forever.

 Very few populous countries permit significant legal immigrations; the United States, Canada and Australia.  Some countries, such as Japan, effectively ban all immigration.  Legal immigration to the United States has varied dramatically over the last 100 years.  Legal immigration peaked around 1907 at 1.4 million per year, then declined to almost nothing due to restrictive laws and the Great Depression.  Immigration has increased steadily since WWII, with a prominent low in 1975 (less than 100,000) and a prominent high (almost 2 million) around 1993.  These lows and highs are mostly paperwork related, not actual movement of bodies; for example, the Immigration Reform and Control Act of 1986 granted legal status to many illegal immigrants, causing a sharp spike.

 In order to develop a more accurate model of population growth, the student must incorporate immigration and emigration in addition to birth and death.  Immigrants and emigrants are added and subtracted from the population (respectively), and then in the next iteration, contribute to more growth of the population by giving birth or reduce growth of the population by dying.

 There are many social issues related to population dynamics that can be explored with these simple models.  For example, which policy will have the biggest long-term impact on reducing population growth (all other factors being equal); reducing the number of immigrants by 50% or lowering the birth rate by 0.1%?

 The simple models of population dynamics that the students work with in this exercise cannot incorporate details and fine structure of real populations; these details are ncessary to create models that will more accurately forecast population growth.  For example, the crude birth rate (number of live births per 1000 individuals) does not take into account that only females of reproductive age (15-45) give birth, and that the birth rates vary among females in this cohort (a cohort is a like group of individuals).  The crude death rate is an average of the wide variety of death rates for the population.  And the resulting number produced by these calculations, total population, does not provide information on the age structure (how many individuals of different ages).  Knowing the age and sex structure of the population, and the birth and death rates of each age and sex, is critical to more advanced population modelling.

 To see some age structure diagrams, also known as "population pyramids", go to the website: http://www.census.gov/ipc/www/idbpyr.html .  Select a country (we recommend comparing Cambodia and Belgium) and summary graph (for three different years), to show both real data and forward models of the population structure of these two countries.  At the other website, http://www.census.gov/ipc/www/idbprint.html  you can see the tabulated results (no graphs) of some sophisticated forward modelling of populations.  Choose mid-year population, a country (such as Cambodia), and an age range (1970-2040).  Most of the modelling parameters are also available.
 

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