Now that humanity has passed 7 billion souls globally, why do we fear the coming declines?
For today
Human population growth
Feeding more people
The promise of technological innovation in agriculture
Quantity and quality of drinking water
The challenge of climate change
Some concepts
We’ll start by calculating the class growth rate. Here are a few concepts that we can use.
Expected reproduction
The lifetime expected reproduction \(R_0\) is the number of offspring expected per female over the course of the lifetime,
\[ R_0 = \sum_a l_a f_a \] where \(l_a\) is the probability of surviving to age \(a\), and \(f_a\) is the expected offspring produced at age \(a\). This is the same “R naught” that you’ve heard a lot about during covid. For a disease, \(R_0\) is the number of new infections expected from a given infection.
Suppose a group of three individuals had these four birth ages:
| Individual | Age (a) | Number | |
|---|---|---|---|
| 1 | 25.0 | 1 | |
| 2 | 30.0 | 1 | |
| 2 | 32.0 | 1 | |
| 3 | 20.0 | 1 | |
| 3 | 30.0 | 1 | |
| 3 | 23.0 | 1 | |
| 4 | NA | 0 | |
| 5 | 33.0 | 1 | |
| 6 | NA | 0 | |
| 7 | 33.0 | 1 | |
| summary | 7 | 28.2 | 8 |
If we assume that survival probability is near 1 (\(l_a = 1\) through all birth ages \(a\)), then \(R_0\) is the total number of offspring divided by the number of parents. There are twice as many parents as there are individuals in the table (until the table includes more than one parent of the same children), in this case 14. For this example, \(R_0 =\) 0.571 and \(R_0\) is less than one, indicating that this group will not replace itself.
The age distribution of reproduction is the number of offspring expected at a given age, normalized by the total (i.e., \(R_0\)), to give a probability mass function.
\[\alpha(a) = \frac{l_a f_a}{R_o}\] Here \(f_a\) is number of offspring born to a parent of age \(a\) divided by the total number of parents, i.e., 14.
The generation time is the average time between generations. It is a weighted average of age \(a\),
\[T = \sum_a a \alpha(a)\]
Here is an age table, with generation time on the bottom row, and an age distribution plot:
| Age (a) | alpha | age X alpha | |
|---|---|---|---|
| 18.0 | 0.000 | 0.000 | |
| 19.0 | 0.000 | 0.000 | |
| 20.0 | 0.125 | 2.500 | |
| 21.0 | 0.000 | 0.000 | |
| 22.0 | 0.000 | 0.000 | |
| 23.0 | 0.125 | 2.875 | |
| 24.0 | 0.000 | 0.000 | |
| 25.0 | 0.125 | 3.125 | |
| 26.0 | 0.000 | 0.000 | |
| 27.0 | 0.000 | 0.000 | |
| 28.0 | 0.000 | 0.000 | |
| 29.0 | 0.000 | 0.000 | |
| 30.0 | 0.250 | 7.500 | |
| 31.0 | 0.000 | 0.000 | |
| 32.0 | 0.125 | 4.000 | |
| 33.0 | 0.250 | 8.250 | |
| 34.0 | 0.000 | 0.000 | |
| 35.0 | 0.000 | 0.000 | |
| summary | 26.5 | NA | 28.250 |
Population growth
Population growth is expessed in discrete time or continuous time. For discrete time, growth occurs at discrete intervals. For continuous time, growth is happening continuously. For discrete time, growth is
\[ n_{t+1} = n_t \lambda \] where \(\lambda = R_0^{1/T}\). For example, if \(R_0 = 1.5\) for a generation time of \(T = 15 yr\), then the population growth rate is 1.03. This discrete-time model has the solution
\[ n_t = n_0 \lambda^t = n_0 R_0^{t/T} \]
Population growth can also be expressed in continuous time,
\[ \frac{dn}{dt} = rn \] where \(n(t)\) is population size at time \(t\), and \(r\) is the per-capita rate of increase. The rate of increase \(r\) is the birth rate \(b\) minus the death rate \(d\),
\[r = b - d\]
The relationship with continuous growth is
\[r \approx \log \lambda\] or
\[\lambda \approx e^r\]
The solution for this exponential population growth model is
\[n(t) = n(0) e^{rt}\] The time required for exponential growth to double in population size if \(r > 0\) is
\[ \log \left( \frac{n(t_2)}{n(0)} \right) = \log (2) = rt_2 \] or \(t_2 = 0.693/r\). In other words, the higher the growth rate \(r\) the shorter the doubling time.
Doubling time is related to the concept of a halflife for radioactive decay, the time it takes to half the concentration of an isotope, where particle emissions are independent of one another. Radiocarbon (\(^{14}C\)) has a halflife of 5730 years.
\[ \log \left( \frac{n(t_2)}{n(0)} \right) = \log (1/2) = r \times 5730 \] or \(r = -0.000121\) yr\(^{-1}\). This is the rate at which radiocarbon atoms are decaying to carbon 12.
For next time
Come prepared to discuss how near-exponential population growth will transition during the 21st century. Identify three countries where changes in demography are likely to collide with climate change. What are the possible outcomes and what can be done to head them off?
Here are the resources from last time.
Population growth
Population 7 Billion Nat Geo recognizes the landmark in 2022 (if this doesn’t work, access January 2011 through Duke Libraries)
2023: On to 8 BillionTHE IMPACT OF POPULATION BOOMS AND BUSTS from Nat Geo
How Do We Feed Nine Billion People from the LA Times
Drought
Drought monitor nationally What is the drought today?
Water wars
The brink of disaster CNN discusses drought and drinking water in the Southwest
[Colorado River crisis]https://www.washingtonpost.com/climate-environment/2023/02/05/colorado-river-drought-explained/)