Population growth patterns and carrying capacity

1How fast can a population grow?

A single yeast cell dropped into warm sugar solution divides roughly every one and a half to two hours. After one day, that single cell could have become several thousand cells; after four days, over a hundred million million; and if nothing stopped it, within two weeks its descendants would outweigh the Earth. That never happens. Something always stops growth. This lesson builds the two mathematical models that ecologists use to describe growth, and uses them on yeast in a flask and on Rwanda's own population.

2The exponential model

When resources are unlimited, each individual adds offspring at the same rate however many there are, so the rate of growth is proportional to the population size:

$$\frac{dN}{dt}=rN\Rightarrow N_t=N_0{e}^{rt}\backslash frac\{dN\}\{dt\}\; =\; rN\; \backslash qquad\; \backslash Rightarrow\; \backslash qquad\; N\_t\; =\; N\_0\backslash ,e^\{rt\}$$

where $N_{0}N\_0$ is the starting population and $N_{t}N\_t$ the population after time $tt$. The population doubles in a fixed time, the doubling time:

$$t_d=\frac{\ln 2}{r}\approx \frac{0.693}{r}t\_d\; =\; \backslash frac\{\backslash ln\; 2\}\{r\}\; \backslash approx\; \backslash frac\{0.693\}\{r\}$$

If you take natural logarithms, $\ln N_t=\ln N_0+rt\backslash ln\; N\_t\; =\; \backslash ln\; N\_0\; +\; rt$: a straight line of slope $rr$. This is why growth data are often plotted on a logarithmic scale, and why the exponential phase is also called the log phase.

3The logistic model

Real populations meet environmental resistance. The logistic model multiplies the exponential rate by the fraction of the carrying capacity that is still unused:

$$\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)\Rightarrow N_t=\frac{K}{1+\left(\frac{K-N_0}{N_0}\right)e^{-rt}}\backslash frac\{dN\}\{dt\}\; =\; rN\; \backslash left(\; \backslash frac\{K\; -\; N\}\{K\}\; \backslash right)\; \backslash qquad\; \backslash Rightarrow\; \backslash qquad\; N\_t\; =\; \backslash frac\{K\}\{1\; +\; \backslash left(\; \backslash frac\{K\; -\; N\_0\}\{N\_0\}\; \backslash right)\; e^\{-rt\}\}$$

When $NN$ is small, $\frac{K-N}{K}\approx 1\backslash frac\{K\; -\; N\}\{K\}\; \backslash approx\; 1$ and growth is almost exponential. As $NN$ approaches $KK$, the fraction approaches 0 and growth stops. Figure 1 draws both models from the same start.

Figure 1: Exponential (J-shaped) and logistic (S-shaped) growth, computed with N₀ = 10, r = 0.5 per day and K = 1000. Shaded: environmental resistance.

4Investigation: growth of a yeast population

Aim: to follow the growth of a yeast population in a closed culture.

Method. Add a small, known amount of dried baker's yeast to 100 cm³ of 5 % glucose solution in a conical flask plugged with cotton wool, and keep it at $25{}^{\circ }\mathrm{C}25\backslash ,^\backslash circ\backslash mathrm\{C\}$. Every 6 hours, swirl the flask, remove a small sample with a sterile pipette, dilute it if it is crowded, and count the cells on a haemocytometer: a thick glass slide with a counting grid of known volume. Count the cells in several squares and take the mean.

Control variables: temperature, the volume and concentration of glucose, and the volume of each sample. Stain with methylene blue to tell living cells (colourless, because they break down the dye) from dead ones (blue).

Safety. Baker's yeast is harmless, but treat every culture as if it might be contaminated: wash your hands, disinfect the bench, never pipette by mouth, and put used cultures in the container your teacher provides for sterilising before disposal.

Figure 2: Yeast in a closed culture (illustrative data), with the four phases shaded.

Figure 2 shows four phases. In the lag phase, the cells are adjusting: making enzymes to use the glucose. In the exponential phase, resources are plentiful and each cell buds repeatedly. In the stationary phase, the birth rate equals the death rate as glucose runs short and ethanol builds up. In a closed flask, nothing removes the waste or adds food, so a death phase follows. A natural population in an open ecosystem, where resources are renewed, tends to stay near $KK$ instead.

5Rwanda's population: which model fits?

Figure 3: Rwanda's population at each census, 1978–2022, and NISR's medium projection for 2052. Data: NISR, Fifth Population and Housing Census, 2022.

Figure 3 shows growth that is neither a clean J nor a clean S. Growth was fast from 1978 to 1991, slowed sharply between 1991 and 2002, the period that included the 1994 Genocide against the Tutsi, and was fast again from 2002. NISR's medium projection, 23.6 million in 2052, assumes that growth will slow as fertility continues to fall. Human populations can raise their own carrying capacity by technology, such as better crop varieties, fertiliser, irrigation and trade, but land and water set limits (Lesson 4).