5 Geometric growth

5.1 Background

The geometric growth model is a simple model of unconstrained population growth. In other words, the population can grow to infinite size and is not affected by competition (population density). In the model, the rate at which a population grows is described by the population growth rate, often represented by the symbol lambda ($\lambda$ ). Higher values of $\lambda$ indicate rapid population growth, while smaller values suggest slow growth. If $\lambda < 1$, the population shrinks in size.

Geometric growth is often used interchangeably with exponential growth, but there is an important distinction. Exponential growth pertains to “continuous time” scenarios, whereas geometric growth involves models where the population changes in discrete time steps, such as yearly intervals.

The key quantities involved in this model are as follows, although it’s important to note that different books or sources may use varying symbols:

$N_t$ = population size at time $t$; and $N_0$, $N_1$ is population size at time 0, time 1 etc.
$\lambda$ = population growth (multiplication) rate over a single time step, $\Delta t$. THis is sometimes called the finite rate of increase.
$R_m$ = population rate of increase over a single time step, $\Delta t$. This is sometimes called the discrete growth factor.
$r_m$ = individual rate of increase over an infinitesimally small time step. This is sometimes called the instantaneous rate of increase or intrinsic rate of increase

$R_m$ is is related to $\lambda$ such that $R_m = \lambda - 1$ (and $\lambda = R_m + 1$).

The relevant equations can be seen in Box 4.1 of the Neal textbook but here we will use the following:

$N_{t+1} = N_t + (R_m N_t)$,

which leads to,

$N_{t+1} = N_t (1 + R_m)$ and, equivalently, $N_{t+1} = N_t \lambda$.

Learning outcomes:

Competence in using Excel formulae for mathematical modeling.
Understanding the the parameters of exponential/geometric growth.
Competence in using mathematical models in Excel to strengthen own understanding of biological processes.
Awareness of rearranging of mathematical formulae to produce different forms of models.
Knowing that the slope of the $ln(N_t)$ vs. $t$ relationship can tell you about population growth rate (it is $ln(\lambda)$).

5.2 Your task

Download and open the Excel file GeometricGrowth.xlsx.

Starting with an initial population size (N) of 10 [at time (t) =0], and with a $\lambda$ of 1.1, use Excel’s equation functions to work out the population size from t=1 through to t=20. E.g. The formula might look something like this “=B8*$F$8”.
Use charts to plot the results. On the horizontal axis (x-axis) you should have time, and on the vertical axis (y-axis) you should have population size (N). A scatterplot would be best for this.
Make another plot of the same data, but this time use a natural logarithmic axis for the population size. The formula in Excel for natural log is LN().
What do you notice about the curves in these different versions of plotting the same data?
Try altering the population growth rate. Try values of 0.8, 1 and 1.2 for example. What happens to the curves? What is special about the value of 1?
Add an trendline with an equation to the log transformed graph. Identify the slope. The slope is $ln(\lambda)$), so you can work out $\lambda$ by “reversing” the $ln$ by taking its exponential. e.g. =EXP(0.0953) in Excel.

Now reset the population growth rate to 1.1. Now lets see if the mathematical rules of so-called “geometric series” work…

If the population starts at t=0, $N_t = N_0 \lambda^t$ . In words, the population at time $t$ is the initial population size multiplied by the finite population growth rate raised to the power of $t$.
Your instructor will explain how this works on the blackboard.
Following these rules, the population size at time $t=5$ ($N_5$), is $10 * 1.1^5$, or $10* 1.61051$, which is $16.1051$. Check that this matches what you got earlier.
This approach can be really useful because it can save lots of time. For example, what is the population after 900 generations? It would be tedious to work this out using the first, “simple” approach, but very easy if you use the geometric series rules and have access to a calculator/computer.

5.3 Questions

In geometric growth, what does a constant growth rate imply for the population’s future growth? Explain with an example.
What makes the geometric growth model “unrealistic” for real-world scenarios? Explain your reasoning.
Give an example of real-world populations where geometric growth might be a suitable model. Explain your reasoning.
Suppose a population’s size decreases by 25% every year due to environmental factors. If there are initially 800 individuals, determine the population size after 4 years.