4 Geometric growth
4.1 Introduction
This exercise explores the geometric growth model in discrete time steps, focusing on how different values of \(\lambda\) affect population dynamics. It aims to develop students’ understanding of growth models, the effects of log transformations, and the limitations of simple models.
4.2 Key Concepts
- Geometric growth assumes constant \(\lambda\), leading to exponential growth or decline.
- Plotting log-transformed population sizes allows easier visualisation of trends.
- Limitations of the model include the assumption of infinite resources and lack of environmental constraints.
4.3 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 \(ln(\lambda)\)).
4.5 Instructions for Facilitating
4.5.1 Excel Setup
- Download File: Ensure students have downloaded the provided Excel file
GeometricGrowth.xlsx. - Initial Setup:
- Start with an initial population size (\(N_0\)) of 10 and a growth rate (\(\lambda\)) of 1.1.
- Use Excel formulas to calculate population size at each time step from \(t = 1\) to \(t = 20\).
- Example formula:
=B8*$F$8, whereB8is the population at timetand$F$8is the growth rate \(\lambda\).
- Creating Charts:
- Plot the population size against time using an x-y scatterplot.
- Create a second plot using the natural logarithm of population size (
LN()) on the y-axis.
4.5.2 Key Observations
- Plot Comparison: The linear plot will show exponential (curved) growth, while the log-transformed plot should display a straight line, where the slope represents \(ln(\lambda)\).
- Altering \(\lambda\): As students experiment with different values for \(\lambda\) (e.g., 0.8, 1, 1.2), they will observe changes in the steepness of the population growth curve. A \(\lambda\) of 1 results in no population growth, while \(\lambda\) values below 1 indicate population decline.
- Trendline Slope: Adding a trendline to the log-transformed graph and determining its slope will give \(ln(\lambda)\). Students can use this slope to compute \(\lambda\) using the Excel function
EXP().
4.5.3 Geometric Series Calculation
- Geometric Series Formula: Introduce students to the geometric series formula, \(N_t = N_0 \lambda^t\). This formula provides a shortcut to calculate population size at any given time without stepping through each time point manually.
- Application: Use this formula to calculate population size for \(t = 5\) and compare it to earlier manual calculations. Then extend it to predict population size after a large number of generations (e.g., \(t = 900\)).
- Where does that equation come from?
As a starting point, consider this equation: \(N_{t+1} = \lambda N_t\).
If we want to calculate \(N_{t+2}\), we would need to plug in \(N_{t+1}\) instead of \(N_t\): \(N_{t+2} = \lambda N_{t+1}\),
and, since we know that \(N_{t+1} = \lambda N_t\),: \(N_{t+2} = \lambda \lambda N_t\).
Similarly, if we wanted to calculate \(N_{t+2}\), we’d end up with: \(N_{t+3} = \lambda \lambda \lambda N_t\).
This can be simplified by raising \(\lambda\) to a suitable power, and using the starting population at time = 0, \(N_0\):
eqn. 9. \(N_{t} = \lambda ^tN_0\).
This should be familiar to those of you that did (or remember!) the concept of geometric series which you have likely covered in earlier maths classes.
4.6 Questions & Model Answers
1. How does the population size change with time for different values of \(\lambda\)?
The growth rate \(\lambda\) drives the population’s exponential increase or decrease.
- For \(\lambda > 1\), the population size increases exponentially.
- For \(\lambda = 1\), the population size remains constant.
- For \(\lambda < 1\), the population size decreases exponentially.
2. Why is it useful to plot log-transformed population size?
Log-transformations linearise exponential relationships. Therefore, the transformation converts the exponential growth trajectory into a straight line allowing for easier interpretation of the growth rate, as the slope corresponds to \(\log(\lambda)\). You can get the \(\lambda\) value by taking the exponential of \(\log(\lambda)\) (in Excel =EXP()).
3. What are the limitations of the geometric growth model? - The model assumes infinite resources, no competition, or environmental constraints, making it unrealistic for real-world populations. Real-world populations face density-dependent factors and environmental stochasticity, which the geometric model does not account for.
4. What would happen if we introduced a carrying capacity to this model? - The population would follow a logistic growth model, where growth slows as the population nears the carrying capacity.
4.7 Teaching Tips:
- Reinforce Mathematical Understanding: Emphasize how rearranging and transforming formulas, such as logarithmic transformations, are useful tools for interpreting population growth models.
- Visual Learning: Encourage students to compare the linear and log-transformed plots side by side to see how exponential growth behaves differently in each case.
- Connect to Ecology: When discussing real-world applications, introduce concepts like carrying capacity and logistic growth to contrast with geometric growth. -. Model limitations: Engage students in discussions about how the model could be adapted to account for real-world variables like environmental factors or density dependence.