6 Estimating Population Growth Rate

6.1 Background

In population biology, one important measure of population dynamics is the population growth rate (\(\lambda\)). This rate tells us how fast a population is growing or shrinking over time. When \(\lambda > 1\), the population grows exponentially; when \(\lambda = 1\), it stays constant, and when \(\lambda < 1\), the population declines.

In this exercise, you will estimate the population growth rate (\(\lambda\)) from real population data. The method involves plotting the population size over time, applying a log transformation to linearise the data, and fitting a linear regression model to estimate \(\lambda\). This method is widely used in ecology to analyse growth trends and make predictions.

Learning outcomes:

  • Competence in using Excel formulae for data transformation and regression analysis.
  • Understanding the role of \(\lambda\) in population growth and its estimation through log-transformed data.
  • Competence in applying mathematical models in Excel to analyse real biological data.
  • Awareness of how log transformations can linearise exponential growth data for easier interpretation.
  • Knowing that the slope of the \(ln(N)\) vs. time relationship represents \(ln(\lambda)\) and can be used to estimate population growth rate.

6.2 Your task

Step 1: Download and Open the Data

  • Download the provided Excel file: EstimatingGrowth.xlsx.
  • Open the file in Excel to view the population data for a species recorded annually over a 25-year period.

Step 2: Plot the Population Size Over Time

  • In Excel, create an x-y scatter plot of the population size (\(N_t\)) on the y-axis and time (Year) on the x-axis.

Step 3: Log-Transform the Population Size

  • Add a new column in Excel for the natural logarithm of the population size using the formula =LN(cell). This transformation helps linearise the exponential growth data.
  • Create a new scatter plot with log-transformed population size (\(\log_e(N_t)\)) on the y-axis and time (Year) on the x-axis.

Step 4: Fit a Linear Regression Model

  • In the log-transformed scatter plot, add a trendline by right-clicking on the data points and selecting “Add Trendline.”
  • Choose “Linear” and ensure you check the box to “Display Equation on Chart.”
  • The slope of the trendline represents \(\log(\lambda)\).

Step 5: Calculate \(\lambda\)

  • Use the slope from the regression equation to calculate \(\lambda\) with the formula: \(\lambda = e^{\text{slope}}\). In Excel, you can do this with the equation =EXP(cell)

6.3 Questions

  • What does the log-transformed plot of population size over time tell you about the population’s growth trend? Does the population appear to grow exponentially?
  • What is the estimated population growth rate (\(\lambda\)) based on your linear regression analysis?
  • What assumptions does this model make about population growth? Discuss any potential real-world factors that might affect the accuracy of your estimate for \(\lambda\).