5 Estimating Population Growth Rate
5.1 Background
In population biology, one important measure of population dynamics is the population growth rate (\(\lambda\)), called the multiplication rate in Neal and also known as the finite rate of increase.
The finite rate of increase (\(\lambda\)) is defined as the ratio of population size at time t + 1 to that at time t (\(N_{t+1}/N_t\)), and it is related to the intrinsic rate of increase \(r_m\) by the expression \(r_m = \ln \lambda\) (so \(\lambda = e^{r_m}\); see the notation box in the Part 2 introduction).
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 \(\ln(\lambda) = r_m\) and can be used to estimate population growth rate.
5.2 Worked example
5.3 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.
- Danish Excel:
LNis the same,EXPisEKSPandSLOPEisHÆLDNING; decimals use a comma and arguments are separated by;. The trendline menu items are “Tilføj tendenslinje” and “Vis ligning i diagram”.
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 (\(\ln(N_t)\)) 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 \(\ln(\lambda) = r_m\). The trendline label rounds the slope; for the exact value use
=SLOPE(ln-column, year-column)(Danish:=HÆLDNING(ln-kolonne; år-kolonne)). Strictly, the slope is the realised average growth rate over the census years; it equals the intrinsic rate \(r_m\) only if growth was not limited by density during that time.
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)
5.4 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 \(\lambda\).