9 Deeper into Logistic Growth

9.1 Background

The logistic and exponential growth models are closely related and serve as fundamental tools in understanding population dynamics.

  • The exponential (geometric) growth model assumes that populations grow without any environmental constraints, leading to unlimited growth.
  • The logistic growth model, on the other hand, incorporates environmental limits, specifically the carrying capacity (\(K\)), which represents the maximum number of individuals the environment can support.

9.1.1 Linking Logistic and Exponential Growth Models

The logistic growth model is represented by the following equation (Eqn. 5.2 in Neal):

\(\frac{\delta N}{\delta t}=r_m N\left(1-\frac{N}{K}\right)\)

In this model, as the population size \(N\) approaches the carrying capacity \(K\), the term \(\left(1 - \frac{N}{K}\right)\) decreases, slowing the population growth rate until it reaches zero when \(N = K\).

However, if we remove the constraint of a carrying capacity (i.e., set \(K = \infty\)), the model simplifies to the exponential growth equation:

\(\frac{\delta N}{\delta t}=r_m N\left(1-\frac{N}{\infty}\right)\)

which simplifies to

\(\frac{\delta N}{\delta t}=r_m N\left(1-0\right)\)

which simplifies to

\(\frac{\delta N}{\delta t}= r_m N\)

This is the familiar geometric (exponential) growth equation (Eqn. 4.6 in Neal), which assumes unlimited resources and continuous population growth.

Take-home Message: The logistic and exponential growth models are closely related. By setting \(K = \infty\), we transition from the logistic model to the exponential model.

Learning outcomes:

  • Increase competence in using Excel for mathematical modeling.
  • Understand the relationship between exponential and logistic growth models.
  • Learn how models can be adjusted to explore different biological phenomena.
  • Develop skills in visualising and interpreting model outputs from different perspectives.
  • Strengthen understanding of biological processes by applying mathematical models.

9.2 Your Task

Download the Excel file Deeper Into Logistic Growth.xlsx.

Look at the BasicLogistic worksheet and work through the following tasks:

Task 1: Population Dynamics (Figure 1)

  1. Experiment with different values of \(r_m\) (e.g., 0.8, 1.2, 1.8, 2.4, 2.7) and observe how the population dynamics change over time in Figure 1.

  2. Use the following terms to describe the dynamics you see:

    • Oscillation, damped oscillation, stable cycle, 2-point cycle, chaotic, unpredictable, predictable.

Task 2: Per Capita Growth Rate vs. Population Size (Figure 2)

  1. Examine Figure 2, which shows the per capita growth rate as a function of population size at time \(t\).

  2. Notice where the line intercepts the x- and y-axes.

    • What are these intercepts?
    • How do these intercepts relate to the values of \(r_m\) and \(K\) that you have set?

  3. Try varying the values for \(r_m\) and \(K\), and note how the graph changes.

  4. On paper, sketch a graph of the per capita growth rate vs. population size for a logistic model with \(r_m = 1.5\) and \(K = 250\). Then, verify your sketch by entering these values into the Excel model.

Task 3: Population Growth Rate vs. Population Size (Figure 3)

  1. Figure 3 shows the overall population growth rate (\(dN/dt\)) — the change in population size per unit time. Adjust the values for \(r_m\) and \(K\) and observe how Figure 3 changes.

  2. Answer the following questions:

    • At what population sizes is the population growth rate 0 (\(dN/dt = 0\))?
    • At what population size is the growth rate maximized?

Task 4: Comparison with Exponential Growth

  1. Now, consider the differences between logistic growth and exponential growth. How would Figures 1, 2, and 3 change when considering exponential growth?

Sketch equivalent graphs for the exponential (geometric) growth model, including:

  • Fig 1: Population size (\(N\)) over time (\(t\)).
  • Fig 2: Per capita growth rate (\(\frac{1}{N} \frac{dN}{dt}\)) vs. population size (\(N\)).
  • Fig 3: Population growth rate (\(\frac{dN}{dt}\)) vs. population size (\(N\))
  1. Look at the Exponential worksheet to see how close you were.

Task 5: Adding a Time Lag

  1. In the TimeLag worksheet, explore how adding a time lag to the logistic model affects population dynamics. This is the equation we are are using: \(\frac{d N}{d t}=r N\left(1-\frac{N_{t-\tau}}{K}\right)\)

  2. Adjust the formula in the Excel sheet to incorporate a time lag in the population size (\(N_{t-\tau}\)). Start with a small \(r_m\) value that results in smooth convergence to \(K\) in the ordinary logistic model.

  3. Add a 1-year time lag and observe how this introduces cycling in the population dynamics. This exercise demonstrates how a simple life history trait (such as a time lag) can introduce complex dynamics, even when population growth rates are low.

9.3 Questions

  • How does increasing or decreasing \(r_m\) affect the shape and behaviour of the population time series in Figure 1? How does it change the per capita growth rate curve in Figure 2 and the population growth rate in Figure 3?
  • What happens to the population dynamics when a time lag is introduced?