8 Deeper into Logistic Growth

8.1 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.

8.1.1 Linking Logistic and Exponential Growth Models

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

$\frac{dN}{dt}=r_m N\left(1-\frac{N}{K}\right)$

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:

$\frac{dN}{dt}=r_m N\left(1-\frac{N}{\infty}\right)$

which simplifies to

$\frac{dN}{dt}=r_m N\left(1-0\right)$

which simplifies to

$\frac{dN}{dt}= r_m N$

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

The Excel workbook uses Neal’s discrete-time versions of these models, in which the population is updated once per time step: $N_{t+1} = N_{t} + r_{m} N_{t}\left(1 - \frac{N_{t}}{K}\right)$ for logistic growth, and $N_{t+1} = N_{t} + r_{m} N_{t}$ for exponential growth. These use the same $r_m$ as the continuous equations, but apply it once per whole time step, so $dN/dt$ is approximated by the change in population size over one time step, $N_{t+1} - N_{t}$. The same simplification applies: setting $K = \infty$ turns the discrete logistic model into the discrete exponential model.

The discrete models are approximations of the continuous ones. For a small population, the discrete model multiplies the population by $1 + r_m$ each time step, whereas the continuous model gives $\lambda = e^{r_m}$ (e.g. $1.8$ vs. $2.23$ for $r_m = 0.8$). The approximation is good when $r_m$ is small and poor when it is large. Stepping forward a whole time unit at a time is what makes the overshoot, oscillations, cycles and chaos you will see in Task 1 possible. Because the population cannot “see” that it is approaching $K$ until the next time step, it can overshoot. The continuous logistic model never does this: it always approaches $K$ smoothly.

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.

8.2 Worked example

8.2.1 Inputs

r_m = 1.2
K = 200
Worksheet: BasicLogistic

8.2.2 Steps

Set r_m and K in the parameter block.
Observe Figure 1 (population through time).
Observe Figure 2 (per-capita growth vs N).
Observe Figure 3 (total growth vs N).

8.2.3 Output and interpretation

In Figure 1 the population rises quickly and settles at $K$, overshooting it only very slightly (to about 201) before levelling off. Figures 2 and 3 show the same model from two other perspectives. You will explore what their shapes and axis intercepts tell you in Tasks 2 and 3.

8.3 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)

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.
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)

Examine Figure 2, which shows the per capita growth rate as a function of population size at time $t$.
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?
Try varying the values for $r_m$ and $K$, and note how the graph changes.
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)

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.
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

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 ($N$)

Look at the Exponential worksheet to see how close you were.

Task 5: Adding a Time Lag

In the TimeLag worksheet, explore how adding a time lag to the logistic model affects population dynamics. This is the equation we are using: $\frac{dN}{dt}=r_m N_t\left(1-\frac{N_{t-\tau}}{K}\right)$, where $\tau$ is the length of the time lag. With $\tau = 1$, density dependence acts on the population size one time step ago rather than the current population size.
Adjust the formula in the Excel sheet to incorporate a time lag in the population size ($N_{t-\tau}$). The dN/dt column (column C) currently uses the current population size. For a 1-step lag, change the formula in cell C11 to =$B$5*B11*(1-(B10/$B$6)), so that the density-dependent term uses the population size from the previous time step. Then fill this formula down to the bottom of the column.
Start with the default $r_m = 0.6$, which gives smooth convergence to $K$ in the ordinary logistic model. With the 1-step lag, you should see the population overshoot $K$ and then show damped oscillations before settling. Now increase $r_m$ to 1.2. What happens to the oscillations?
Compare this with the ordinary logistic model (no lag) at the same $r_m$ values (e.g. using the BasicLogistic worksheet). This exercise demonstrates how a simple life history trait (such as a time lag) can make a population overshoot and oscillate at growth rates that would otherwise give smooth convergence to $K$. With a large enough lag or growth rate, the oscillations become sustained cycles.

8.4 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?

The peak of the parabola in Figure 3 gives the maximum sustainable yield (MSY): the largest number of individuals that could be harvested each time step without the population declining. What is the MSY for a population with $r_m = 0.8$ and $K = 200$, and at what population size should the population be kept to achieve it? Why might harvesting at exactly the MSY be risky in practice?

In the discrete model, $N_{t+1} = N_t + dN/dt$. Using what you know about Figure 3, what shape would a plot of $N_{t+1}$ against $N_t$ have? Where would it cross the line $N_{t+1} = N_t$, and why?

What happens to the population dynamics when a time lag is introduced? A time lag could be caused by a long gestation period, or by the need to take a year out before breeding again.

8.5 Takeaways

Logistic and exponential growth are linked by the carrying capacity term.
Figures of $N$ vs. time, per-capita growth rate vs. $N$ and $dN/dt$ vs. $N$ are three views of the same model; you should be able to move between them.
In discrete time, high $r_m$ causes overshoot, oscillations, cycles and chaos.
Time lags can destabilize otherwise stable dynamics.