Exponential vs Logistic Growth Simulator

The graph plots population size N against time (0 to 50, in arbitrary units). Every run starts from a small population of 5 individuals at time 0. The controls are:

The box above the sliders has three readouts. model names the current model. dN/dt shows the growth equation in use: rN for exponential, rN(K−N)/K for logistic. max growth at reads N = K/2 for logistic, or — (no ceiling) for exponential.

On the graph, a grey dashed line marks K and is labelled with its value. In logistic mode a dotted teal line marks K/2 (fastest growth), which is where the S-curve is steepest. The y-axis tops out a little above K, so changing K rescales it.

In exponential mode the curve shoots past K and is cut off at the top of the plot, so it runs flat along the top edge. That flat stretch is only the edge of the graph. An exponential population never levels off.

Try r = 0.10 and then 0.50 in logistic mode. A larger r gets the population to K sooner, but it still levels off at the same K.

The AP Biology formula sheet gives both growth models, with the maximum per-capita growth rate written as $r_{max}$:

In the exponential model, $dN/dt$ grows in proportion to N, so the bigger the population, the faster it grows. This gives a J-shaped curve. In the logistic model, the factor $(K-N)/K$ slows growth as N rises, so the curve is S-shaped (sigmoid). The product $N(K-N)$ is largest when $N = K/2$, so a logistic population adds the most individuals per unit time at half the carrying capacity. Growth stops at K, where births and deaths balance.

Problem: A population of beetles has $r_{max} = 0.25$ per week and a carrying capacity of 300. Use the logistic model to find the population growth rate when N = 60, 150 and 270. Then find the growth rate at N = 150 if growth were exponential.

Step 1: N = 60. $\frac{dN}{dt} = 0.25 \times 60 \times \frac{300 - 60}{300} = 15 \times 0.8 = 12$ beetles per week.

Step 2: N = 150. $\frac{dN}{dt} = 0.25 \times 150 \times \frac{150}{300} = 37.5 \times 0.5 = 18.75$ beetles per week.

Step 3: N = 270. $\frac{dN}{dt} = 0.25 \times 270 \times \frac{30}{300} = 67.5 \times 0.1 = 6.75$ beetles per week.

Step 4: Exponential at N = 150. $\frac{dN}{dt} = 0.25 \times 150 = 37.5$ beetles per week, twice the logistic value. At $N = K/2$ the factor $(K-N)/K$ is 0.5.

Interpret: the growth rate goes up and then down (12, then 18.75, then 6.75). It peaks at N = 150, which is K/2. N itself never stops rising; only its growth rate falls.

Check it in the simulator: leave r = 0.25 and K = 300 (the defaults) on Logistic. The curve is steepest where it crosses the K/2 line at N = 150, at roughly time 16. Switch to Exponential with the same settings, and the curve passes 300 at about the same time and keeps climbing.

Population growth comes up in Unit 8 (Ecology), in the population ecology and density-dependence topics. Both equations are on the formula sheet. Expect questions that ask you to calculate dN/dt from given values, identify J-shaped and S-shaped curves, or explain why growth slows as resources run short. A free-response question may show a graph that levels off and ask you to estimate K and give a density-dependent reason for the slowdown.