Population Growth Curves Simulation

About this experiment

Compare two populations that start with the same size and intrinsic growth rate. The exponential model assumes unlimited resources. The logistic model reduces per-capita growth as the population approaches a fixed carrying capacity. Scrub through time or play the curves, then test what happens when the initial population already exceeds capacity.

Learning goals

Compare exponential growth with density-dependent logistic growth
Interpret carrying capacity and the sign of population growth
Relate initial conditions and growth rate to the shape of the population curve

How to use it

1Play the default comparison and watch the logistic curve flatten as it approaches capacity.
2Restart time and lower the intrinsic growth rate. Compare the population at day 10.
3Set initial population to 800 and capacity to 500. Check the sign of logistic growth.
4Set initial population equal to capacity and explain why the logistic population remains constant.

The model

\frac{d N}{d t} = r N, N (t) = N_{0} e^{r t}

Exponential model with constant per-capita growth

\frac{d N}{d t} = r N (1 - \frac{N}{K})

Logistic growth with carrying capacity K

N (t) = \frac{K}{1 + ( K / N _{0} - 1 ) e ^{- r t}}

Closed-form logistic solution for positive initial population

Assumptions

Population is treated as a continuous quantity, so fractional values represent model expectations rather than partial organisms.
Growth rate and carrying capacity stay constant. There is no migration, seasonal forcing, age structure or stochastic variation.
Resource limitation acts immediately, with no delay or overshoot. Changing a parameter starts a new trajectory from time zero.

Try this in class

Q1At what population does logistic growth reach its maximum positive rate?
Q2Why does a population initially above capacity decline toward capacity?
Q3Does the logistic population reach carrying capacity in finite time in this ideal model?

Frequently asked questions

Why does the exponential line leave the graph?

Exponential growth is unbounded. The vertical scale is kept near carrying capacity so the logistic dynamics remain visible; the exponential readout continues to show its calculated value.

Can the logistic model predict every real population?

No. Real populations have variable environments, delays, interactions and random events. This model isolates the effect of an immediate density-dependent growth rate.