Population-growth project

Consider a population of size \(N(t)\). Exponential growth assumes a constant per-capita growth rate \(r\):

\[\frac{dN}{dt}=rN,\qquad N(t)=N_0e^{rt}.\]

When resources limit growth, a logistic model introduces carrying capacity \(K\):

\[\frac{dN}{dt}=rN\left(1-\frac NK\right).\]

Worked example

With \(N_0=100\), \(r=0.2\) and \(K=1000\), growth is initially close to exponential but slows as \(N\) approaches \(K\).

Questions to investigate

Compare exponential and logistic trajectories, vary \(r\) and \(K\), and identify the biological assumptions responsible for the different long-term behaviour.

Project outcome. Connect a biological assumption about resource limitation to a change in the differential equation and its predicted population trajectory.