Logistic growth
Logistic growth extends exponential growth by including a reduction in per-capita growth as population size increases.
The logistic equation
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).\]The parameter \(r\) is the intrinsic growth rate and \(K\) is the carrying capacity.
Meaning of the density term
When \(N\ll K\), the factor \(1-N/K\) is close to 1, so growth is approximately exponential. As \(N\) approaches \(K\), this factor approaches 0 and growth slows.
Equilibria
Setting \(dN/dt=0\) gives
\[N^*=0,\qquad N^*=K.\]For \(r>0\), \(N=K\) is the stable positive equilibrium.
Maximum total growth. The quantity \(rN(1-N/K)\) is largest at \[N=\frac K2.\]
Key idea. Logistic growth represents negative density dependence: population growth slows as population size approaches carrying capacity.