Logistic growth
Exponential growth assumes that the per-capita growth rate remains constant however large the population becomes. Logistic growth introduces a simple biological correction: as population density increases, growth becomes more difficult.
From exponential growth to logistic growth
Exponential growth uses
\[\frac{dN}{dt}=rN.\]To make growth density-dependent, multiply by
\[1-\frac{N}{K}.\]This gives the logistic equation:
\[\boxed{\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)}.\]What the symbols mean
| Symbol | Meaning |
|---|---|
| \(N(t)\) | population size at time \(t\) |
| \(r\) | intrinsic per-capita growth-rate parameter |
| \(K\) | carrying capacity in this model |
| \(1-N/K\) | density-dependent factor that reduces per-capita growth |
What does carrying capacity mean?
The parameter \(K\) is the population level at which births and deaths balance in this simplified model, so net population growth is zero.
It can represent the effect of limited food, space, nesting sites or other resources, but it should not always be interpreted as a fixed physical maximum. In real ecosystems, environmental conditions can change and therefore carrying capacity can change too.
Understand the density-dependent factor first
The key term is
\[1-\frac{N}{K}.\]| Population level | Density factor | Consequence |
|---|---|---|
| \(N\ll K\) | close to 1 | growth is approximately exponential |
| \(N=K/2\) | \(1/2\) | per-capita growth is half its low-density value |
| \(N=K\) | 0 | net growth is zero |
| \(N>K\) | negative | the model predicts population decline toward \(K\) |
Per-capita growth rate
Divide the logistic equation by \(N\):
\[\boxed{\frac{1}{N}\frac{dN}{dt}=r\left(1-\frac{N}{K}\right)}.\]Unlike exponential growth, the per-capita growth rate is no longer constant. It decreases linearly as population size increases.
Why total growth does not simply decrease
Total population growth is
\[G(N)=rN\left(1-\frac{N}{K}\right).\]At very small \(N\), there are few individuals reproducing, so total growth is small. Near \(K\), there are many individuals but strong density limitation, so total growth is again small.
Between these extremes, total growth reaches a maximum.
Maximum total growth occurs at \(K/2\)
Expand the growth function:
\[G(N)=rN-\frac{r}{K}N^2.\]Differentiate with respect to \(N\):
\[G'(N)=r-\frac{2r}{K}N.\]Setting \(G'(N)=0\) gives
\[\boxed{N=\frac{K}{2}}.\]At this population size,
\[G_{\max}=\frac{rK}{4}.\]The logistic population curve through time
For an initial population \(N_0>0\), the solution is
\[\boxed{N(t)=\frac{K}{1+\left(\frac{K-N_0}{N_0}\right)e^{-rt}}}.\]When \(0 The curve has three intuitive stages. At low density, competition is weak and the population accelerates. Around \(N=K/2\), total growth is greatest. Above \(K/2\), density dependence becomes strong enough that total growth begins to slow, even though the population is still increasing. An equilibrium occurs when For the logistic model, so Assume \(r>0\). IfWhy is the curve S-shaped?
Equilibria
Why is \(K\) stable?
then \(dN/dt<0\), so the population moves downward toward \(K\).
Therefore \(K\) attracts nearby positive population states and is a stable equilibrium.
What about \(N=0\)?
If \(N=0\), the deterministic model remains at zero. But for a small positive population, \(dN/dt>0\), so the population moves away from zero. Thus, for \(r>0\), zero is an unstable equilibrium.
What if the initial population is above \(K\)?
The logistic model does not require \(N_0
so \(dN/dt<0\). The population declines toward \(K\).
A numerical example
Suppose
\[N_0=100,\qquad K=1000,\qquad r=0.4\text{ per year}.\]Initially, the density factor is
\[1-\frac{100}{1000}=0.9.\]Therefore the initial total growth rate is
\[\frac{dN}{dt}=0.4(100)(0.9)=36\text{ individuals per year}.\]If instead \(N=900\), the density factor is only \(0.1\), and
\[\frac{dN}{dt}=0.4(900)(0.1)=36.\]The same total growth can occur on opposite sides of \(K/2\), but for different reasons: few individuals with weak competition versus many individuals with strong competition.
Exponential versus logistic growth
| Feature | Exponential | Logistic |
|---|---|---|
| per-capita growth | constant \(r\) | \(r(1-N/K)\) |
| density dependence | absent | present |
| long-term behaviour for \(r>0\) | unbounded growth | approaches \(K\) |
| low-density behaviour | exponential | approximately exponential |
Biological assumptions
The logistic model assumes a simple instantaneous relationship between population density and per-capita growth. Individuals are effectively treated as equivalent, and environmental conditions determining \(r\) and \(K\) are assumed constant.
Real populations may have age structure, time delays, migration, seasonal environments, spatial structure, stochastic fluctuations and interactions with other species.
Why logistic growth matters in mathematical biology
The logistic equation is one of the simplest ways to introduce negative density dependence. Its structure appears in population ecology, tumour-growth models, microbial growth approximations and many extensions involving competition, harvesting and stochastic population dynamics.
It is also a useful bridge from a simple linear model to nonlinear differential equations because the product \(N(1-N/K)\) makes the growth rate depend on the current state.