Logistic model

The logistic model is a population growth model that includes resource limitation: per-capita growth slows down as the population approaches a maximum carrying capacity. Improves on the Malthusian model which predicts unbounded exponential growth.

The differential equation:

$\frac{dp}{dt}=k_{1}p-k_3\frac{p(p-1)}{2}$

The first term is Malthusian growth; the second is a competition term proportional to the number of pairwise interactions $\left(\genfrac{}{}{0ex}{}{p}{2}\right)=\frac{p(p-1)}{2}$.

Standard form

Rearranging:

$\frac{dp}{dt}=-\frac{k_3}{2}\left[p^2-\left(1+\frac{2k_1}{k_3}\right)p\right]=-Ap(p-p_1)$

where $A=k_3/2$ and $p_1=1+2k_1/k_3$ is the carrying capacity (the equilibrium where growth stops).

Often written more cleanly as:

$\frac{dp}{dt}=rp\left(1-\frac{p}{K}\right)$

with $r$ the intrinsic growth rate and $K$ the carrying capacity. Same equation, different parameterization.

Equilibria

The logistic equation has two equilibria, values where $dp/dt=0$:

• $p=0$ (no population; unstable for $r>0$: any small positive perturbation grows away from zero).
• $p=K$ (carrying capacity; asymptotically stable).

Between them, the population grows:

• If $0<p<K$: $\frac{dp}{dt}>0$, population grows toward $K$.
• If $p>K$: $\frac{dp}{dt}<0$, population shrinks back toward $K$.

So $p=K$ is an asymptotically stable equilibrium: populations starting near $K$ converge to $K$. Meanwhile $p=0$ is unstable for $r>0$, any small perturbation grows.

For a worked example with $\frac{dp}{dt}=p(2-p)$: equilibria at $p=0,2$. The arrows on the slope field point upward in $0<p<2$, downward for $p>2$.

Solution

The logistic ODE is separable. Solving:

$p(t)=\frac{K}{1+\left(\frac{K}{p_0}-1\right)e^{-rt}}$

Behavior:

• Starts at $p_0$ when $t=0$.
• Approaches $K$ as $t\to \infty$.
• The graph is the S-curve (sigmoidal): slow start, accelerating middle, decelerating approach to capacity.

Where it’s used

• Population biology: growth in bounded habitats.
• Epidemiology: the SIR model uses logistic-style equations.
• Tumor growth: cancer cells proliferate logistically rather than exponentially.
• Adoption of technology: S-curve diffusion of innovations.
• Neural networks: the logistic function $\sigma (x)=1/(1+e^{-x})$ is the same shape, used as an activation function.

The S-curve is the generic shape for any process with self-limiting growth, so it shows up wherever growth is bounded.

Why it’s better than Malthusian

The Malthusian model says: growth rate is constant, population explodes forever. Reality: growth slows as resources run out. The logistic model captures this with one extra parameter ($K$) and matches real-world data far better for medium-to-long-term predictions. Drop the competition term (early-stage growth) and you’re back to the Malthusian model.