Malthusian model

The Malthusian model is the simplest population growth model: the rate of change of the population is proportional to the population itself.

$\frac{dp}{dt}=kp$

where $p(t)$ is the population at time $t$ and $k$ is a constant proportionality factor. If $k>0$, the population grows; if $k<0$, it decays.

Named for Thomas Malthus (1798), who used it to argue that populations grow geometrically while food supplies grow only arithmetically — predicting (incorrectly, as it turned out) inevitable famine.

Solution

The ODE is separable:

$\frac{1}{p}dp=kdt$

Integrate:

$\ln |p|=kt+C$

Exponentiate:

$p(t)=p_0{e}^{kt}$

where $p_0=p(0)$ is the initial population.

The solution is exponential growth (when $k>0$) or exponential decay ($k<0$). The doubling time (when growing) is $\ln 2/k$; the half-life (when decaying) is the same value.

Derivation from rates

Start from the conceptual decomposition: $\frac{dp}{dt}=\text{(growth rate)}-\text{(death rate)}$. If both are proportional to $p$:

$\frac{dp}{dt}=k_{1}p-k_{2}p=(k_1-k_2)p=kp$

with $k=k_1-k_2$. Per-capita growth rate $k$ is constant, independent of population size — that’s the Malthusian assumption.

Why it’s wrong (limitations)

The model breaks down at large populations. Real populations face:

• Resource limits: more individuals → less food per individual → lower per-capita growth rate.
• Crowding effects: disease spreads faster, predators concentrate, competition intensifies.
• Carrying capacity: the environment can sustain only so many.

These effects mean the per-capita growth rate decreases with population size, not stays constant. The Malthusian model overpredicts long-term growth.

Where it’s still useful

For early growth (when the population is small relative to environmental limits), the Malthusian model is an excellent approximation. Used for:

• Bacterial cultures in their exponential phase before nutrients run out.
• Cancer cells in early proliferation.
• Compound interest: a bank balance grows Malthusian-style with continuous compounding.
• Radioactive decay: $\frac{dN}{dt}=-\lambda N$, exactly Malthusian with $k=-\lambda$.

For more realistic population modeling, see Logistic model — adds a competition term to limit growth.

Generalization

A generalized Malthusian allows time-dependent rates: $\frac{dp}{dt}=k(t)p$. Solution by Integrating factor or directly:

$p(t)=p_0\exp \left({\int }_0^{t}k(s)ds\right)$

This handles seasonal birth rates, time-varying decay rates, etc.

For a model with two interacting species (predator-prey style), see Lotka-Volterra equations (not yet in this vault).