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 realistic long-term modeling the Logistic model adds a competition term to cap 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.