How Differential Equations Model Everything from Populations to Physics

Have you ever wondered why falling objects accelerate or why population growth depends on its size? The answer lies in differential equations—a powerful mathematical tool that captures continuous change in the real world.

Population Models

Malthusian Population Model

The classic Malthusian model assumes that the growth rate of a population is proportional to its current size, expressed by a simple first‑order differential equation. This model, introduced by Thomas Malthus in 1798, ignores factors such as immigration.

Logistic Growth Model

The logistic model incorporates environmental carrying capacity, slowing growth as the population approaches a maximum limit.

Immigration Model

This variant adds terms for incoming and outgoing individuals to the basic growth equation.

Decay Models

Radioactive Decay

Radioactive decay follows a differential equation analogous to population decline, with a negative constant representing the decay rate.

Serial Decay (Decay Chain)

When a decay product is itself unstable, a chain of first‑order equations describes the successive transformations.

Parallel Decay

A single nuclide may decay via multiple pathways, each with its own decay constant.

Newton's Cooling/Heating Law

The rate of temperature change of an object is proportional to the difference between the object's temperature and the ambient temperature.

Infectious Disease Models

SIS Model

The SIS model includes only Susceptible (S) and Infected (I) compartments, allowing recovered individuals to become susceptible again.

SIR Model

The SIR model adds a Recovered (R) compartment. Its three differential equations describe the flow between S, I, and R.

S : Number of susceptible individuals.

I : Number of currently infected individuals.

R : Number of recovered or removed individuals; the total S+I+R remains constant.

SEIR Model

The SEIR model introduces an Exposed (E) compartment for individuals who are infected but not yet infectious.

SIRD Model

The SIRD model separates the removed compartment into Recovered and Dead, accounting for mortality.

SVIR Model

The SVIR model incorporates vaccination, adding a Vaccinated (V) compartment.

Chemical Reaction Kinetics

First‑order reactions follow a differential equation where the rate is proportional to the amount of reactant remaining; second‑order reactions involve a different proportionality.

Catenary Curve

The shape of a flexible cable hanging between two supports is described by a first‑order differential equation relating the cable profile to the vertical load and tension.

Newton's Second Law

For a freely falling object, Newton's second law yields a second‑order differential equation linking position, mass, and gravitational acceleration.

Lorenz Equations (Chaotic System)

The Lorenz system consists of three coupled, nonlinear first‑order differential equations that model atmospheric convection and exhibit chaotic behavior.

Lotka‑Volterra Competition Model

The Lotka‑Volterra equations describe competition between two species, with parameters for intrinsic growth rates, inter‑species inhibition, promotion, and mortality.

Conclusion

The differential‑equation models presented span biology, chemistry, physics, engineering, and economics, offering a versatile toolkit for understanding and predicting dynamic phenomena in both natural and human‑made systems.