Field · 3 models
Population and Epidemic Models
These are the classic differential-equation models of mathematical biology: a population limited by its environment, an epidemic moving through a closed population, and two species locked in a predator–prey cycle. Each is solved numerically and checked against a quantity the equations say must hold.
Models in this field
- OS-028Logistic growthThe S-curve of a population meeting its carrying capacity, against unchecked exponential growth.
- OS-029SIR modelSusceptible, infected, recovered: the three-compartment ODE behind epidemic curves and R₀.
- OS-030Lotka–VolterraTwo coupled equations whose predator and prey populations circle forever on closed orbits.
Models with a built-in check
Each model carries a mathematical fact the simulation can be tested against: logistic growth has a closed-form solution; the SIR model conserves the total population and obeys a final-size equation; the Lotka–Volterra system conserves a quantity along every orbit. The pages report the numerical drift from those facts, which tells you how far to trust the curves.
The models
- Logistic growth: exponential growth bent by a carrying capacity, and the harvest level at which a population collapses.
- SIR epidemic model: susceptible, infected and recovered compartments, R₀ and herd immunity.
- Lotka–Volterra: predator and prey populations that cycle forever — in the equations.
One family of equations
All three models share a structure: each group grows or shrinks at a per-capita rate that depends on the size of the other groups. In logistic growth the rate falls linearly as the population approaches its carrying capacity; in the SIR model the infection rate is proportional to the number of susceptible and infected people meeting; in Lotka–Volterra the prey’s loss is the predator’s gain. The product terms — S·I, x·y — are the mass-action assumption: encounters happen in proportion to how many of each kind there are, as if everyone mixed at random. Change that assumption and the models change character, which is exactly what the network and agent-based versions do.
Deterministic averages, random reality
These equations describe large populations where chance averages out. Small populations behave differently: an outbreak can fizzle before it starts, and a prey species can go extinct at the bottom of a cycle. The stochastic, individual-based counterparts — an epidemic on a contact network and agent-based predator–prey — show what the smooth curves hide. The time stepping used here is explained in Euler vs Runge–Kutta.