Populations Have Structure
A population is not just a number — it has age structure. A country of 50 million with mostly young people will grow very differently from one with mostly elderly. The Leslie matrix model captures this by tracking how many individuals exist in each age group and how fertility and survival rates determine the flow between groups across generations.
The Leslie Matrix
Patrick Leslie formalized age-structured models in 1945 using matrix algebra. The first row holds fertility rates — how many offspring each age class produces. The sub-diagonal holds survival probabilities — the chance of moving from one age class to the next. Multiply this matrix by the current population vector and you get the next generation's age distribution.
Convergence to Stability
One of the most beautiful results in population mathematics is that regardless of the initial age distribution, the population eventually converges to a stable age structure. This structure depends only on the fertility and survival rates, not on how the population started. The convergence rate depends on the ratio of the two largest eigenvalues of the Leslie matrix.
Demographic Momentum
Even if fertility drops to replacement level instantly, a young population will continue growing for decades — this is demographic momentum. The large cohorts of young people have yet to enter their reproductive years. This simulator lets you see this momentum in action: change fertility rates and watch how the age structure takes generations to respond.