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Home/Biophysics, Fluids & Geoscience/Leslie Matrix Age-Structured Population

Leslie Matrix Age-Structured Population

A Leslie matrix advances an age-structured population by applying fertility rates in the first row and survival rates on the subdiagonal. Repeated multiplication reveals long-run growth governed by the dominant eigenvalue λ and age proportions governed by the corresponding eigenvector.

Who it's for: Population ecology, conservation biology, matrix models, and applied linear algebra courses.

Key terms

  • Leslie matrix
  • Age structure
  • Dominant eigenvalue
  • Stable age distribution
  • Fertility
  • Survival

The displayed λ is estimated from late total-population ratios. After transients, the age proportions approach the dominant eigenvector.

Live graphs

Fertility and survival

0.25
1.35
0.55
0.52
0.74
0.32

Initial age vector

120
80
25

Measured values

Dominant λ1.1166
Stable juveniles52.7%
Stable adults24.5%
Stable old22.8%

How it works

Age-structured Leslie matrix model with fertility, survival, dominant eigenvalue, and stable age distribution.

Key equations

n_{t+1} = L n_t, L = [[F1,F2,F3],[S0,0,0],[0,S1,S2]]
Dominant eigenvalue λ gives long-run growth; eigenvector gives stable age distribution

Frequently asked questions

What does λ mean?
λ is the asymptotic per-step multiplication factor. Values above 1 imply growth, values below 1 imply decline.
Why do age proportions stabilize?
When one eigenvalue dominates, repeated matrix multiplication aligns the population vector with its eigenvector.