Population Dynamics

Linear Algebra (Matrices) · Axiom Academy

Sort the rabbits into ages — kits , juniors , adults — and one year of births, survival, and aging is exactly one matrix step . Three things you can do with it. Advance the population one year Multiplying by L is the ecology: each adult makes kits, kits become juniors, juniors become adults, some adults survive. Set this year's census, hit Advance, and watch the columns of L move the rabbits. The long-term growth rate is an eigenvalue Run L for many years and the total settles into multiplying by ONE number each step — the dominant eigenvalue . Above 1 the colony booms, below 1 it dies out. Tune how many kits each adult raises and watch cross the line. It always settles into the same mix Here is the surprise: start with any blend of ages and run the years. The proportions stop changing — the population locks onto one fixed age mix, the stable distribution . That mix is the eigenvector for . Pick a wild start and watch it converge. One matrix, two readouts: its dominant eigenvalue is the long-run growth rate (boom, steady, or bust), and its eigenvector is the stable mix everything drifts toward. The same two numbers run demography (will a country s population grow?), conservation quotas , Markov chains , and Google s PageRank — anywhere a fixed rule is applied over and over.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.