Population Growth Models

Combinatorics · Axiom Academy

REAL WORLD Population Growth Models How recurrence relations predict the future of species In the year 1202, mathematician Leonardo Fibonacci posed a deceptively simple question: Starting with one pair of baby rabbits... Each month, every mature pair produces one new pair of baby rabbits Rabbits mature after one month and then reproduce every month How many pairs of rabbits will there be after one year? This medieval brain teaser wasn't just about rabbits—it introduced one of the most important sequences in mathematics, biology, and computer science. The answer involves a recurrence relation , a pattern where each term depends on previous terms. Let's trace the population growth month by month and discover the hidden mathematical pattern. Let's visualize how the rabbit population grows. Watch carefully for the pattern: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... Notice anything? Each number is the sum of the previous two! This is the famous Fibonacci sequence, and it's defined by a simple recurrence relation. You've seen the Fibonacci sequence in action. But why does this pattern work for rabbit populations? Why does F(n) = F(n-1) + F(n-2)? Beyond Fibonacci: General Growth Models The Fibonacci model is elegant, but real populations don't grow quite like immortal rabbits. Let's explore a more realistic discrete-time growth model : where r is the growth rate per generation

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