Discrete Logistic Model

Mathematical Modeling · Axiom Academy

LESSON Discrete Logistic Model A fundamental nonlinear model with surprisingly rich dynamics - from stable equilibria to chaos 1. Motivation: Population with Limited Resources Consider a population that reproduces in discrete generations (like insects or salmon). Simple exponential growth assumes unlimited resources, but in reality, resources are finite . When population is small, growth is approximately exponential. As population approaches the carrying capacity K , resources become scarce and growth slows. The discrete logistic model captures this density-dependent growth. Exponential growth (unlimited resources) vs. the need for a limiting factor 2. The Discrete Logistic Equation The discrete logistic equation models population with a growth rate that decreases as population increases: The factor (1 - x/K) reduces growth as x approaches K Normalized Form (x scaled to [0,1]) Setting x' = x/K gives the canonical form 3. Fixed Points and Their Meaning A fixed point x* satisfies f(x*) = x*, meaning the population stays constant from one generation to the next. Zero population remains at zero. This is always a fixed point, but is it stable? A positive equilibrium exists only when r > 1. Represents sustainable population level. A fixed point is stable if nearby orbits converge to it. The key criterion involves the derivative f'(x*) = r(1 - 2x*). The fixed point x* is locally stable if and only if |f'(x*)| < 1 Extinction stable for low growth rates

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