Logistic Growth

Mathematical Modeling · Axiom Academy

Modeling population dynamics with carrying capacity - when growth meets limits 1. Motivation: Limited Resources Exponential growth assumes unlimited resources, but reality is different. As a population grows, individuals compete for food, space, and other necessities. This density-dependent effect slows growth as the population approaches its environmental limits. Assumes unlimited resources. Growth rate proportional only to population size. Predicts unbounded growth. Accounts for limited resources. Growth rate decreases as population approaches capacity. Predicts bounded growth. The logistic equation modifies exponential growth by adding a braking term that reduces the growth rate as population N approaches the carrying capacity K: The term (1 - N/K) is the "braking factor" that slows growth The logistic model has two fundamental parameters that characterize the population dynamics: The maximum per capita growth rate when resources are unlimited. Units: 1/time. Determines how fast the population can grow under ideal conditions. The maximum sustainable population size given available resources. Units: individuals. Represents the environmental limit on population. Interpreting the Per Capita Rate The per capita growth rate (1/N)(dN/dt) = r(1 - N/K) decreases linearly from r (when N = 0) to 0 (when N = K). This linear relationship is a key feature of the logistic model. 4. Solving the Logistic Equation

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