Discrete Population Models

Mathematical Modeling · Axiom Academy

EXAMPLE Discrete Population Models Work through four population modeling scenarios from simple to complex A bacterial culture in a petri dish doubles every hour under ideal conditions. Starting with 1000 bacteria, we want to model the population growth and predict future populations. The bacteria have unlimited nutrients and space (for now). where is the growth rate and is the initial population Example 1 Complete: Simple Exponential Growth Model: describes unlimited growth Equilibrium: Only N* = 0 exists (trivial equilibrium) Stability: Unstable when r > 1 (population escapes from zero) Behavior: Exponential growth: population doubles every time step No resource constraints - assumes infinite food/space No carrying capacity - unrealistic for real populations Predicts infinite population which is physically impossible Only valid for short time scales in resource-rich environments A fish population in a lake has a natural growth rate r = 2.5 but is limited by available resources. The lake can support a maximum of K = 1000 fish (carrying capacity). Currently there are 100 fish. How will the population evolve? Model Formulation: Discrete Logistic Equation The factor reduces growth as population approaches K Example 2 Complete: Discrete Logistic Model Model: Adds carrying capacity K to limit growth Equilibria: N* = 0 (unstable) and N* = K(1-1/r) = 600 (stable) Stability: Determined by |f'(N*)| compared to 1 Behavior: Population oscillates but converges to 600 fish

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