Continuous Models Summary

Mathematical Modeling · Axiom Academy

A comprehensive review of Unit 3: Differential equations, population dynamics, epidemic models, and qualitative analysis techniques. Continuous models use differential equations to describe rates of change, capturing how quantities evolve smoothly over time rather than in discrete steps. Exponential and logistic models are fundamental building blocks for understanding growth, decay, and resource-limited dynamics in natural and engineered systems. Systems of ODEs model interacting quantities , allowing us to capture predator-prey relationships, disease spread, and other multi-variable dynamics. Phase plane analysis reveals qualitative behavior without requiring explicit solutions, showing equilibria, stability, and long-term system trajectories. Fundamental equation: dN/dt = kN where k is the growth (k > 0) or decay (k < 0) rate constant. Solution: N(t) = N(0)e^(kt) shows exponential growth or decay from initial condition. Doubling time: For growth, the time to double is t_d = ln(2)/k , independent of current population. Applications: Radioactive decay, bacterial growth in unlimited resources, compound interest. Equation: dN/dt = rN(1 - N/K) introduces carrying capacity K that limits growth. Key insight: Growth rate decreases as population approaches K, creating self-limiting dynamics. S-shaped curve: Solutions exhibit slow initial growth, rapid middle growth, and leveling off near K. Equilibria: N = 0 (unstable) and N = K (stable) determine long-term behavior.

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