Differential Equations Decoded 16 Logistic Growth Why Populations Stop Exploding

Free to reuse. Free to remix. No attribution required. Make your own at https://www.patreon.com/cw/MadSciHub QUICK SUMMARY The logistic equation fixes the biggest lie in exponential growth by adding a carrying capacity brake. It is the universal equation for any growth process that starts explosive and runs into a wall, from bacteria to markets to tumors. KEY CONCEPTS 1. The Exponential Lie - Pure exponential growth predicts a single bacterium outweighs Earth in 48 hours. 2. The Logistic Equation - dP/dt equals r times P times one minus P over K. 3. The S-Curve - Flat, steep, flat. Same shape in bacteria, iPhones, COVID, and tumors. 4. Inflection at K Over Two - The steepest point is exactly halfway to capacity. 5. Stable vs Unstable Equilibria - K is a bowl that pulls population in. Zero is a hill. 6. Bioelectric K - Michael Levin showed carrying capacity is partly set by voltage patterns. DEFINITIONS - Logistic Equation: dP/dt equals r P times one minus P over K - Carrying Capacity K: The ceiling a population cannot exceed in the long run. - Intrinsic Growth Rate r: How fast population would grow unchecked. - Inflection Point: Where the curve is steepest; for logistic always at P equals K over two. - Partial Fractions: An algebra trick to integrate the logistic equation. HOW IT WORKS 1. Start with dP/dt equals r P times one minus P over K. 2. Separate variables. 3. Apply partial fractions. 4. Integrate both sides. 5. Use initial condition P zero to solve for A. 6. Solve algebraically for P of t. 7. Final form: P of t equals K divided by one plus the ratio K minus P zero over P zero times e to the negative r t. KEY ARGUMENTS 1. Pure exponential is accurate about six hours, then lies. 2. Real populations run out of food, space, and patience. 3. Verhulst proposed in 1838 the simplest correction. 4. The logistic equation impersonates exponential when P is small and zero growth when near K. 5. The inflection is at K over two because two forces balance there. 6. Bacteria, iPhones, epidemics, and tumors fit the same logistic curve. 7. Tumor growth can be reversed by restoring correct voltage maps. KEY TAKEAWAYS - The logistic equation is universal for growth that starts exponential and hits a ceiling. - The inflection point lives at K over two, always. - Zero is unstable, K is stable. - Same equation describes organisms and markets because structure is universal. MEMORY HOOKS - A single bacterium outweighs Earth in 48 hours without a wall. - K is a bowl, zero is a hill. SOURCE https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3128408/ #DifferentialEquations #LogisticGrowth #Calculus #PopulationDynamics #AppliedMath #ExamPrep #madscilecture #decoded #differentialequations #math #science