Population Growth Models: Exponential and Logistic

Exponential growth $dP/dt = kP$ is the right model when resources are unlimited, but every real system has a carrying capacity — the maximum population the environment can sustain. The logistic equation $dP/dt = kP(1 - P/K)$ corrects the exponential model by adding a negative feedback term: as $P$ approaches $K$, the growth rate slows to zero. The resulting sigmoid (S-curve) solution is ubiquitous: in biology it describes population growth; in finance it models adoption rates of new products, market saturation of a trading strategy, and the recovery of asset prices toward fundamental value. The qualitative analysis of this ODE — finding equilibria, classifying their stability, and reading the sign of $P'$ — is the prototype of the phase-line analysis used throughout dynamical systems and quantitative risk management.

Learning objectives. By the end of this note you should be able to: