Stochastic evolution of population probabilities — exact solution via Kolmogorov equations
Master equation: dP(n,t)/dt = λ(n-1)P(n-1,t) + μ(n+1)P(n+1,t) - [λ(n)+μ(n)]P(n,t). This is the Kolmogorov forward equation for a continuous-time Markov chain. For simple birth-death (linear rates), the steady state is geometric when μ>λ. For logistic growth λ(n) = λ(1-n/K), the distribution is sharply peaked near K. Exact solution via generating functions G(z,t) = Σ P(n,t)z^n satisfies a PDE. The bar chart shows P(n,t) evolving in time.