Engine–Brake Dynamics in Logistic and Solow-Type Growth Models

This preprint presents a unified calculus framework for analyzing growth dynamics in biological populations and economic capital accumulation. Both systems are formulated as a balance between a growth “engine” and a limiting or depreciating “brake.” The paper derives and analyzes the classical logistic population model and a Solow-type capital accumulation model, establishes equilibria and local stability conditions, and examines convergence behavior. Closed-form solutions and steady-state characterizations are provided, along with illustrations of transient J-shaped adjustment paths under exogenous shocks. The work is primarily expository and aims to clarify structural parallels between these two canonical growth models.