The exponential model cannot hold forever — a real lake has limits on space, light and nutrients. By introducing the carrying capacity \(K\) we arrive at the logistic equation \(\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)\): equilibrium points and stability, the phase portrait, the sigmoidal curve with its three stages, estimation of \(r\) and \(K\) from data with nls(), and alternative models (θ-logistic, Gompertz).
Part of the series: Population Dynamics — From the Derivative to the Logistic Equation
