Population Models

A rigorous comparison of exponential, logistic, harvesting, and data-calibrated population models.

Equal input steps create multiplicative output change. An increasing curve stays above a horizontal asymptote, grows slowly at first, and then rises rapidly. Three marked points show a constant multiplication factor rather than a constant difference.

Precise definition

The exponential model $P'=rP$ assumes constant per-capita rate $r$ and has $P=P_0e^{rt}$. The logistic model $P'=rP(1-P/K)$ introduces carrying capacity $K>0$; equilibria are $0$ and $K$, and positive solutions approach $K$ when $r>0$.

Notation and mathematical language

Population $P$ has organism units, $r$ has inverse-time units, and $K$ has population units. The per-capita growth rate is $P'/P$. A constant harvest $H$ changes the equation to $P'=rP(1-P/K)-H$ and can create, merge, or remove positive equilibria.

Conceptual picture

Exponential growth compounds because every individual contributes at the same rate. Logistic growth reduces that rate linearly with density: growth is fastest in absolute terms at $P=K/2$. Carrying capacity is a model parameter, not an immutable biological ceiling.

Fully worked example

Interpretation and application

Population equations support ecology, epidemiology, and resource planning. Long-range forecasts can be highly sensitive to parameter changes and structural assumptions. Report uncertainty and scenarios rather than presenting one fitted trajectory as a causal certainty.