Population Models

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.

Conditions and key results

The models assume a closed, continuously varying population, constant parameters, and no age or spatial structure. Fitting a curve shows association with observed data; it does not prove density dependence causes the pattern. Counts are discrete, although a continuous approximation may be suitable at large scale.

A reliable strategy

1. Define population, time unit, feasible domain, and the mechanisms included or omitted.
2. Choose exponential or logistic structure from assumptions, not merely from a curve shape.
3. Find equilibria and phase-line stability before solving; estimate parameters with documented data if required.
4. Solve or simulate, verify units and initial data, and interpret predictions only within the calibrated range.

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.

Common mistakes

Verification and reasonableness

• Substitute the logistic formula and check $P(0)=P_0$.
• Confirm equilibria, monotonicity, and the limit $K$ for positive initial data.
• Compare residuals with data and inspect whether parameters or errors vary systematically over time.

Practice

1. For $P'=0.2P$, what is doubling time?
2. Where is logistic absolute growth greatest?
3. What are logistic equilibria for $r>0$?

Further deduction

With constant harvest, the maximum natural logistic growth is $rK/4$, attained at $P=K/2$. If $H>rK/4$, no positive equilibrium exists and the model predicts eventual decline toward extinction; at equality there is a semistable threshold. This mathematical warning depends on constant parameters and harvest, but it shows how equilibrium analysis can reveal risk before explicit solution.

Parameter units provide an audit. In $P'=rP(1-P/K)$, both $P$ and $K$ share population units, making $P/K$ dimensionless, while $r$ has inverse-time units so the right side has population per time. Adding a harvest $H$ requires population-per-time units. Unit mismatch exposes many incorrectly transcribed models before solution.

Estimating $r$ from two early measurements assumes the exponential approximation is appropriate on that interval. A straight line in a plot of $\log P$ versus time supports constant per-capita growth, whereas curvature suggests density dependence, time-varying conditions, or measurement issues.

Related topics

• Autonomous Equations
• Equilibrium Solutions
• Separable Differential Equations