Logistic Population Model

Definition

The logistic population model is an autonomous differential equation that describes population growth subject to a resource-constrained environment. It is defined by: $\frac{dP}{dt} = rP\left(1 - \frac{P}{M}\right)$

• How to read: “The derivative d P d t equals r times P times the quantity one minus the ratio of P to M.”
• Meaning: Population change rate equals intrinsic growth times current size times the fraction of carrying capacity still available. Growth slows as $P$ approaches $M$.

where $P$ is the population, $r$ is the intrinsic growth rate, and $M$ is the carrying capacity.

Why It Matters

The logistic model exposes the inevitable ‘carrying capacity’ of any finite environment; ignoring its limits leads to catastrophic overshoots and the eventual collapse of ecosystems, economies, and organizations.

Core Concepts

• Carrying Capacity ($M$): The maximum population size that the environment can sustain indefinitely. It acts as a stable equilibrium.

  • How to read: “The carrying capacity M.”
  • Meaning: Carrying capacity—the upper limit where resources balance births and deaths; the population asymptotically settles here.

• Growth Dynamics:

  • If $P < M$, growth is positive ($\frac{dP}{dt} > 0$).
    • How to read: “If P is less than M, then the derivative d P d t is greater than zero.”
    • Meaning: Below capacity, the $(1 - P/M)$ factor is positive, so the population still increases.
  • If $P > M$, growth is negative ($\frac{dP}{dt} < 0$).
    • How to read: “If P is greater than M, then the derivative d P d t is less than zero.”
    • Meaning: Above capacity, overcrowding drives the population back down toward $M$.

• Maximum Growth Rate: Occurs at $P = M/2$, where the derivative $\frac{dP}{dt}$ is maximized.

  • How to read: “The population P equals M divided by two.”
  • Meaning: Maximum growth rate occurs at half capacity—$\frac{dP}{dt}$ is largest here, the inflection point of the sigmoid S-curve.

• Sigmoid Curve: The characteristic S-shaped trajectory of the solution as it approaches the stable equilibrium $M$.

Connected Concepts

• Autonomous Differential Equations
• Equilibrium
• Predator-Prey Models
• Exponential Growth Model
• Exponential Decay Model