Logistic Growth and Carrying Capacity

The Logistic Growth Model

  • Density dependence: per capita growth rate declines as population size N increases.
  • Carrying capacity K: the maximum population size that the environment can sustain.
  • Logistic differential equation:
    dNdt=rNK−NK\frac{dN}{dt} = rN \frac{K - N}{K}
  • Behavior over N:
    • When N is small compared to K, growth is approximately exponential with rate r.
    • As N approaches K, the per capita growth rate slows and growth tends to zero at N = K.
  • Maximum growth occurs at N = K/2:
    (dNdt)max=rK4\left(\frac{dN}{dt}\right)_{\text{max}} = \frac{rK}{4}
  • Resulting growth curve is sigmoidal (S-shaped): rapid growth at intermediate N, slowing as resources become limiting.
  • When N = 0 or N = K, dN/dt = 0.

Carrying Capacity (K)

  • Defined as the maximum population size the environment can sustain.
  • Varies with the abundance of limiting resources (energy, shelter, nutrients, water, prey, etc.).
  • Can vary across space and time.

Density Dependence and Per Capita Growth

  • Per capita growth rate declines with increasing N due to limited resources and increased competition.
  • If growth slows with density, either birth rate decreases, death rate increases, or both.
  • Per capita growth rate can be expressed as:
    1NdNdt=rK−NK\frac{1}{N}\frac{dN}{dt} = r\frac{K - N}{K}

Real Populations and Assumptions

  • Logistic model assumes instantaneous adjustment to carrying capacity; real populations may exhibit time lags.
  • Overshoot can occur when reproduction continues after resources become limiting.
  • Some laboratory populations fit logistic growth under constant conditions; others overshoot or fluctuate in natural settings.
  • The model is a starting point for more complex models and is useful in conservation biology (recovery after bottlenecks, sustainable harvests, extinction risk).

Practical Implications and Examples

  • For a population with K = 1{,}500 and r = 1.0:
    • Maximum growth rate occurs at N = 750, yielding (\left(\dfrac{dN}{dt}\right)_{\text{max}} = \dfrac{rK}{4} = 375) individuals per year.
    • The logistic curve differs from exponential growth by leveling off as N approaches K.
  • The model helps estimate how quickly a population can rebound and what population size is sustainable under given resources.

Notes on Real-World Relevance

  • The logistic model is a conceptual tool, not a perfect predictor for all species.
  • It informs conservation decisions, such as setting harvest limits and identifying minimum viable populations.