Logistic equations model population growth with limited resources. They show how populations increase rapidly at first, then slow down as they approach a maximum sustainable size called the carrying capacity.

Direction fields and solutions help visualize logistic growth patterns. These tools reveal how populations behave over time, approaching or diverging from the carrying capacity based on initial conditions and growth rates.

The Logistic Equation

Carrying capacity in logistic growth

• Maximum sustainable population size in a given environment
  • Depends on available resources (food, water, shelter)
  • Limited by factors such as space and competition (territory, mates)
• Population growth slows as it nears carrying capacity ($K$)
  • Increased competition for scarce resources
  • Density-dependent factors regulate growth (disease, predation)
• At carrying capacity, population stabilizes
  • Births and deaths are balanced
  • Population remains constant at equilibrium (steady state)

Direction fields for logistic equations

• Visual representation of solution behavior for differential equations
  • Arrows show rate of change (slope) at each point
  • Horizontal arrows signify constant population
  • Vertical arrows indicate increasing (up) or decreasing (down) population
• Logistic equation: $\frac{dP}{dt} = rP(1 - \frac{P}{K})$
  • $P$: population size
  • $r$: intrinsic growth rate
  • $K$: carrying capacity
• Constructing a direction field:
  1. Select points $(t, P)$ in the $t$-$P$ plane
  2. Calculate slope at each point using logistic equation
  3. Plot arrows with corresponding slopes
• Interpreting direction fields:
  • Locate carrying capacity $K$ (horizontal arrows)
  • Examine solution behavior near $K$
    • Arrows pointing to $K$: population approaching carrying capacity
    • Arrows pointing away from $K$: population diverging from carrying capacity
• Identify equilibrium points where population growth rate is zero

Solutions of logistic equations

• Solving logistic equations using separation of variables
  1. Rearrange equation: $\frac{dP}{P(1 - \frac{P}{K})} = rdt$
  2. Integrate both sides
  3. Solve for $P(t)$ to find general solution
• General solution: $P(t) = \frac{K}{1 + Ce^{-rt}}$
  • $C$: constant determined by initial population $P(0)$
• Interpreting solutions in population growth context:
  • As $t \to \infty$, $P(t) \to K$ (population approaches carrying capacity)
  • $C$ determines initial population relative to $K$
    • $C > 1$: initial population below $K/2$
    • $0 < C < 1$: initial population above $K/2$
  • $r$ affects speed of population growth towards $K$
• Real-world applications of logistic model:
  • Bacterial growth in limited nutrient medium (petri dish)
  • Invasive species dynamics in new environments (kudzu, zebra mussels)
  • Epidemic spread within a susceptible population (COVID-19, influenza)
  • Population dynamics in ecology and environmental science

Characteristics of logistic growth

• S-shaped curve represents population growth over time
• Initial phase resembles exponential growth when resources are abundant
• Growth rate slows as population approaches carrying capacity
• Logistic equation is a fundamental model in population dynamics