Exponential Growth and Decay in Population

Key Concepts

• Equation for Population Growth:
  [ \frac{dp}{dt} = k \times p ]
  • ( dp/dt ): Rate of population growth
  • ( k ): Relative growth rate
  • ( p ): Size of the population at time ( t )

Conceptual Understanding

• Proportional Growth:
  • If population size doubles, the growth rate also doubles.
  • Example:
    • 1000 bacteria grow at 50 cells/hour;
    • 2000 bacteria grow at 100 cells/hour.

Deriving the General Formula

1. Starting equation:
   [ dp = k imes p imes dt ]
2. Separate variables:
   [ \frac{1}{p} dp = k dt ]
3. Integrate both sides:
   • Left side: ( \ln(p) )
   • Right side: ( kt + c )
4. Exponentiate to solve for ( p ):
   [ p = C e^{kt} ]
5. Find initial population ( p_0 ):
   • When ( t = 0 ):
     [ p(0) = C e^{0} = C \rightarrow p_0 ]
6. Final equation:
   [ p(t) = p_0 e^{kt} ]

Example Problem: Rabbit Population

• Given: Rabbit population on an island starting in year 2000.
• Calculate Initial Population:
  • ( p_0 = 1500 ) when ( t = 0 )
• Using Data Point:
  • Population in 2001 (t = 1) is 1577.
  • ( 1577 = 1500 e^{k} )
  • Solve for ( k ):
    • ( k = \ln(1.0513) \approx 0.05 )
• General Formula:
  [ p(t) = 1500 e^{0.05t} ]

Population Estimates

• Estimate for 2010:
  • ( t = 10 ):
    [ p(10) = 1500 e^{0.5} \approx 2473 \text{ rabbits} ]
• Doubling Time:
  • To find when population doubles to 3000:
    [ 2 = e^{0.05t} ]
  • Solve:
    [ t = \frac{\ln(2)}{0.05} \approx 13.86 \text{ years} ]
  • Round to:
  • Approx. 14 years (by year 2014).

Summary

• Understanding exponential growth is crucial for modeling populations.
• The derived formula ( p(t) = p_0 e^{kt} ) is key for predictions and calculations.