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Population Growth

Population Growth. Instantaneous and Finite Rates. If interest rate = 10% per year and you start with $100, you will have $110 at the end of the year. 10% is a finite rate . If interest rate = 5% per half year, you will have $105 after 6 mo. and $110.25 after 1 yr. 5% is a finite rate.

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Population Growth

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  1. Population Growth

  2. Instantaneous and Finite Rates • If interest rate = 10% per year and you start with $100, you will have $110 at the end of the year. 10% is a finite rate. • If interest rate = 5% per half year, you will have $105 after 6 mo. and $110.25 after 1 yr. 5% is a finite rate. • The time interval for calculating interest can be reduced to 0. The interest rate is now an instantaneous rate. • Similarly, rates of population growth can be measured per year, per month, per reproductive season etc. They can also be measured instantaneously.

  3. Converting finite and instantaneous rates • finite rate = einstantaneous ratewhere e = 2.71828 • instantaneous rate = ln finite rate • Let  = finite rate; r = instantaneous rate •  = er • r = ln  • Note: ln is Loge

  4. Geometric Rate of Increase:  • Appropriate for species with nonoverlapping generations or for seasonal breeders. •  = Nt+1/Nt

  5. l(lambda) • l = Nt+1/Nt • Nt+1 = Ntl

  6. Example of Geometric Growth •  = Nt+1/Nt • From below:  = 12/6 = 2

  7. Example of Geometric Growth •  = Nt+1/Nt • From below:  = 12/6 = 2 • Use Nt = N0lt to calculate the population size in 4 years (2005). • N4 = 3(2)4 = 3(16) = 48

  8. Continuous Rates • Nt= N0ert • Appropriate for species with overlapping generations.

  9. Geometric and Exponential Growth • Geometric growth is J-shaped growth described by the equation Nt = N0lt. It increases in increments because reproduction is in increments. • Exponential growth is J-shaped growth described by the equation Nt= N0ert. It increases continuously, producing a smooth curve.

  10. Continuous Growth • At time t, the population size is:Nt= N0ert • At any instant in time, the change in population is:rN

  11. dN dt = rN Change in population size at an instant in time Carrying Capacity Population Size dN dt K-N K = rN Logistic Equation

  12. The End

  13. l(lambda) • N1 = N0l • N2 = N1l • N3 = N2l • N4 = N3l

  14. l(lambda) • N1 = N0l • N2 = N1l • N3 = N2l • N4 = N3l • N257 = ?

  15. l(lambda) • N1 = N0l • N2 = N1l • N3 = N2l • N4 = N3l Notice that N1 is in both of these two equations.

  16. N2 = N0ll = N0l2 l(lambda) • N1 = N0l • N2 = N1l • N3 = N2l • N4 = N3l

  17. l(lambda) • N1 = N0l • N2 = N1l • N3 = N2l • N4 = N3l N2 = N0ll = N0l2 N3 = N0l2l = N0l3

  18. l(lambda) • N1 = N0l • N2 = N1l • N3 = N2l • N4 = N3l N2 = N0ll = N0l2 N3 = N0l2l = N0l3 N4 = N0l3l = N0l4 Nt = N0lt

  19. Finite and Continuous Rates • Finite rate = einstantaneous rate = er • Nt = N0lt replace with er • Nt= N0ert • Appropriate for species with overlapping generations.

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