Geometric Growth Non-Overlapping Generations Reproduction is Pulsed λ: Geometric Rate of Increase λ < 1: λ = 1: λ > 1: Exponential Growth Overlapping Generations Reproduction is Continuous r: Per Capita Rate of Increase r < 0: r = 0: r > 0:
Geometric Growth: Calculation of Geometric Rate of Increase (λ) λ = N t+1 ______________ N t
Phlox drummondii 8 N 0 = 996 N 1 = 2,408 λ = Calculating Geometric Rate of Increase (λ)
Geometric Growth: Projecting Population Numbers N 0 = 996 Phlox drummondii 8 λ = 2.42 N 2 = N 1 = 2,408 N 5 =
STEADILY INCREASING POPULATIONS Non-Continuous Reproduction (Geometric Growth) Fig in Molles 2006 N t = N o λ t
Problem A: The initial population of an annual plant is 500. If, after one round of seed production, the population increases to 1,200 plants, what is the value of λ?
Problem B. For the plant population described in Problem A, if the initial population is 500, how large will be population be after six consecutive rounds of seed production?
Problem C: For the plant population described above, if the initial population is 500 plants, after how many generations will the population double?
UNLIMITED POPULATION GROWTH B: (Exponential Growth) Fig in Molles 2006 (e = 2.718)
dN dT UNLIMITED POPULATION GROWTH B Exponential Growth (Rate of Population Growth) dN ___ dT = Rate
Fig in Molles 2006 EXPONENTIAL POPULATION GROWTH: Rate of Population Growth dN ___ dT dN ___ dT dN ___ dT
dN __ dT = r max N EXPONENTIAL POPULATION GROWTH: Rate of Population Growth Intrinsic Rate of Increase Population Size Rate of Population Growth
r = bN - dN Per Capita Rate of Increase Per Capita Birth Rate Per Capita Death Rate Meaning of r
EXPONENTIAL POPULATION GROWTH: Predicting Population Size dN __ dT = r max N N t =N o e r max t (e = 2.718)
Problem D. Suppose that the worldwide population of whooping cranes, with initial population of 22 birds, is increasing exponentially with r max =.0012 individuals per individual per year. How large will the population be after 100 years? After 1000 years?
Problem E. How many years will it take the whooping crane population described above to reach 1000 birds? LN(AB) = LN(A) + LN(B)LN(A/B) = LN(A) – LN(B) LN(A B ) = B LN(A)LN(e) =
Problem F. “Doubling Time” is the time it takes an increasing population to double. What is the doubling time for the whooping crane population described above?
Problem E. Refer to the whooping crane population described earlier. How fast is the population increasing when the population is 100 birds? How fast is the population increasing once the population reaches 500 birds?
Problem F. How large is the whooping crane population when the rate of population change is 5 birds per year? When the rate of population change is 20 birds per year?
LOGISTIC GROWTH: Rate of Population Change Fig in Molles 2006
N T Carrying Capacity (K): Sigmoid Curve: 82 LOGISTIC GROWTH: Carrying Capacity
Figs in Molles (Logistic Population Growth) LOGISTIC GROWTH: Rate of Population Change dN ___ dT
LOGISTIC GROWTH: Rate of Population Change dN ____ dT r max N = ( ) 1 - N K “Brake” Term