Ch. 53 Exponential and Logistic

GrowthObjective:

SWBAT explain how competition for resources limits exponential growth and can be described by the logistic growth

model.

Unrealistic! Does not take into account

limiting factors (resources and competition). However, a good model for showing upper

limits of growth and conditions that would facilitate growth.

Exponential Growth

Exponential Growth

EquationChange inpopulation

sizeBirths

Immigrantsentering

populationDeaths

Emigrantsleaving

population

N B Dt

Per capita (individual)

B bND mN

N bN mNt

Per capita growth rate

r b m

Nt

rN

dNdt

rmaxN

Under ideal conditions, growth rate is at its max

Exponential

growth results in a J curve.

Exponential Graph

Number of generations

Popu

latio

n si

ze (N

)

0 5 10 15

2,000

1,500

1,000

500

dNdt

dNdt

= 1.0N

= 0.5N

Can occur when:

Populations move to a new area.

Rebounding after catastrophic event (Cambrian explosion)

Real Life Examples

Year

Elep

hant

pop

ulat

ion

8,000

6,000

4,000

2,000

01900 1910 1920 1930 1940 1950 1960 1970

Takes into account limiting factors. More realistic. Population size increases until a carrying capacity

(K) is reached (then growth decreases as pop. size increases). point at which resources and population size are in

equilibrium. K can change over time (seasons, pred/prey

movements, catastrophes, etc.).

Logistic Growth

Logistic Growth

Equation

dNdt

(K N)Krmax N

Logistic growth

results in an S-shaped curve

Logistic Graph

Number of generations

Population growthbegins slowing here.

Exponentialgrowth

Logistic growth

Popu

latio

n si

ze (N

)

0 5 1510

2,000

1,500

1,000

500

0

K = 1,500

dNdt

= 1.0N

dNdt

= 1.0N

1,500 – N1,500( )

Real Life Examples

Time (days) Time (days)

(a) A Paramecium population in the lab

(b) A Daphnia population in the lab

Num

ber o

f Paramecium

/mL

Num

ber o

f Daphnia

/50

mL

1,000

800

600

400

200

00 5 10 2015 0 16040 60 80 100 120 140

180

150

120

9060

30

0

Note overshoot