Transcript Population Ecology - Department of Environmental Studies

Population Ecology
ES 100
10/23/06
Announcements:
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Problem Set will be posted on course website
today.
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Start early!
Due Friday, November 3rd
Midterm: 1 week from today
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Last year’s midterm is posted on website
This year: will require a bit more thinking
Mathematical Models
Uses:
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synthesize information
look at a system quantitatively
test your understanding
predict system dynamics
make management decisions
Population Growth
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t = time
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N = population size (number of individuals)
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dN = (instantaneous) rate of change in population
dt
size
r = maximum/intrinsic growth rate (1/time)
= b-d (birth rate – death rate)
Population Growth
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Lets build a simple model (to start)
dN
=r*N
dt
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Constant growth rate  exponential growth
Assumptions:
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Closed population (no immigration, emigration)
Unlimited resources
No genetic structure
No age/size structure
Continuous growth with no time lags
Projecting
Population Size
Nt = N0ert
N0 = initial population size
Nt = population size at time t
e  2.7171
r = intrinsic growth rate
t = time
Doubling Time
t double
ln( 2)

r
When Is Exponential Growth a Good
Model?
•r-strategists
•Unlimited resources
•Vacant niche
Let’s Try It!
The brown rat (Rattus norvegicus) is known to have
an intrinsic growth rate of:
0.015 individual/individual*day
Suppose your house is infested with 20 rats.
 How long will it be before the population doubles?
 How many rats would you expect to have after 2
months?
Is the model more sensitive to N0 or r?
Population size (N)
Can the population really
grow forever?
What should this curve look
like to be more realistic?
Time (t)
Population Growth
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Logistic growth
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Population Density:
# of individuals of a certain
species in a given area
Assumes that density-dependent factors affect
population
Growth rate should decline when the population
size gets large
Symmetrical S-shaped curve with an upper
asymptote
Population Growth
 How do you model logistic growth?
 How do you write an equation to fit that S-shaped
curve?
 Start with exponential growth
dN
=r*N
dt
Population Growth
 How do you model logistic growth?
 How do you write an equation to fit that S-shaped curve?
 Population growth rate (dN/dt) is limited by carrying
capacity
dN
N
= r * N (1 –
)
dt
K
What does (1-N/K) mean?
Unused Portion of K
If green area represents carrying capacity,
and yellow area represents current population size…
K = 100 individuals
N = 15 individuals
(1-N/K) = 0.85 population is growing at 85% of the
growth rate of an exponentially increasing population
Population Growth
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Logistic growth
Lets look at 3 cases:
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Result?
N=K (population size is at carrying capacity)
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N
)
K
N<<K (population is small compared to carrying capacity)
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dN
= r * N (1 –
dt
Result?
N>>K (population exceeds carrying capacity)
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Result?
Population Size as a Function of Time
K
Nt 
 rt
1  [(K  N0 ) / N0 ]e
At What Population Size does the
Population Grow Fastest?
growth rate (dN/dt) is slope of the S-curve
 Maximum value occurs at ½ of K
 This value is often used to maximize sustainable
yield (# of individuals harvested)
/time
 Population
Bush
pg. 225
Fisheries Management:
MSY (maximum sustainable yield)
 What
is the maximum # of individuals that can be
harvested, year after year, without lowering N?
= rK/4 which is dN/dt at N= 1/2 K
 What
 What
happens if a fisherman ‘cheats’?
happens if environmental conditions
fluctuate and it is a ‘bad year’ for the fishery?
Assumptions of Logistic Growth
Model:
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Closed population (no immigration, emigration)
No genetic structure
No age/size structure
Continuous growth with no time lags
Constant carrying capacity
Population growth governed by intraspecific competition
Lets Try It!
Formulas:
dN
N

 rN 1  
dt
 K
K
Nt 
1  [(K  N0 ) / N0 ]e rt
A fisheries biologist is maximizing her fishing yield by maintaining
a population of lake trout at exactly 500 fish.
Predict the initial population growth rate if the population is stocked
with an additional 600 fish. Assume that the intrinsic growth rate
for trout is 0.005 individuals/individual*day .
How many fish will there be after 2 months?