Population Ecology
ES 100
8/21/07
Remember from last time:
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Population ecology
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Life Tables
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Cohort-based vs. Static
Identifying vulnerable growth stages
Age-specific birth rate
Computing fitness, net reproductive rate and generation
time
Population projections
Today:

Metapopulation Theory




Immigration and Emigration
Source and Sink Populations
Maintaining Genetic Diversity
Population Models

Exponential and Logistic growth

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
Assumptions
Doubling time
When should this model be used?
Is the Population Increasing, or
Decreasing?
 Fitness
is one indication….. But…
 Populations vary dramatically over time
(boom/bust cycles)
 Individuals move in (immigration) and out
(emigration) of populations
 Metapopulations
(18.5 Bush)
Nt+1 = Nt + (B-D) + (I-E)
Threatened Species:Western Snowy
Plover
Before 1970, 53 breeding locations in CA (including Santa Barbara)
Now, 8 breeding sites support 78% of the CA metapopulation
Populations across the landscape
Metapopulation: sum of multiple interacting subpopulations
sub-population A
sub-population C
sub-population B
sub-population D
Populations across the landscape
Genetic diversity is maintained by exchange of genes
between the sub-populations
sub-population A
sub-population C
sub-population B
sub-population D
Populations across the landscape
Most mating occurs within a sub-population
sub-population A
sub-population C
sub-population B
sub-population D
Populations across the landscape
Some habitat patches are better than others
hot and dry
most ideal
many
predators
few nesting
sites
Populations across the landscape
Sub-populations can be source populations or
sink populations
sink
hot and dry
sink
source
most ideal
many
predators
few nesting
sites
sink
Populations across the landscape
In source population habitats:
•
•
living conditions are good, so births meet or exceed deaths
competition may be great, forcing some members out
sink
hot and dry
sink
source
most ideal
many
predators
few nesting
sites
sink
Populations across the landscape
If a sub-population goes extinct, it can be revived by
recruits from a source population….
But sinks are important too!
sink
source
source of
recruits
locally
extinct
Controls on immigration
mainland
Distance to source population
Lots of immigration
Little immigration
Obstacles
•
•
Mountains
Waterways
mountains
hills
Sample Metapopulation Data
Age
Stage
0-1
1-2
2-3
3-4
4-5
Number of individuals
sub-population A sub-population B sub-population C
60
24
14
10
7
25
30
26
20
13
Total
4
12
10
4
1
dN/dt =
Nx -Nx-1
89
-----------
66
-23
50
-16
34
-16
21
-13
•Is this population assessment static or cohort based?
•Which sub-population(s) are sources? Sinks?
•Can you develop a life table for each sub-population?
•Can you develop a life table for the total population?
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
•
N = population size (number of individuals)
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dN = rate of change in population size (ind/time)
dt
r = maximum/intrinsic growth rate (1/time)
= fractional increase, per unit time, when resources are
unlimited
Population Growth
•
Lets build a simple model (to start)
dN
dt
•
•
=r*N
Constant growth rate  exponential growth
Assumptions:
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•
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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
ln( 2)
t double 
r
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?
When Is Exponential Growth a Good
Model?
•r-strategists
•Unlimited resources
•Vacant niche
Environmental Stochasticity
 Our
exponential growth model is deterministic
 Outcome
is determined only by model inputs
 Intrinsic growth rate varies with ‘good’ and ‘bad’
environmental conditions:
 Often we know the mean growth rate r and the
variance in the growth rate,  r2
 These
can be incorporated into our model!
Herd Size (Deterministic Model)
2500
Herd Size
2000
1500
1000
500
0
0
2
4
6
8
10
Year
Herd Size with Environmental Stochasticity
3000
Herd Size
2500
2000
1500
1000
500
0
0
2
4
6
Year
8
10
Plover Population Model with
Stochasticity
Nur, Page and Stenzel: POPULATION VIABILITY ANALYSIS FOR PACIFIC COAST SNOWY PLOVERS
What Controls Population Size and
Growth Rate (dN/dt)?
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Density-dependent factors:
•Intra-specific
competition
•food
# of individuals of a certain
species in a given area
•Space
•contagious
disease
•waste production
•Interspecific competition
•Other species interactions!
•Density-independent
•disturbance,
factors:
environmental conditions
•hurricane
•flood
•colder
Population Density:
than normal winter
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


Logistic growth
Lets look at 3 cases:

Result?
N=K (population size is at carrying capacity)


N
)
K
N<<K (population is small compared to carrying capacity)


dN
= r * N (1 –
dt
Result?
N>>K (population exceeds carrying capacity)

Result?
Population Size as a Function of Time
K
Nt 
 rt
1  [( K  N 0 ) / N 0 ]e
Last Time…
Metapopulation Theory
Immigration
and Emigration
Source and Sink Populations
Maintaining Genetic Diversity
Population
Models
Exponential
Assumptions
Doubling
time
When should this model be used?
Logistic
growth
How
does it account for density dependent factors?
What is the difference between dN/dt and r?
3
cases:
N<<K (exponential growth)
N=K (no growth)
 N>>K (exponential decline)
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
“recruitment” depends on current population size
Lets Try It!
Formulas:
dN
N

 rN 1  
dt
K

K
Nt 
1  [( K  N 0 ) / N 0 ]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?