Population Ecology ES 100 8/21/07. Remember from last time: Population ecology Life Tables...
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Transcript of Population Ecology ES 100 8/21/07. Remember from last time: Population ecology Life Tables...
Population Ecology
ES 100
8/21/07
Remember from last time:
• Population ecology• Life Tables
• 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
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 populationsMetapopulations (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 sub-populations
sub-population A
sub-population B
sub-population D
sub-population C
Populations across the landscape
Genetic diversity is maintained by exchange of genes between the sub-populations
sub-population A
sub-population B
sub-population D
sub-population C
Populations across the landscape
Most mating occurs within a sub-population
sub-population A
sub-population B
sub-population D
sub-population C
Populations across the landscape
Some habitat patches are better than others
hot and dry
most ideal
manypredators
few nesting
sites
Populations across the landscape
Sub-populations can be source populations or sink populations
hot and dry
manypredators
few nesting
sitesmost ideal
source
sink
sink
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
hot and dry
manypredators
few nesting
sitesmost ideal
source
sink
sink
sink
Populations across the landscape
locally extinctsource of
recruits
source
sink
If a sub-population goes extinct, it can be revived by recruits from a source population….
But sinks are important too!
Controls on immigration
Distance to source population
main
lan
d
Lots of immigration
Little immigration
Obstacles• Mountains• Waterways
mountains
hills
Age Stage
sub-population A sub-population Bsub-population C Total dN/dt =
Nx-Nx-1
0-1 60 25 4 89 -----------
1-2 24 30 12 66 -23
2-3 14 26 10 50 -16
3-4 10 20 4 34 -16
4-5 7 13 1 21 -13
Number of individuals
•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?
Sample Metapopulation Data
Mathematical Models
Uses:• synthesize information• look at a system quantitatively• test your understanding• predict system dynamics• make management decisions
Population Growth
• t = time
• N = population size (number of individuals)
• = rate of change in population size (ind/time)
• r = maximum/intrinsic growth rate (1/time) = fractional increase, per unit time, when resources are unlimited
dN dt
Population Growth
• Lets build a simple model (to start)
= r * N
• Constant growth rate exponential growth• Assumptions:
• Closed population (no immigration, emigration)• Unlimited resources• No genetic structure• No age/size structure• Continuous growth with no time lags
dN dt
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
rtdouble
)2ln(=
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 deterministicOutcome is determined only by model inputs Intrinsic growth rate varies with ‘good’ and ‘bad’
environmental conditions:Often we know the mean growth rate and the
variance in the growth rate,
These can be incorporated into our model!
r2rσ
Herd Size with Environmental Stochasticity
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10
Year
Herd Size
Herd Size (Deterministic Model)
0
500
1000
1500
2000
2500
0 2 4 6 8 10
Year
Herd Size
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)?
• Density-dependent factors:
Population Density:# of individuals of a certain # of individuals of a certain
species in a given areaspecies in a given area
•Intra-specific competition•food•Space
•contagious disease•waste production•Interspecific competition•Other species interactions!
•Density-independent factors:•disturbance, environmental conditions
•hurricane•flood•colder than normal winter
Time (t)
Pop
ula
tion s
ize (
N)
Can the population really grow forever?
What should this curve look like to be more
realistic?
Population Growth
• Logistic growth
• 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 Density:# of individuals of a certain # of individuals of a certain
species in a given areaspecies in a given area
Population Growth
How do you model logistic growth?
How do you write an equation to fit that S-shaped curve?
Start with exponential growth
= r * N= r * NdN 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 dt = r * N (1 – )= r * N (1 – )N
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 individualsN = 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:
N<<K (population is small compared to carrying capacity)
Result?
N=K (population size is at carrying capacity)
Result?
N>>K (population exceeds carrying capacity)
Result?
= r * N (1 – )= r * N (1 – )N K
dN dt
Population Size as a Function of Time
rtt eNNK
KN −−+
=]/)[(1 00
Last Time…Metapopulation Theory
Immigration and EmigrationSource and Sink PopulationsMaintaining Genetic Diversity
Population ModelsExponential
AssumptionsDoubling timeWhen should this model be used?
Logistic growthHow 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?
Population 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)/tim
eBush 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 happens if a fisherman ‘cheats’?
What happens if environmental conditions fluctuate and it is a ‘bad year’ for the fishery?
Assumptions of Logistic Growth Model:
• 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!
⎟⎠
⎞⎜⎝
⎛ −=K
NrN
dt
dN1 rtt eNNK
KN −−+
=]/)[(1 00
Formulas:
A fisheries biologist is maximizing her fishing yield by maintaininga 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?