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Population Ecology. ES 100 10/23/06. Announcements:. Problem Set will be posted on course website today. Start early! Due Friday, November 3rd Midterm: 1 week from today Last year’s midterm is posted on website This year: will require a bit more thinking. Mathematical Models. Uses: - PowerPoint PPT Presentation
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Population Ecology
ES 100
10/23/06
Announcements:
• Problem Set will be posted on course website today. • Start early!• Due Friday, November 3rd
• Midterm: 1 week from today• Last year’s midterm is posted on website• This year: will require a bit more thinking
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)
• = (instantaneous) rate of change in population size
• r = maximum/intrinsic growth rate (1/time) = b-d (birth rate – death rate)
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(
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?
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
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
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?