integral projection models continuous variable determines survival growth reproduction easterling,...
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![Page 1: Integral projection models Continuous variable determines Survival Growth Reproduction Easterling, Ellner and Dixon, 2000. Size-specific elasticity: applying](https://reader036.vdocument.in/reader036/viewer/2022081908/56649f1b5503460f94c30a22/html5/thumbnails/1.jpg)
Integral projection models
Continuous variable determines
SurvivalGrowth Reproduction
Easterling, Ellner and Dixon, 2000. Size-specific elasticity: applying a new structured populationmodel. Ecology 81:694-708.
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The state of the population
0 1 2 3 4 5 6
0.0
2.5
size
fre
qu
en
cy
Stable distributions for n=50 and n=100
0 1 2 3 4 5 6
0.4
1.0
size
Re
pro
du
ctiv
e v
alu
eRelative reproductive value for n=50 and n=100
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Integral Projection Model
dxtxf(x,y)yxpty ),(]),([)1,( nn
=Number of size y individuals at time t+1
Probability size x individuals Will survive and become size y individuals
Number of size x individuals
at time t
Babies of size y made by size x individuals
Integrate over all possible sizes
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Integral Projection Model
dxtxyxkty ),()],([)1,( nn
=Number of size y individuals at time t+1
Number of size x individuals
at time t
The kernel(a non-negative surface representingAll possible transitions from size x to size y)
Integrate over all possible sizes
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survival and growth functions
),()(),( yxgxsyxp
s(x) is the probability that size x individual survives
g(x,y) is the probability that size x individuals who survive grow to size y
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survival s(x) is the probability that size x individual
survives
logistic regressioncheck for nonlinearity
bxaxsxs ))(1/)(log(
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growth function g(x,y) is the probability
that size x individuals who survive grow to size y
meanregressioncheck for nonlinearity
variance
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growth function
22 )(2/))((
)(2
1),( xxye
xyxg
-4 -2 0 2 4 6 80
0.05
0.1
0.15
0.2
0.25
0.3
0.35
size y, at time t+1
prob
abili
ty d
ensi
ty
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Comparison to Matrix Projection Model
Populations are structured Discrete time model Population divided into
discrete stages Parameters are estimated
for each cell of the matrix: many parameters needed
Parameters estimated by counts of transitions
• Populations are structured
• Discrete time model• Population characterized
by a continuous distribution
• Parameters are estimated statistically for relationships: few parameters are needed
• Parameters estimated by regression analysis
Matrix Projection Model Integral Projection
Model
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Comparison to Matrix Projection Model
Recruitment usually to a single stage
Construction from observed counts
Asymptotic growth rate and structure
• Recruitment usually to more than one stage
• Construction from combining •survival, growth and
fertility functions into one integral kernel
• Asymptotic growth rate and structure
Matrix Projection Model Integral Projection
Model
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Comparison to Matrix Projection Model
Analysis by matrix methods
• Analysis by numerical integration of the kernel
• In practice: make a big matrix with small category ranges
• Analysis then by matrix methods
Matrix Projection Model Integral Projection
Model
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Steps
read in the data statistically fit the model components combine the components to compute
the kernel construct the "big matrix“ analyze the matrix draw the surfaces