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AN INTRODUCTION TO INTEGRAL PROJECTION MODELS (IPMS)
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Size
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2 4 6 8Size (t)(a)
0.22
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0.68
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Size
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2 4 6 8Size (t)(b)
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2 4 6 8Size (t)(c)
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0.06Cory Merow
IPMsProcess-based demography:
• Accurate stage structure
• Decompose life history to desired level of detail
• Link vital rates to covariates
• Heterogeneity among individuals
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
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0.31
0.29
0.68
0.03
0.55
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Size
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2 4 6 8Size (t)(b)
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Size
(t+1
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2 4 6 8Size (t)(c)
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What is an IPM? Lefkovich matrix for a long-lived perennial shrub
Transition Probability Ti
me
t+1
Time t
time t
time t+1
Could change color schemeAnd .56 to 1.2
What is an IPM?
50231423
=18131448
Transition Probability
What is an IPM?
Transition Probability
What is an IPM?
More stages = more heterogeneity among individuals
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Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
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0.6
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Size
(t+1
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2 4 6 8Size (t)(b)
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
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0.55
0.04
0.51
0.45
0.56
0.02
0.29
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0.6
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Size
(t+1
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2 4 6 8Size (t)(b)
0.00
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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0.06
Transition Probability
What is an IPM?
Matrix models and IPMs arrive at matrices for different reasons
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
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0.56
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0.29
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0.0
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Size
(t+1
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2 4 6 8Size (t)(b)
0.00
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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0.06
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Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
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Size
(t+1
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2 4 6 8Size (t)(b)
0.00
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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0.06
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
0.79
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0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
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0.0
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Size
(t+1
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2 4 6 8Size (t)(b)
0.00
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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0.06
Workflow
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
0.79
0.01
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0.31
0.29
0.68
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0.55
0.04
0.51
0.45
0.56
0.02
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Size
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2 4 6 8Size (t)(b)
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
Workflow
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Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
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Size
(t+1
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2 4 6 8Size (t)(b)
0.00
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Size
(t+1
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2 4 6 8Size (t)(c)
0.00
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0.06
Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
Life history
time
growth
survival
fecundity
germination
flowering probability
germinantsurvival
Workflow
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
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0.55
0.04
0.51
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0.56
0.02
0.29
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0.0
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2 4 6 8Size (t)(b)
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Size
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2 4 6 8Size (t)(c)
0.00
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Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
Data: Growth
Data: Survival
Data: Fecundity
Germination probability
Workflow
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Size
(t+1
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2 4 6 8Size (t)(a)
0.22
0.79
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0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
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2 4 6 8Size (t)(b)
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2 4 6 8Size (t)(c)
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Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
Vital Rate Regression: Growth
mean = b0 + b1size+ b2size2
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Size (t)
Size
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Vital Rate Regression: Growth
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Mean Growth
Size (t)
Size
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●
mean = b0 + b1size+ b2size2
variance = b3 + b4size
Size
(t+1
)2
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82 4 6 8
Size (t)
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Size (t)
Size
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Mean Growth
Size (t)
Size
(t+1
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●
Vital Rate Regression: Growth
mean = b0 + b1size+ b2size2
variance = b3 + b4size
Size
(t+1
)2
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82 4 6 8
Size (t)
0.00
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0.04
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2 4 6 8
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Size (t)
Size
(t+1
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8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
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0.4
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(t+1
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2 4 6 8Size (t)(b)
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2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Full kernel
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2 4 6 8
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Mean Growth
Size (t)
Size
(t+1
)
●
Vital Rate Regression: Growth
g(x, y) = 12πσ (x)2
exp (y−µ(x))2
2σ (x)2
"
#$
%
&'
Workflow
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
The model
Number of individuals of each size
2 4 6 80.00
00.
010
Size
Den
sity
Stable size distribution
2 4 6 8
12
34
56
Size
Offs
prin
g
Reproductive values
2 4 6 8
86
42
Size (t)
Size
(t+1
)
Elasticity
2 4 6 8
86
42
Size (t)Si
ze (t
+1)
Sensitvity
• t = time • x = size at t• y = size at t+1• nt(x) = size distribution at t• nt+1(y) = size distribution at t+1
• K(x,y) = full kernel• P(x,y) = growth/survival
kernel• F(x,y) = fecundity kernel
The model• t = time • x = size at t• y = size at t+1• nt(x) = size distribution at t• nt+1(y) = size distribution at t+1
• K(x,y) = full kernel• P(x,y) = growth/survival
kernel• F(x,y) = fecundity kernel
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
The model
nt+1 = A nt (Matrix)
nt+1(y) = K(y, x) nt (x)dxall sizes∫ (IPM )
• t = time • x = size at t• y = size at t+1• nt(x) = size distribution at t• nt+1(y) = size distribution at t+1
• K(x,y) = full kernel• P(x,y) = growth/survival
kernel• F(x,y) = fecundity kernel
The model
nt+1 = A nt (Matrix)
nt+1(y) = K(y, x) nt (x)dxall sizes∫ (IPM )
• t = time • x = size at t• y = size at t+1• nt(x) = size distribution at t• nt+1(y) = size distribution at t+1
• K(x,y) = full kernel• P(x,y) = growth/survival
kernel• F(x,y) = fecundity kernel
50231423
=18131448
The model
nt+1 = A nt (Matrix)
nt+1(y) = K(y, x) nt (x)dxall sizes∫ (IPM )
nt+1(y) = P(x, y) +F(x, y)[ ] all sizes∫ nt (x) dx
• K(x,y) = full kernel• P(x,y) = growth/survival
kernel• F(x,y) = fecundity kernel
• t = time • x = size at t• y = size at t+1• nt(x) = size distribution at t• nt+1(y) = size distribution at t+1
The model
nt+1 = A nt (Matrix)
nt+1(y) = K(y, x) nt (x)dxall sizes∫ (IPM )
nt+1(y) = P(x, y) +F(x, y)[ ] all sizes∫ nt (x) dx
size(y)t+1 = growth(size x→ y) +offspring(size x→ y)[ ]all sizes∫ size(x)t dx
• K(x,y) = full kernel• P(x,y) = growth/survival
kernel• F(x,y) = fecundity kernel
• t = time • x = size at t• y = size at t+1• nt(x) = size distribution at t• nt+1(y) = size distribution at t+1
We need functions for…• Growth• Survival• Reproduction
We have the option of splitting these in to finer detail if the data are available and the life history
requires it
Life Historyn(y, t +1) = P(x, y) +F(x, y)[ ]
Ω∫ n(x, t) dx
Example 1: Long-lived perennial plant
P(x,y) = (survival probability at size x) * (growth from x to y)= s(x) * g(x,y)
Life Historyn(y, t +1) = P(x, y) +F(x, y)[ ]
Ω∫ n(x, t) dx
Example 1: Long-lived perennial plant
P(x,y) = (survival probability at size x) * (growth from x to y)= s(x) * g(x,y)
F(x,y) = (mean # seeds of size x parent) * (establishment probability)(probability of size y offspring from size x parent)
= fseeds(x) * pestab*frecruit(y)
Life Historyn(y, t +1) = P(x, y) +F(x, y)[ ]
Ω∫ n(x, t) dx
Example 1: Long-lived perennial plant
P(x,y) = (survival probability at size x) * (growth from x to y)= s(x) * g(x,y)
F(x,y) = (mean # seeds of size x parent) * (establishment probability)(probability of size y offspring from size x parent)
= fseeds(x) * pestab * frecruit(y)
Life Historyn(y, t +1) = P(x, y) +F(x, y)[ ]
Ω∫ n(x, t) dx
Example 1: Long-lived perennial plant
P(x,y) = (survival probability at size x) * (growth from x to y)= s(x) * g(x,y)
F(x,y) = (mean # seeds of size x parent) * (establishment probability)(probability of size y offspring from size x parent)
= fseeds(x) * pestab * frecruit(y)
Workflow
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
Vital Rate Regression: Growth – g(x,y)Metcalf et al. 2008
Ozgul et al. 2010
Ferrer-Cervantes et al. 2012Jongejans et al. 2011
Hegland et al. 2010
Size (t)
Size
(t+1
)
Merow et al. 2014
Vital Rate Regression: Survival – s(x)
Metcalf et al. 2008
Metcalf et al. 2009Salguero-Gomez et al. 2012Dahlgren et al. 2011
Jongejans et al. 2011 Merow et al. 2014
Size
Vital Rate Regression: Flowering – pflower(x)Metcalf et al. 2008 Salguero-Gomez et al. 2012
Jongejans et al. 2011
Hegland et al. 2010
Dahlgren et al. 2011 Rose et al. 2005
Size
Vital Rate Regression: Fecundity – fseeds(x)Easterling et al. 2000 Metcalf et al. 2009 Ferrer-Cervantes et al. 2012
Jongejans et al. 2011 Dahlgren et al. 2011Rose et al. 2005
Size
Vital Rate Regression: Fecundity – frecruit(x,y)
Usually...
frecruit (y) =12πσ 2
exp (y−µ)2
2σ 2
"
#$
%
&'
Vital Rate Regression: Fecundity – frecruit(x,y)
Easterling et al. 2000
Usually...
frecruit (y) =12πσ 2
exp (y−µ)2
2σ 2
"
#$
%
&'
but sometimes...µ(x) = ax + b
frecruit (x, y) =12πσ 2
exp (y− (ax + b))2
2σ 2
"
#$
%
&'
Workflow
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Analysis
IPM Kernel
Vital Rate Regressions
Data
Life History
Analysis• Want the same things from IPMs as from matrix models
• Eigenvalues
• Eigenfunction (vectors)
• Can do all the same analyses with IPMs as matrix models
• Elasticity/sensitivity
• Forward projections
• Stochastic dynamics
• Life table response experiments
• Passage time, Life expectancy
• Etc…
λ
Full kernel function
size(y)t+1 = growth(size x→ y) +offspring(size x→ y)[ ]all sizes∫ size(x)t dx
nt+1(y) =
logit(asx + bs )* 12π (agσ x + bgσ )2
exp(x − (agµx + bgµ ))
2(agσ x + bgσ )2
"
#$$
%
&'' +
exp(af #x + bf # ) * 12πσ 2
exp(x − (af x + bf ))
2
2ο 2
"
#$$
%
&''
(
)
*****
+
,
-----
Ω∫ nt (x)dx
Numerical integration
IPMs discretize for numerical integration
Midpoint rule
Numerical integration
Evaluate kernel at midpoint of each cell to obtain a large matrix
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2Si
ze (t
+1)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Numerical integration
Evaluate kernel at midpoint of each cell to obtain a large matrix
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2Si
ze (t
+1)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
nt+1(y) = K(y, x) nt (x)dxΩ∫
↓nt = K nt+1
Full kernel function
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
~Nicolé et al. 2011 Easterling et al. 2000
Godfray et al. 2002Rees et al. 2002
Dalgliesh et al. 2011
Merow et al. 2014
Analysis
2 4 6 80.00
00.
010
Size
Den
sity
Stable size distribution
2 4 6 8
12
34
56
Size
Offs
prin
g
Reproductive values
2 4 6 8
86
42
Size (t)
Size
(t+1
)
Elasticity
2 4 6 88
64
2
Size (t)
Size
(t+1
)
Sensitvity
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Kernel
Summary - Why IPMs?
Process-based demography
• Continuous stages
• Heterogeneity among individuals
• Decompose life history to desired level of detail
• Built on regressions and matrices
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Summary - Why IPMs?
Process-based demography
• Continuous stages
• Heterogeneity among individuals
• Decompose life history to desired level of detail
• Built on regressions and matrices
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(a)
0.22
0.79
0.01
0
0.31
0.29
0.68
0.03
0.55
0.04
0.51
0.45
0.56
0.02
0.29
0.7
0.0
0.2
0.4
0.6
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(b)
0.00
0.05
0.10
0.15
8
6
4
2
Size
(t+1
)
2 4 6 8Size (t)(c)
0.00
0.02
0.04
0.06
Questions?
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