AN INTRODUCTION TO - GitHub Pages · AN INTRODUCTION TO INTEGRAL PROJECTION MODELS (IPMS) 8 6 4 2...

AN INTRODUCTION TO INTEGRAL PROJECTION MODELS (IPMS)

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

0.06Cory Merow

IPMsProcess-based demography:

• Accurate stage structure

• Decompose life history to desired level of detail

• Link vital rates to covariates

• Heterogeneity among individuals

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

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

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

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

Transition Probability

What is an IPM?

Matrix models and IPMs arrive at matrices for different reasons

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

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

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

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

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

Life history

time

growth

survival

fecundity

germination

flowering probability

germinantsurvival

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

Data: Growth

Data: Survival

Data: Fecundity

Germination probability

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

mean = b0 + b1size+ b2size2

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2 4 6 8

2

4

6

8

Size (t)

Size

(t+1

)

●

Vital Rate Regression: Growth

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2 4 6 8

2

4

6

8

Mean Growth

Size (t)

Size

(t+1

)

●

mean = b0 + b1size+ b2size2

variance = b3 + b4size

Size

(t+1

)2

46

82 4 6 8

Size (t)

0.00

0.02

0.04

0.06

0.08

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2 4 6 8

2

4

6

8

Size (t)

Size

(t+1

)

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2 4 6 8

2

4

6

8

Mean Growth

Size (t)

Size

(t+1

)

●

Vital Rate Regression: Growth

mean = b0 + b1size+ b2size2

variance = b3 + b4size

Size

(t+1

)2

46

82 4 6 8

Size (t)

0.00

0.02

0.04

0.06

0.08

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2 4 6 8

2

4

6

8

Size (t)

Size

(t+1

)

●

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

Full kernel

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2 4 6 8

2

4

6

8

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?