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.

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

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

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

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

survival s(x) is the probability that size x individual

survives

logistic regressioncheck for nonlinearity

bxaxsxs ))(1/)(log(

growth function g(x,y) is the probability

that size x individuals who survive grow to size y

meanregressioncheck for nonlinearity

variance

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

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

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

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

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